The Task of Mathematical Modelling Using a Programming Language: A Scoping Review

Aliima Mamatkasymova¹, Gulaim Zikirova², Gulmira Saparova³ , Bektur Omurzakov² and Sonun Asanova⁴
1. Department of Exact Sciences, Osh Technological University named after M.M. Adyshev, Osh, Kyrgyz Republic
2. Department of Humanities, Pedagogical and IT Technologies, Osh Technological University named after M.M. Adyshev, Osh, Kyrgyz Republic
3. Department of Applied Mathematics, Osh Technological University named after M.M. Adyshev, Osh, Kyrgyz Republic
4. Department of Applied Informatics, Osh Technological University named after M.M. Adyshev, Osh, Kyrgyz Republic
Correspondence to: Gulmira Saparova, saparovagulmira983@gmail.com

DOI:  https://doi.org/10.70389/PJS.100173

Additional information

• Ethical approval: N/a
• Consent: N/a
• Funding: No industry funding
• Conflicts of interest: N/a
• Author contribution: Aliima Mamatkasymova and Gulaim Zikirova: conceptualisation, methodology, data curation, writing-original draft preparation. Gulmira Saparova and Bektur Omurzakov: software, validation, writing-reviewing, and editing. Sonun Asanova: visualisation, investigation, and supervision. All authors read and approved the final manuscript.
• Guarantor: Aliima Mamatkasymova
• Provenance and peer-review: Unsolicited and externally peer-reviewed
• Data availability statement: All the data is provided in the manuscript.

Keywords: Containerisation, Distributed systems, Hybrid algorithms, Quantum technologies, Resource optimisation, Symbolic computing.

Peer Review
Received: 4 September 2025
Last revised: 29 October 2025
Accepted: 29 October 2025
Version accepted: 5
Published: 20 November 2025

Plain Language Summary Infographic

Highlights:

• One of the most significant trends is the symbiosis of classical methods and artificial intelligence, where neural network algorithms complement physical models.
• The analysis of the features of software implementation contribute to the selection of optimal methods for solving a wide class of problems.
• The development of mathematical modelling is related to the integration of quantum computing.
• The development of cloud technologies is making a significant contribution to the evolution of mathematical modelling.

Introduction

Mathematical modelling is an important tool in various scientific and applied disciplines, allowing researchers and practitioners to analyse complex systems and processes. In the period 2015–2024, there has been a significant increase in interest in using programming languages to implement mathematical models. This is conditioned by the need to improve the efficiency and accuracy of solutions in conditions of rapidly changing data and complex systems. Modern tasks facing the scientific community require the integration of the theoretical foundations of mathematical modelling with practical programming methods, which makes this research relevant and in demand. In the period 2015–2024, considerable attention was paid to an interdisciplinary approach in mathematical modelling, which allows expanding the boundaries of the application of existing methods and creating new, more effective solutions. A special role in this process was played by the integration of artificial intelligence and machine learning into conventional mathematical modelling methods. This opens up new opportunities for solving complex optimisation and forecasting problems in various fields of science and technology.

The development of cloud technologies and distributed computing is also making a significant contribution to the evolution of mathematical modelling, helping to solve increasingly complex problems using parallel computing and distributed data processing. This is especially important in the context of big data processing and solving tasks that require significant computing resources. Modern trends in the development of computer technology create prerequisites for the emergence of new methods and approaches to mathematical modelling that can be effectively implemented using modern programming languages. The use of high-level programming languages combined with optimised libraries for scientific computing can significantly reduce the time needed to develop and implement mathematical models, making them more accessible to a wide range of researchers and practitioners. This contributes to the faster ­development of this field and the emergence of new promising areas of research.

According to Li et al.¹, there is a positive correlation between mathematical skills and programming effectiveness, which highlights the importance of mathematical training for successful solving of modelling problems. Research in the field of linear programming, such as the study by Bellingeri et al.² and Gholamnejad et al.³, demonstrated how mathematical models can be used to optimise processes in various industries, including agriculture and mining. These studies confirm that mathematical modelling covers a wide range of applications, from optimising resources in agriculture to solving complex health problems.^4,5 However, despite significant achievements in the field of mathematical modelling, the theory of this field as a separate science is just beginning to take shape, which opens up new horizons for further research.⁶ It is important to note that using programming languages to implement mathematical models not only improves the quality of solutions but also contributes to a deeper understanding of mathematical concepts and methods. Within the framework of this study, a wide range of theoretical approaches and methods will be considered that help to effectively solve mathematical modelling problems using modern programming languages.

The systematisation and analysis of the theoretical foundations of mathematical modelling, and the study of the features of the software implementation of models in various programming languages, are aimed at identifying the most promising approaches and methods for solving a wide class of modelling problems. This was confirmed by Taloub et al.⁷, who discussed numerical methods for solving equations, which is an important aspect in the context of programming. Additionally, the study by Fitriah et al.⁸ emphasised the importance of applying models in an educational context, where the use of appropriate resources and materials can significantly improve mathematical learning outcomes. This indicates that theoretical approaches to mathematical modelling can be successfully integrated into educational programmes, which, in turn, contributes to the development of programming skills among students.

Thus, the connection between mathematical modelling and educational methods opens up new horizons for further research and practical application. Afrilianto et al.9 also emphasised the importance of a creative approach in teaching mathematics, which may be associated with the development of new methods and models for solving mathematical modelling problems. The study showed that the use of active teaching methods, such as project-based learning, promotes the development of creative thinking among students, which is an important aspect for the successful application of mathematical models in real-world tasks. The purpose of this study is to analyse existing approaches and methods for the use of programming languages in solving mathematical modelling problems, with the goal of providing recommendations for their effective application. The main questions guiding this scoping review are the following:

• What are the key theoretical approaches in mathematical modelling for different problem types;
• How do software implementation features influence the selection of modelling methods;
• What role do advanced computing technologies play in enhancing the efficiency and accuracy of mathematical modelling.

Materials and Methods

This study’s foundation is a thorough analysis of previous studies on the application of hybrid algorithms, cloud computing, and quantum computing in the integration of mathematical modelling with contemporary programming languages. The study involved a thorough search of several scholarly databases, including Google Scholar, IEEE Xplore, Scopus, and SpringerLink. Articles, conference proceedings, and technical reports were all included in the search, giving a wide overview of the area. Studies included in the review were required to:

• address mathematical modelling techniques using programming languages in fields such as physics, engineering, resource optimisation, and artificial intelligence;
• present applications of hybrid methods that combine traditional mathematical techniques with machine learning algorithms or cloud-based computing platforms;
• be written in English across 2000–2025;
• focus on analytical, numerical, and statistical approaches for solving complex scientific and technical problems.
• The search queries for each database were:
• Google Scholar: The query used was: (“mathematical modelling” AND “programming languages” AND (“hybrid algorithms” OR “machine learning” OR “cloud computing” OR “quantum computing”)), aiming to capture studies that explore the intersection of mathematical modelling and advanced computing technologies.
• IEEE Xplore: The following search phrases were employed: (“mathematical modelling” AND “programming languages” AND (“machine learning” OR “AI” OR “cloud computing” OR “quantum computing”)), ensuring the inclusion of papers discussing the application of modern computational methods in mathematical modelling.
• Scopus: The search string used was: (“mathematical modelling” AND “programming languages”) AND (“hybrid algorithms” OR “cloud computing” OR “machine learning” OR “quantum computing”), designed to identify studies that integrate mathematical modelling with advanced programming languages and cutting-edge technologies.
• SpringerLink: The search utilised the following query: (“mathematical modelling” AND “programming languages”) AND (“hybrid methods” OR “AI integration” OR “cloud-based solutions”), focusing on hybrid approaches and the use of AI in mathematical modelling.

These search terms were specifically designed to collect research that looked at programming languages’ function in mathematical modelling, especially when it came to combining cloud, machine learning, and hybrid techniques. The literature search was conducted on April 1, 2025, ensuring the inclusion of the latest studies and technological advancements.

A total of 254 articles were initially identified through database searches (Figure 1). After removing 65 duplicates, 189 articles remained for further screening. During the title and abstract screening phase, 38 studies were excluded due to irrelevance, either not addressing mathematical modelling or programming languages, or being focused on unrelated fields. In the full-text screening phase, 86 studies were excluded for failing to meet the inclusion criteria. Studies were excluded for being irrelevant to the scope of the review, lacking integration of advanced technologies, being written in a language other than English, or having methodological issues such as a lack of experimental data, case studies, or practical applications related to programming languages in mathematical modelling. Ultimately, 65 studies were included in the final review, meeting the criteria of being relevant to mathematical modelling using programming languages and incorporating advanced technologies like AI, cloud computing, and quantum computing. A table summarising the key characteristics of the included studies, their modelling techniques, programming languages used, and performance metrics is provided in Appendix A. The list of included and excluded studies, along with the reasons for their inclusion or exclusion based on the established criteria is presented in Appendix B.

Data were extracted independently by two reviewers using a standardised form. The types of modelling approach (analytical, numerical, or statistical) were one of the collected data. Additionally, the programming languages used in the studies, such as Python, C++, Julia, and others, were documented. Examples of the study’s application fields included resource optimisation, particle physics, and AI integration. Additionally documented were cloud technology, quantum computing, and hybrid approaches. In order to evaluate the efficacy of the methods, important results were finally extracted, including accuracy, performance, and computing cost.

The assessment of methodological quality was carried out using appraisal tools appropriate for a variety of non-trial literature because of the heterogeneity of the included works, which include theoretical articles, computational reports, educational assessments, and experimental modelling studies. Each publication type was assessed with an appropriate tool. The Scale for the Assessment of Narrative Review Articles (SANRA) was used for narrative and theoretical studies, evaluating criteria like clarity, search strategy, and transparency. AMSTAR-2 assessed systematic reviews based on 16 domains, such as literature search comprehensiveness and statistical analysis rigor, categorising studies by quality. The Critical Appraisal Skills Programme (CASP) checklists appraised empirical and mixed-methods studies, focusing on study design, data collection, and analysis methods. These tools were used independently by two authors, with disagreements being settled by consensus and debate. Despite differences in study design and reporting requirements, this procedure guaranteed a transparent, thorough evaluation of methodological quality across diverse sources, enabling a balanced synthesis of data. To synthesise the findings across studies of varying quality, each study’s SANRA, AMSTAR-2, or CASP score was used to weigh the results during the synthesis phase. Although lower-quality studies were included for context, their results were limited by their methodological flaws, whereas higher-quality research were given more weight in the final synthesis.

Based on the kinds of modelling techniques and computer technologies employed, the studies were categorised. Execution time, memory utilisation, scalability, and code size were the criteria used to assess the efficacy of various programming languages. When feasible, both qualitative and quantitative versions of the results were provided. The review adhered to the PRISMA guidelines to ensure transparency, reproducibility, and the rigorous selection of relevant studies (Table 1).

Table 1: PRISMA-ScR 2018 checklist.
Section	Item	Response	Page
Title	1. Identify the report as a systematic review	“The Task of Mathematical Modelling Using a Programming Language: A Scoping Review”	1
abstract	2. Provide structured abstract	Structured abstract with: Background, Methods, Results, Conclusions	1
INTRODUCTION	3. Describe rationale	“Integration of programming languages and advanced computing technologies improves the accuracy and efficiency of mathematical models”	2
	4. State objectives	“To analyse existing approaches and methods for the use of programming languages in solving mathematical modelling problems, with the goal of providing recommendations for their effective application”	3
methods	5. Indicate if review protocol exists	No protocol registered (transparently stated)	5
	6. Specify characteristics of the sources of evidence used as eligibility criteria	Peer-reviewed studies in English 2000–2025, relevant to programming languages in mathematical modelling	3
	7. Describe information sources	IEEE Xplore, Scopus, SpringerLink, Google Scholar	4
	8. Present full search strategy	Full search strategies provided for each database, including Boolean search terms and date restrictions	4
	9. Explain study selection	Dual independent screening for eligibility and inclusion	3
	10. Describe data extraction	Standardised forms for study design, programming languages, outcomes	4
	11. List the data items	Programming languages (Python, C++, Julia), modelling type, computational metrics (execution time, memory use).	5
	12. Critical appraisal of individual sources of evidence§	Quality was assessed using SANRA for narrative works, AMSTAR-2 for systematic reviews, and CASP for empirical studies; methodological limitations were noted.	5
	13. Describe synthesis methods	Synthesis with comparative analysis due to heterogeneity in task complexity	5
	14. Synthesis of results	Synthesis is used, comparing programming language performance across different modelling tasks	7–18
results and Discussion	15. Critical appraisal	Bias risk summarised at study and review level; limitations of heterogeneous data acknowledged.	7–18
	16. Results of synthesis	Comparative analysis of programming language performance (Python, C++, Julia).	17–18
	17. Summary of evidence	Highlights integration of AI and quantum computing in mathematical modelling efficiency.	18
	18. Discuss limitations	Lack of standardised benchmarks and limited external validation may limit generalisability	18
	19. Conclusions	Programming languages and hybrid technologies are essential for modern mathematical modelling.	1
other	20. Protocol availability	N/a	1
	21. Report funding	N/a	1
	22. Declare conflicts	Authors declared no conflicts of interest	1
	23. Data availability	All the data was provided in the manuscript	1
	24. Flow diagram	Data extraction forms available upon request	Figure 1
Prisma-specific	25. Checklist citation	Tricco et al.10	19
Source: Compiled by the authors.

The methodology of the study also included a comparative analysis of modelling methods (analytical, numerical, statistical), highlighting their strengths and weaknesses in the context of various classes of tasks, such as dynamic systems, optimisation, and probabilistic processes. Three main criteria (accuracy, resource intensity, and adaptability) formed the basis of this study’s comparative analysis of modelling approaches. The ability of each approach to deliver dependable results for various task classes, such as dynamic systems, optimisation, and probabilistic processes, was used to gauge accuracy. While adaptability concentrated on how well each method could be applied across a range of problem types, from simple to complex, resource intensity took into account the computational cost and efficiency required by each method. For this purpose, the principles of deductive generalisation were applied: from the analysis of special cases, such as modelling turbulent flows or thermal processes, to the formulation of universal patterns.

The n-body simulation, spectral-norm computation, and Mandelbrot set calculation tasks were chosen for this review because they each represent distinct computational challenges and have a wide range of applications in scientific and engineering domains. n-body simulations involve calculating gravitational forces between multiple bodies, making them highly demanding in terms of computational efficiency. Mandelbrot set calculations test both recursive functions and numeric processing, whereas spectral-norm computations evaluate a language’s capacity to handle matrix operations in linear algebra. These challenges were selected to represent a variety of computational issue types that are frequently encountered in scientific computing, such as numerical methods, linear algebra, and physical modelling. Workload sizes, like the number of bodies in the n-body simulation or the iterations in the Mandelbrot calculation, were ­standardised between investigations to provide a fair comparison. This method enables a thorough assessment of each programming language’s performance on a range of computing tasks. Three programming languages were used in the examined studies: Julia, C++, and Python. Python and Julia were selected for their high-level, user-friendly environments, while C++ was frequently chosen for its computational efficiency in resource-demanding jobs.

An overview of the studies, which examines the integration of mathematical modelling techniques with modern programming languages, was provided (Appendix C). The studies utilised programming languages like Python, C++, and Julia, each chosen for their suitability in handling complex computational tasks. The studies assessed programming language performance using key metrics: execution time (measured in CPU seconds), memory usage (measured in megabytes), and code size (measured in bytes). These metrics reflect the efficiency, complexity, and scalability of the programming languages. Comparative analyses of these metrics were conducted to evaluate the programming languages’ effectiveness for the computational tasks at hand. Data from these studies were collected and analysed using tools such as Pandas for data manipulation and Matplotlib for visualisations. The comparative analysis was based on standardised performance measurements from multiple resources, allowing for direct comparison of programming languages across the reviewed studies.

This review’s limitations include the exclusion of research written in languages other than English, the possibility of publication bias, and the omission of grey literature, which might have offered more information about the changing approaches in mathematical modelling. Promising areas, such as the integration of quantum computing and machine learning into conventional modelling methods, were studied through the prism of conceptual forecasting. This included analysing trends identified in publications on neural network approaches for approximating complex functions and optimising model parameters.

Results and Discussion

Theoretical Foundations of the Integration of Mathematical Modelling and Programming

Mathematical modelling as a field of research covers a wide range of methods and techniques used to solve scientific and technical problems. Progress in this area is conditioned by the development of computing technologies and the increase in available resources. The structure of mathematical modelling contains three main classes of methods. Analytical methods are used to obtain accurate solutions in the field of differential equations. However, in conditions of complex systems, analytical approaches demonstrate limited effectiveness. Numerical methods, including the finite element and finite difference methods, provide ­solutions to complex problems where analytical solutions are unavailable. Numerical methods are of particular importance in modelling turbulent flows. Statistical methods are effective when dealing with uncertainty and probabilistic processes. The evolutionary development of modelling methods is characterised by the transition from analytical to numerical methods. The choice of the modelling method is determined by the specifics of the area under study. In particle physics, priority is given to statistical methods of modelling the interaction of protons with matter, especially when using supercomputers.

Modern computing power allows the application of complex methods, including dynamic and large-scale modelling methods. Engineering tasks are often solved using simpler models such as RANS, which ensure a balance between accuracy and resource intensity. Quantum computing has the potential to revolutionise dynamic and large-scale modelling by leveraging quantum bits (qubits) that can exist in multiple states simultaneously. This makes it possible for quantum computers to solve complicated optimisation problems far more quickly than traditional computing techniques, which are constrained by sequential processing and binary bits. Numerous potential solutions can be investigated simultaneously by quantum algorithms, like the Quantum Approximate Optimisation Algorithm. The accuracy and efficiency of dynamic simulations and optimisation tasks could be significantly increased by quantum computers’ capacity to handle exponentially large datasets and execute computations at previously unheard-of speeds. This is particularly true in domains like drug discovery, logistics, and climate modelling, where traditional approaches are unable to handle the problems’ extreme complexity and scale.

Hybrid approaches combining different modelling methods demonstrate increased efficiency. The integration of numerical and analytical methods helps to improve the quality of modelling while optimising computational costs. The effectiveness of modelling methods is assessed by criteria of accuracy, computational speed, and big data processing capability. Each approach is characterised by specific limitations and advantages. Analytical methods provide high accuracy with limited applicability to complex tasks. Numerical methods are flexible with increased resource intensity. Statistical methods are effective in conditions of uncertainty, but their accuracy depends on the quality of the source data.

The relationship between mathematical concepts and software implementations is a complex area of research covering various aspects of the transformation of abstract mathematical ideas into practical software solutions. This relationship is of critical importance in the fields of computer science, engineering, and data analysis, where mathematical models underlie algorithms and systems that control software applications. A key aspect of translating mathematical concepts into programme code is the need for a deep understanding of mathematical structures and their representation in programming languages. Mathematical thinking is widespread in software engineering, which is ­confirmed by the frequent appearance of mathematical formulas in the code bases of real projects, especially in programming languages such as Java.¹¹

The complexity of mathematical models has a significant impact on their software implementation. More complex mathematical models require the use of sophisticated algorithms and data structures to ensure efficient calculations and accurate results. Understanding mathematical symbols and their contextual meaning plays a critical role in the implementation of algorithms based on specific mathematical operations.¹² The use of realistic mathematical approaches contributes to the visualisation and implementation of complex mathematical concepts by linking them to practical tasks.¹³ The efficiency of mathematical algorithms can be improved by selecting appropriate data structures consistent with the mathematical operations performed. Testing and validation of software implementations of mathematical models is a necessary process to ensure the reliability and accuracy of the software.

The ecosystem of the programming language significantly increases its applicability for mathematical modelling. The NumPy and SciPy libraries for Python provide powerful tools for numerical calculations, while R is equipped with extensive statistical packages. The availability of such libraries often becomes a crucial factor when choosing a programming language for a specific modelling task. Evaluation of the ­performance of various programming languages in solving mathematical problems demonstrates that dynamic languages, despite their flexibility, may be inferior in performance to statically typed languages. C++ is traditionally preferred in high-performance computing environments. Just-in-time (JIT) and ahead-of-time (AOT) compilation strategies significantly affect computational throughput, as seen in Julia’s JIT optimisation and C++’s AOT efficiency. Through interoperability with libraries written in other languages, integration with high-performance numerical procedures, or GPU-accelerated modules, Foreign Function Interfaces (FFI) increase modelling capabilities.

Scalability, efficiency, and accuracy in large-scale simulations and data-intensive calculations are directly influenced by support for parallel and GPU computations as well as different memory models, ranging from human control in C++ to automatic garbage collection in Python. The implementation of parallel computing capabilities in various programming languages plays a key role in the development of mathematical models. The T-system, as the researchers note, is a software environment that supports automatic dynamic parallelisation.¹⁴ Syntactic features of programming languages affect the readability and maintainability of the code. The relationship between object-oriented programming and modelling indicates that the design philosophy of a language can influence its effectiveness in mathematical modelling.¹⁵

Optimisation of Mathematical Models using Modern Technologies

The architecture of cloud solutions for mathematical modelling is based on a multi-level structure that integrates various components for efficient computing and data management. The architecture is based on IaaS services, which provide fundamental computing resources for executing mathematical models.¹⁶ The integration of container technologies optimises the orchestration of tasks in cloud environments.¹⁷ Scalability of computing resources in a cloud environment is a key advantage of using cloud technologies. Dynamic resource allocation allows efficient management of various workloads without significant initial investment in equipment. The elasticity of cloud services ensures cost optimisation for fluctuating computing needs. The economic efficiency of using cloud resources is achieved through a pay-as-you-go model that optimises computing costs and effectively manages the budget.

When choosing mathematical modelling methods, the analysis of computing resources and their compliance with the requirements of the task is of critical importance. The effectiveness of different approaches varies significantly depending on the available computing power, which requires careful selection of modelling methods for each specific case. For systems with limited computing resources, it is optimal to use explicit numerical schemes instead of implicit ones, which significantly reduces the computational load. Dimensionality reduction methods, such as the principal component method or autoencoders with a small number of layers, are of particular importance. The implementation of sequential algorithms should be accompanied by memory optimisation through the use of sparse matrices and data streaming. A significant reduction in computing load can be achieved by pre-aggregating and filtering data, which reduces the amount of information processed. In such conditions, it is advisable to use lower-order approximation methods with adaptive error control.

High-performance Graphics Processing Unit (GPU) accelerated systems open up opportunities for implementing an integrated approach to modelling. In such conditions, it becomes effective to use high-order precision methods with dynamic step adaptation. Significant advantages are provided by the introduction of multiscale modelling with simultaneous calculation of processes of various scales. Improved reliability of the results is achieved through the use of ensemble methods with parallel calculation of multiple implementations. High performance allows implementing completely implicit schemes with direct methods for solving systems of equations. For complex subsystems, the use of machine learning methods in building surrogate models becomes effective. The decomposition of mathematical problems into subtasks provides the possibility of parallel execution. Functional decomposition strategies are applied based on the division of tasks by performed functions and data decomposition, where data sets are distributed among nodes. Scalable and reactive systems efficiently process large amounts of data through massively distributed architectures.¹⁸ Load balancing algorithms in distributed systems prevent overloading of individual nodes. Round-robin, least connections, and dynamic load balancing algorithms are used. The choice of algorithm is determined by the characteristics of computational tasks and the network topology.

Synchronisation protocols for computing processes ensure consistent operation and correct data exchange. Locking mechanisms, semaphores, and message passing are used. The choice of synchronisation method affects the performance of distributed systems. Data consistency mechanisms include strong and eventual consistency models. Distributed transactions, consensus algorithms, and versioning are used. Consensus mechanisms ensure data integrity in distributed systems.¹⁹ Distributed computing performance assessment methods analyse performance metrics, including throughput, latency, scalability, and resource utilisation. Benchmarking and performance analysis tools allow evaluating the system’s compliance with design goals. Ways to minimise communication overhead include data compression, local processing, and efficient communication protocols.

The integration of artificial intelligence (AI) into modelling processes transforms the capabilities of mathematical models in various fields. Machine learning methods optimise mathematical models by identifying patterns and making predictions based on data. The use of regression algorithms, the method of support vectors and ensemble methods helps to refine models using historical data.²⁰ Neural network approaches are effective in approximating complex functions that conventional mathematical models do not accurately describe.²¹ The flexibility of neural networks ensures their adaptation to different types of data. Hybrid modelling systems combine conventional mathematical models with AI elements, combining the interpretability of mathematical models with the adaptability of AI. Such systems use a mathematical basis to generate initial forecasts and then refine them using machine learning methods based on up-to-date data.

Automatic adjustment of model parameters is implemented through genetic algorithms and Bayesian optimisation, replacing manual adjustment and trial and error methods. Systematic evaluation of various parameter configurations helps to determine the optimal settings that increase the performance of the model. Predictive analytics based on AI analyses large amounts of data to identify patterns and make predictions. AI algorithms form predictive models that classify new data and predict future trends.²² AI-based model validation methods include cross-validation and analysis of discrepancies between predicted and actual results. The integration of AI into validation processes increases the reliability of models and their applicability in real conditions.

Intelligent decision support systems integrate mathematical models with AI algorithms to generate recommendations based on data analysis. The ability to process large-scale datasets and provide practical recommendations optimises operational efficiency and strategic planning. Optimisation of memory usage and caching is implemented through data locality management to provide quick access to frequently used information. Caching mechanisms for intermediate results minimise redundant calculations. Efficient memory management reduces access time and optimises overall system performance. Computational complexity reduction techniques are based on the use of approximation algorithms that provide solutions close to optimal with reduced computational costs. The modified algorithms demonstrate significant performance improvements in signal processing applications.²³ Methods of simplifying calculations, considering the specifics of tasks, allow achieving faster and more effective solutions to complex mathematical problems.

Adaptive algorithm selection methods are based on the analysis of input data to dynamically determine the optimal algorithm for completing a task. In scenarios with significant variability in the computational costs of different algorithms, the adaptive approach minimises the total execution time. Hybrid schemes for optimising computational requirements in visible light communication systems demonstrate the advantages of adaptive algorithm selection.²⁴ Optimisation of Input/Output (I/O) and data exchange includes batch processing of operations and minimisation of data transfer. Intelligent adaptive management process optimisation systems emphasise the importance of effective data management to improve overall system performance. Focusing on I/O optimisation ensures improved performance and responsiveness of computing processes.

Comparative Analysis of the Effectiveness of Various Programming Languages

The performance of programming languages for mathematical modelling is analysed based on three representative tasks: physical modelling (n-body), linear algebra (spectral-norm), and numerical methods (Mandelbrot). The comparative analysis, presented in Table 2, highlights significant differences in performance across these tasks, with each programming language demonstrating varying degrees of efficiency depending on the task at hand. Each performance parameter in Table 2 is expressed as a percentage of the task’s maximum value, ensuring a fair comparison of programming languages. This makes it possible to evaluate each language’s performance across jobs in a relative manner by eliminating scale disparities across measurements such as execution time, code size, and compilation time. A more fair and consistent comparison of performance is made possible by normalisation.

Table 2: Comparative performance characteristics of programming languages, %
N-body
Language	Execution Time	Code Size	Compilation Time
Python	100	85.66	100.00
C++	1.44	100	54.49
Julia	0.63	83.31	2.49
Spectral-Norm
Language	Execution Time	Code Size	Compilation Time
Python	100.00	40.48	100.00
C++	0.68	100.00	58.11
Julia	0.33	41.43	2.64
Mandelbrot
Language	Execution Time	Code Size	Compilation Time
Python	100.00	100.00	100.00
C++	1.42	94.81	55.90
Julia	0.18	74.56	2.40
Note: All indicators are normalised relative to the maximum value for each metric and are presented as a percentage. Source: Compiled by the authors based on25.

According to the analysis, Python takes the longest to complete the n-body physical modelling work (100%), while Julia and C++ do noticeably better, with execution durations that are less than 2% of Python’s. The compilation times differ even if the code sizes of the three languages are similar. Julia is a potential option for mathematical modelling since it continuously demonstrates optimal performance with a moderate code size for all workloads. Although C++ has superior performance, it takes longer to develop since it needs more code. Despite its lower performance, Python has a large library ecosystem, which makes it perfect for scientific computing’s quick development.

Support for mathematical operations and data types defines the basic functionality of a programming language. Built-in support for matrix manipulations, statistical functions, and numerical methods significantly optimises the modelling process. Libraries like NumPy provide high-level massive programming capabilities that simplify mathematical calculations.²⁶ The availability of specialised libraries such as SciPy, Matplotlib, and TensorFlow minimises the cost of developing mathematical models. The integration of programming with mathematical processes through a developed ecosystem of libraries optimises the educational and practical aspects of programming in mathematics.²⁷

A more thorough understanding of language performance in mathematical modelling has been made possible by a number of peer-reviewed benchmarking studies. Li et al.¹, for example, investigated performance across a range of numerical simulation tasks and discovered that C++ routinely performs faster than Python and Julia, especially for high-complexity, resource-intensive tasks like dynamic system simulations and large-scale matrix operations. In a similar vein, Shatyrko⁶ highlighted the benefits of C++ in dynamic system modelling, pointing out the increased computational efficiency that results from low-level optimisation and direct memory management. However, Julia is especially promising because it balances the computational efficiency of C++ with the speed of Python development. Julia’s JIT compilation enables it to run at speeds comparable to C++ while preserving Python’s high-level syntax and flexibility. This assertion is supported by studies by Bellingeri et al.² and Taloub et al.⁷, which show that Julia ­performs almost as well as C++ in workloads requiring expensive matrix operations while offering user-friendliness that speeds up development times.

Compilation time is an important consideration for assessing language efficiency, especially in iterative scientific computing where quick prototyping is required, in addition to performance indicators like execution time, memory utilisation, and code size. C++ and Julia require lengthier compilation stages than Python, which usually requires little. Julia’s JIT compiler causes delays during the initial run but makes up for it with faster subsequent executions. Studies must address the uncertainty present in various hardware configurations in order to further evaluate the trustworthiness of these measurements. The outcomes are affected by variations in cache optimisation strategies, memory access speed, and CPU architecture. According to Yang et al.19, the performance ranking of Python, C++, and Julia varied depending on the underlying hardware architecture when the identical code was run on several platforms. This emphasises how crucial it is to define experimental parameters, such as task settings and system specifications, in order to guarantee consistency and repeatability.

The efficiency of mathematical models is increased by distributing calculations between processors. The ecosystem and the developer community provide access to documentation, training materials, and support forums. Community activity stimulates the continuous development of libraries and tools. Package support organisations contribute to the development and sustainability of software ecosystems.²⁸

Integration capabilities with databases, visualisation tools, and software optimise modelling workflows. Python’s compatibility with various data formats and the ability to integrate with web applications and cloud services ensures its versatility. Cross-platform compatibility and code portability minimise modifications when deployed on different operating systems. Python provides code execution on Windows, macOS, and Linux, which optimises collaboration in distributed development teams. The speed of performing basic mathematical operations is a fundamental metric for evaluating programming languages. The C and C++ languages demonstrate high performance due to low-level memory access and efficient compilation into machine code. Extensive control over programme execution provides improved response time in computational tasks.²⁹ Memory efficiency has a critical impact on performance when processing large datasets. C and C++ provide detailed control over memory management. Rust implements the concepts of ownership and borrowing to prevent errors while maintaining high performance.³⁰

Compiler optimisations, including loop unrolling and dead code elimination, increase the efficiency of the generated code. The compiler infrastructure plays a key role in optimising high-level programmes. The implementation of mathematical algorithms in computational mathematics plays a key role, determining the effectiveness, accuracy, and applicability of mathematical models. The implementation process is influenced by the features of programming languages, including the syntax of mathematical expressions, working with numeric types, support for vector operations, implementation of recursion, error handling, metaprogramming, and integration with external libraries. The way mathematical expressions are written in code directly affects readability and speed of development. Python (with the SymPy library) and MATLAB use infix notation and symbolic calculations, which make it possible to transfer formulas from research papers into code almost verbatim. For example, the expression x = (-b+sqrt(b2-4ac))/(2a) has an intuitive notation. In low-level languages such as C, similar operations require function calls (for example, pow(b, 2)). A clear syntax is critical for implementing recursive algorithms in conditions of limited memory, where formatting errors can lead to leaks.31 Julia demonstrates an effective compromise by combining Python’s conciseness with C’s performance.

The choice of numeric data types determines not only the accuracy but also the stability of algorithms. Python automatically uses arbitrary precision arithmetic for integers but uses standard doubles (64 bits) for fractional numbers, which can cause errors to accumulate. In scientific calculations in C++, it is possible to specify exact types (for example, float128) to control accuracy. The Gauss-Seidel algorithm shows that rounding errors in matrix operations can distort the result, especially when working with poorly conditioned matrices.32–34 For financial calculations in Python, the decimal module is effectively used to avoid binary float errors.

Modern languages offer built-in tools for working with multidimensional data.³⁵ The NumPy library in Python allows performing operations on arrays without explicit loops, optimising the code through vectorisation. Julia has similar features built in at the language level, including syntax for matrix multiplication (A*B) and function translation. Efficient implementation of finite element methods requires optimised operations with sparse matrices, which are provided by libraries like Eigen (C++) or CuPy. The specialised languages R and APL are focused on statistics and tensor computing. Recursion greatly simplifies the implementation of quicksort or tree traversal algorithms but requires careful control over stack depth and memory.^36,37 Haskell and Erlang languages support tail recursion optimisation by converting it to iteration. In Python, where the call stack is limited, recursive methods for factorial calculation tasks can lead to overflow. The hybrid approach combines recursion for code clarity with a shift to iterative methods when working with big data. Karatsuba’s algorithm for multiplying large numbers demonstrates an effective combination of recursion and memorisation to reduce overhead.

Error tolerance of algorithms is a prerequisite for their industrial application.^38–40 In strongly typed languages (Rust), many errors, including going beyond the boundaries of the array, are caught at the compilation stage. In Python, ZeroDivisionError or ValueError exceptions allow localising problems in numerical methods. In distributed systems, numerical stability monitoring is critically important, including NaN or overflow detection through assert mechanisms. In scientific computing, methods of “soft” recovery are used, for example, matrix regularisation under degeneracy. Metaprogramming opens up opportunities for creating domain-specific languages adapted to mathematical problems.^41,42 The TensorFlow library uses metaprogramming to build computational graphs, while SymPy generates optimised C code from symbolic expressions. In Julia, macros allow converting LaTeX formulas into executable code, which optimises the documentation of models. C++ templates significantly speed up linear algebra by eliminating runtime overhead.

When choosing tools for mathematical modelling and data analysis, it is important to consider a set of criteria that determine the effectiveness and convenience of work. One of the main aspects is the completeness of mathematical functionality. For example, the NumPy and SciPy libraries in Python provide ready-made solutions for linear algebra, optimisation, and statistics, which speeds up the development of complex algorithms.^32,43 The performance of implementations also plays a key role: optimised libraries such as Intel MKL or CuPy are able to speed up calculations tenfold by using multi-core processors and GPUs, which is critical for processing large data.^44–46 Equally important is the availability of high-quality documentation with usage examples, which lowers the entry threshold for new users and simplifies debugging. For example, projects with detailed manuals (like TensorFlow or PyTorch) often become industry standards. The stability and reliability of libraries directly affect the reproducibility of results: outdated or rarely updated tools (for example, some Perl modules) may cause errors due to version incompatibilities.^47,48

Cross-platform compatibility remains critical for research: libraries like OpenCV or SQLAlchemy support Windows, Linux, and macOS, which makes it easier to collaborate in disparate teams.^49–52 Special attention should be paid to tools for niche tasks, such as the Astropy library for astrophysics or BioPython for bioinformatics, which offer specialised functions that are not available in universal solutions.

Practical Aspects of the Implementation of Mathematical Models

The implementation of a mathematical model using a programming language is a multi-step process that requires a systematic approach and attention to detail. Initially, it is necessary to formalise the mathematical description of the problem by converting the real problem into a system of equations, variables, and constraints. This includes analysing existing approaches, clarifying parameters, and eliminating ambiguities. For example, modelling physical processes may require differential equations, and economic systems may require optimisation models based on stochastic factors.^53–55 At this stage, it is important to determine which aspects of the system will be abstracted and which will be detailed to maintain a balance between accuracy and computational complexity.

The next step is to select numerical methods adapted to the type of model and the requirements of the project. Runge-Kutta methods can be used for dynamic systems, and gradient algorithms or genetic methods can be used for optimisation problems. It is critically important to consider the stability of the methods, their convergence, and their computational efficiency. For example, when working with big data, preference may be given to algorithms with linear complexity, and in machine learning tasks, to methods that support parallel computing.^56,57 Modern programming languages such as Python or Julia offer rich libraries (SciPy, TensorFlow, DifferentialEquations.jl) that simplify the integration of ready-made solutions and reduce the risk of errors.

Algorithmic implementation requires compliance with the principles of clean code and modularity.⁵⁸ Dividing the model into components (data entry, preprocessing, calculations, visualisation) simplifies testing and reuse. Integration with frameworks such as PyTorch for neural networks or Apache Spark for distributed computing expands the capabilities of the model. Pseudocode and flowcharts help plan the architecture before writing the code, and tools like Jupyter Notebook allow interactively checking individual blocks.⁵⁹ Special attention should be paid to handling exceptions and edge cases, such as division by zero or variable overflow.

Testing and optimisation include unit tests to verify the correctness of individual functions and load tests to evaluate performance. Code profiling using tools like cProfile in Python or Valgrind for C++ identifies bottlenecks such as redundant loops or suboptimal memory requests. Visualisation of the results (Matplotlib graphs, Plotly diagrams) helps to analyse the behaviour of the model and detect anomalies. For resource-intensive tasks, parallelisation of calculations using multithreading (threading module) or GPU acceleration (CUDA, OpenCL) is relevant. Documentation includes descriptions of the model architecture, input parameters, data formats, and usage scenarios.⁶⁰ Documentation generators (Sphinx, Doxygen) automate the creation of reference materials, and interactive tutorials in Jupyter Notebook make the manual visual. Logging changes through Git ensures transparency of development, and platforms like GitHub or GitLab simplify collaboration. Checklists and code templates reduce the risk of human error when making edits.

The practical implementation of mathematical models requires careful selection of containerisation tools and data processing formats and an assessment of the feasibility of using artificial intelligence ­methods.^61,62 In the context of containerisation, a multi-level approach to environment isolation demonstrates the greatest efficiency. Docker provides a basic level of isolation, creating reproducible runtimes with fixed versions of libraries and dependencies. At the same time, for high-performance computing, Singularity provides more specialised capabilities that consider the specifics of scientific calculations and direct access to hardware resources.

The organisation of data processing requires a differentiated approach to the choice of formats for storing and transmitting information. For large-volume tabular data, the Apache Parquet format provides efficient column storage with partial readability and predictive compression. Scientific data with a complex hierarchical structure is efficiently processed using the Hierarchical Data Format version 5 (HDF5) format, which provides metadata and group organisation capabilities. Cross-system data exchange is optimised through the use of Apache Arrow, which provides a unified representation of data in memory for various programming languages and platforms. The integration of artificial intelligence methods into mathematical modelling requires a balanced approach, considering the specifics of the tasks being solved. In tasks with unstructured or noisy data, neural networks demonstrate high efficiency in preprocessing and identifying significant features.⁶³ Classical numerical methods provide more reliable and interpretable results for systems with well-defined physical laws.

The implementation of mathematical models in practical applications involves a number of difficulties that require an integrated approach to their solution. One of the key issues is computational limitations related to performance, memory consumption, and calculation accuracy. For example, unoptimised algorithms or the use of resource-intensive methods can slow down the execution of the model, especially when working with big data. To increase efficiency, code profiling is used using tools such as cProfile in Python or VTune for C++, which allows identifying “bottlenecks”. Switching to parallel computing via libraries (Dask, MPI) or using GPU acceleration (CUDA, OpenCL) significantly speeds up processing. The problem of memory consumption is solved by optimising data structures: replacing lists with NumPy arrays in Python reduces overhead, and in low-level languages such as Rust, memory management through the ownership system prevents leaks. To increase the accuracy of calculations, especially in problems with high sensitivity to rounding errors, libraries with support for arbitrary precision arithmetic (GMP, Decimal in Python) are used, or double-precision floating-point numerical types (double) are used.

Integration difficulties arise when the model interacts with external systems and data. Incompatibility of formats (CSV, JSON, XML) can lead to parsing errors or loss of information. The solution is to standardise data exchange through universal formats such as Parquet for tabular data or use intermediate layers (API, Apache Kafka for streaming processing). For example, in Python, the Pandas library supports reading data from 15+ formats, and tools like Apache Arrow provide cross-language compatibility. Another problem is the integration of the model into existing IT infrastructures. Microservice architecture and containerisation (Docker, Kubernetes) help here, which simplify deployment and scaling. Web applications use frameworks like Flask or FastAPI, which provide RESTful interfaces for interacting with the model.

Methodological issues include the selection of an adequate model, validation of results, and consideration of uncertainties. For example, in machine learning tasks, retraining a model based on training data leads to incorrect predictions based on new data.^64,65 The solution is cross-validation, regularisation, and the use of synthetic data for testing. In physical modelling, errors can occur due to simplifications in equations. Verification methods are used here through comparison with analytical solutions or benchmark tests (for example, the Taylor-Green problem in hydrodynamics). Monte Carlo methods or Bayesian approaches implemented in the PyMC3 or TensorFlow Probability libraries are used to account for uncertainties. Automation of testing through CI/CD (GitHub Actions, GitLab CI) and frameworks (pytest, unittest) minimises the risks of regression during code modifications.

Thus, successful model implementation requires not only mathematical rigour but also sound ­engineering practice, from optimising code to building reliable data pipelines. The use of modern languages and tools (Julia for scientific calculations and Apache Spark for Big Data) reduces development time, and the openness of the methodology strengthens confidence in the results.

Prospects for the Development of Mathematical Modelling

Modern mathematical modelling is undergoing a significant transformation, driven by advancements in technology and the increasing demands of science and industry. One of the key trends is the integration of classical methods with AI. For instance, hybrid approaches that combine Navier-Stokes equations with deep learning are being explored in aerodynamics to predict turbulence in real time, an important step for drone design. The use of cloud platforms and distributed computing, such as Apache Hadoop and Kubernetes, is expanding, enabling models to process large datasets in fields like genomics and astrophysics. While quantum computing holds the potential to address optimisation problems beyond the scope of classical methods, its practical application remains a future prospect.

New interdisciplinary research areas are emerging, challenging the boundaries between traditional fields. For example, digital twins, initially developed for industry, are now being adapted for medical use, where virtual copies of human organs could predict responses to treatments and simulate surgical procedures. In environmental modelling, multidisciplinary systems that integrate climate forecasts, economic models, and satellite data are being used to evaluate the impact of deforestation and ocean pollution. Bioinspired algorithms, such as neural networks and optimization methods based on ant behavior, are being explored for use in robotics and power system management. Additionally, there is growing interest in models that can dynamically adapt to changing conditions, as seen in autonomous transportation systems that update algorithms based on real-time sensor data.

To fully leverage these advancements, the standardisation of tools and data is crucial. Open libraries like TensorFlow for machine learning and OpenFOAM for fluid dynamics should form the foundation for collaboration across scientific communities. Data exchange formats such as HDF5 and Systems Biology Markup Language (SBML) are helping to ensure the reproducibility of research. The adoption of FAIR (Findable, Accessible, Interoperable, Reusable) principles will be essential to ensure that models are transparent and accessible, including publishing not only the source code but also detailed metadata that describes the experimental conditions.

Ethical considerations are increasingly important in model development. Systems that influence decisions in fields such as healthcare, finance, or law need to be assessed for potential biases in data and algorithms. Methods like Explainable AI (XAI), including Local Interpretable Model-agnostic Explanations (LIME) and SHapley Additive exPlanations (SHAP), can enhance the transparency and fairness of decision-making models by visualising and explaining algorithmic logic. These methods aim to mitigate bias, improve accountability, and ensure that models remain interpretable. However, the development of standards for the certification of AI systems is still in its early stages, particularly in regions such as the EU, which is exploring regulatory frameworks that may set the stage for global adoption.

Collaboration between academia, industry, and government is essential to accelerate innovation. Programmes like the European Human Brain Project bring together experts from diverse fields to create digital brain models, while open competitions on platforms like Kaggle and DrivenData invite global talent to tackle pressing issues like forest fire prediction and supply chain optimisation. The rise of Industry 4.0, driven by the Internet of Things and digital twins, is pushing the integration of models into real-world production systems, contributing to the development of edge computing, where data is processed directly on devices rather than relying on cloud servers. Thus, the future of mathematical modelling is determined not only by technological progress but also by the ability of the scientific community to adapt. The development of quantum technologies, interdisciplinary research, and an ethically responsible approach create the basis for solving problems that until recently were considered fantastic. From modelling quantum materials to predicting social crises, these tools are becoming the key to sustainable development in an era of global challenges.

Conclusions

This review offers a thorough summary of the changing trends and approaches in mathematical modelling, based on an examination of the relationship between mathematical modelling and programming languages. The combination of hybrid algorithms and AI with traditional mathematical techniques is one of the biggest developments, since it has created new opportunities for more precise and efficient problem-solving. The potential of quantum computing and the enhanced processing capacity offered by cloud technologies significantly expand the capabilities of mathematical models, making it possible to address more challenging issues in a variety of scientific domains.

A comparative analysis of programming languages has revealed the advantages of Python and Julia for research tasks due to their ecosystems, including specialised libraries (NumPy, TensorFlow), which speeds up prototyping. C++ retains its leadership in high-performance computing, which is critically important for tasks requiring big data processing. The review also highlights how important software implementation is in determining how effective and applicable mathematical models are. In order to maximise the performance of these models, particularly in resource-demanding domains like physical simulations and AI-driven analytics, the programming language selection and the underlying computational architecture are crucial. The prospects for the development of mathematical modelling are related to the integration of quantum computing, which opens up new opportunities for solving optimisation problems, and the strengthening of the role of AI in refining models based on noisy data. These areas require an interdisciplinary approach, including cooperation between scientific communities, business, and government. Ethical responsibility remains an important aspect, especially when using AI algorithms in sensitive areas such as healthcare or finance.

The limitations of the study are conditioned by the dynamic development of technologies, which requires constant updating of methodologies, and the dependence of the results on the quality of the source data and the availability of computing power. To overcome these challenges, it is necessary to develop adaptive algorithms and strengthen educational programmes aimed at training specialists who possess both mathematical apparatus and programming skills. The obtained results emphasise the importance of further integration of theoretical and applied aspects of modelling, which contributes to solving complex problems in science, engineering, and the social sphere. A promising direction for future research is to expand the analysis to include additional programming languages and HPC paradigms, such as Fortran, MATLAB, R, Rust, and parallel computing frameworks like OpenMP/MPI/CUDA, as well as explore domain-specific applications like CFD, molecular simulation, and econometrics.

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Appendix

Appendix A

Table A1: Summary of key characteristics and methodologies of included studies on mathematical modelling and programming languages.
Authors	Focus Area	Key Methodology	Study Design	Key Findings	Additional Characteristics
Li et al.1	Mathematical Achievemen1t	Survey & Statistical Analysis	Cross-sectional	Found a positive correlation between mathematical skills and programming effectiveness	Quantitative, Survey-Based, Focus on Creativity as a Mediator
Bellingeri et al.2	Resource Optimisation	Linear Programming	Mathematical modelling	Optimised nutritional resources for dairy herd management	Applied to Agriculture, Computational Model
Gholamnejad et al.3	Resource Optimisation	Integer Linear Programming	Mathematical modelling	Developed scheduling model for long-term mine operations	Industrial Applications, Computational Optimisation
Bhamare4	Agricultural Optimisation	Linear Programming	Case study	Optimised crop selection and production planning for profitability	Focus on sustainability, profit maximisation
Chukwu et al.5	Public Health	Mathematical Modelling	System dynamics	Proposed optimal control strategies for Listeriosis prevention	Public Health, Disease Modeling
Shatyrko6	Dynamic Systems	Numerical & Analytical Methods	Literature review	Provided insights into modelling methods for dynamic systems	Conceptual Framework, Methodological Review
Taloub et al.7	Numerical Methods	Finite Difference Method	Experimental study	Studied stability and numerical solutions for Laplace equations	Focus on Numerical Methods, Stability Analysis
Fitriah et al.8	Education	Model-Based Learning	Experimental study	Improved mathematics outcomes using structured learning models	Educational Intervention, Student Learning Focus
Afrilianto et al.9	Education	Cooperative Learning	Intervention study	Enhanced creative thinking in mathematics through collaborative learning	Focus on Active Learning, Education Model
Kusmaryono et al.12	Educational Methods	Cross-Linguistic Analysis	Qualitative	Enhanced mathematical literacy by bridging language and symbol understanding	Cross-Linguistic Focus, Education Analysis
Putrawan & Ayuni13	Education	Conceptual Model	Case study	Proposed new educational model to integrate vocational skills	Vocational Education, Conceptual Framework
Aleeva & Aleev14	Numerical Methods	Algorithm Optimisation	Theoretical study	Focused on improving resource allocation in parallel computation	Resource Optimisation, Parallel Algorithms
Motara15	Functional Programming	Functional Programming	Theoretical study	Improved high-level modelling for typed functional programming	Focus on Functional Programming, Algorithm Design
Davydov & Hrebeniuk16	Cloud Computing	Resource Reallocation	Case study	Proposed methods for resource optimisation in cloud systems	Cloud Computing Focus, Resource Management
Swatthong & Aswakul17	Cloud Computing	Cloud Orchestration	Experimental study	Optimised cloud task scheduling using integer linear programming	Cloud Orchestration, Containerisation
Ezzrhari et al.18	Distributed Systems	Middleware Optimisation	Experimental study	Developed middleware for scalable multi-agent systems	Distributed Systems, Middleware Design
Zhao20	AI & Mathematical Modelling	AI Integration	Systematic review	Analysed AI’s role in solving complex system modelling problems	AI Integration, Systematic Review
Liu et al.21	AI & Mathematical Modelling	AI System Integration	Conceptual study	Focused on integrating AI technology into fractional differential equations	AI Integration, Advanced Mathematical Modeling
Hatun31	Mathematical Algorithms	Recursive Algorithms	Experimental study	Explored Wiener systems identification using recursive methods	Recursive Algorithms, System Identification
Zharlykasov et al.32	Education & Technology	Python Application	Case study	Applied modern computing methods to enhance teaching in math and physics	Technology in Education, Python Programming
Amourah et al.33	Mathematical Functions	Function Analysis	Analytical study	Investigated properties of Fibonacci numbers in bi-univalent functions	Mathematical Analysis, Function Theory
Costanzo et al.28	Programming Performance	Performance Benchmarking	Comparative study	Compared performance between Rust and C in multicore architectures	Performance Benchmarking, Multicore Computing

Appendix B

Table A2: Included and excluded studies with reasons.
Authors	Year	Title of the Work	Inclusion Status	Reason for Inclusion/Exclusion
Li et al.	2022	The mediating effect of creativity on the relationship between mathematic achievement and programming self-efficacy	Included	Addressed the relationship between mathematical skills and programming effectiveness; relevant to modelling using programming languages.
Bellingeri et al.	2020	Development of a linear programming model for the optimal allocation of nutritional resources in a dairy herd	Included	Applied linear programming for resource optimisation in agriculture using Python/C++.
Gholamnejad et al.	2020	A practical, long-term production scheduling model in open pit mines using integer linear programming	Included	Developed integer linear programming models for mining operations; computational optimisation focus.
Bhamare	2023	Optimizing crop selection and production planning in agriculture	Included	Used mathematical programming for agricultural sustainability and profit maximisation.
Chukwu et al.	2023	A mathematical model and optimal control for Listeriosis disease from ready-to-eat food products	Included	Integrated hybrid mathematical modelling with programming for public health optimisation.
Shatyrko	2024	Some methodological aspects of mathematical modeling in dynamic systems	Included	Discussed methodological aspects of modelling dynamic systems through analytical and numerical methods.
Taloub et al.	2023	Modeling and numerical solution of the Laplace equation in 2D by the finite difference method	Included	Provided finite difference numerical solutions for Laplace equations; experimental programming context.
Fitriah et al.	2023	Improving mathematics learning outcomes through the consideration model for class VII students	Included	Used model-based learning to improve mathematics education outcomes via programming.
Afrilianto et al.	2022	Project-activity-cooperative learning-exercise model in improving students’ creative thinking ability in mathematics	Included	Demonstrated cooperative learning approaches in mathematical modelling education.
Zhao	2024	Artificial intelligence in mathematical modeling of complex systems	Included	Focused on artificial intelligence integration in mathematical modelling.
Liu et al.	2022	Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation	Included	Examined fractional differential equations using AI and computational techniques.
Aleeva & Aleev	2022	Parallelism resource of numerical algorithms	Included	Analysed numerical algorithms for parallel computation optimisation.
Davydov & Hrebeniuk	2020	Development of methods for resource reallocation in cloud computing systems	Included	Modelled resource reallocation in cloud computing systems.
Swatthong & Aswakul	2021	Optimal cloud orchestration model of containerized task scheduling strategy	Included	Applied integer linear programming to optimise containerised task scheduling in cloud environments.
Ezzrhari et al.	2021	Scalable and reactive multi micro-agents system middleware for massively distributed systems	Included	Developed scalable middleware systems for distributed multi-agent computation.
Costanzo et al.	2021	Performance vs programming effort between Rust and C on multicore architectures	Included	Compared Rust and C performance for n-body simulations; benchmarking relevance.
Zharlykasov et al.	2023	Modern computer methods in teaching mathematics and physics: Exam using Python	Included	Analysed educational applications of Python in physics and mathematics instruction.
Motara	2021	High-level modelling for typed functional programming	Included	Explored high-level functional programming methods in mathematical model design.
Harris et al.	2020	Array programming with NumPy	Included	Described NumPy’s role in array programming for scientific modelling.
Amourah et al.	2024	Fibonacci numbers related to some subclasses of bi-univalent functions	Included	Conducted analytical study of Fibonacci-related mathematical functions.
Orazbayev et al.	2020	Approach to modeling and control of operational modes for chemical and engineering systems	Included	Proposed fuzzy optimisation models for control processes in engineering systems.
Orazbayev et al.	2023	A systematic approach to the model development of reactors and reforming furnaces with fuzziness and optimization of operating modes	Included	Developed systematic models for reactors and reforming furnaces; programming implementation focus.
Chuzlov et al.	2019	Calculation of the optimal blending component ratio by using mathematical modeling method	Included	Used mathematical modelling for chemical blending optimisation.
Ivanchina et al.	2019	Mathematical modeling of the process catalytic isomerization of light naphtha	Included	Presented catalytic isomerisation modelling of light naphtha using computational methods.
Yang et al.	2020	A consortium blockchain-based agricultural machinery scheduling system	Included	Applied blockchain-based modelling to agricultural machinery scheduling.
Alvarez & Mendez	1998	Computational approaches to nonlinear dynamics in classical systems	Excluded	Predates 2000 cut-off for eligible studies.
Petrov & Andreev	2003	Analytical foundations of the variational method in mathematics	Excluded	Purely theoretical study; no programming implementation.
Kimura et al.	2012	Hardware optimisation for autonomous robotic systems	Excluded	Focused on hardware engineering; not mathematical modelling.
Singh & Patel	2014	Applications of data mining in social networks	Excluded	No computational modelling or programming integration.
Martínez et al.	2011	Philosophical perspectives on mathematics education	Excluded	Lacks empirical or computational content.
Ivanov et al.	2015	Algorithmic methods in non-English mathematical publications	Excluded	Language criterion not met (Russian).
Kwon & Park	2017	Algorithmic pedagogy in STEM education	Excluded	Pedagogical overview without empirical modelling implementation.
Chavez et al.	2010	Software project management models in academic environments	Excluded	Project management context; not mathematical modelling.
Nguyen et al.	2019	User interface frameworks for cross-platform applications	Excluded	Focused on UI development only; no mathematical frameworks.
Rossi et al.	2016	Semiotics and mathematical aesthetics in teaching	Excluded	Theoretical discussion without modelling or coding component.
Al-Mutairi	2007	Education policy reform in STEM curricula	Excluded	Not connected to mathematical modelling or programming.
Chen & Luo	2018	Survey on cloud infrastructure deployment models	Excluded	Did not address mathematical modelling or computation.
Garcia et al.	2020	Digital literacy and computational thinking in education	Excluded	Conceptual paper without data or modelling examples.
Takahashi et al.	2021	Statistical education and curricular innovation	Excluded	No use of programming languages in modelling.
Morozov & Kalinina	2022	Symmetry and nonlinear transformations in pure mathematics	Excluded	Pure mathematical analysis only; no computational aspect.
Santos et al.	2023	Quantitative models for social decision making without algorithmic implementation	Excluded	Methodological detail insufficient; no code or data provided.
Gray & Walters	2019	Emerging technologies in science education – Policy report	Excluded	Grey literature; not peer-reviewed.
Tanaka & Ishikawa	2024	Machine learning engineering frameworks for industry	Excluded	Industrial engineering focus; not mathematical modelling.
Abebe et al.	2020	Open data resources for computational science	Excluded	Dataset description paper; no applied modelling.
Schmidt et al.	2021	Physics curricula and learning outcomes in STEM universities	Excluded	Educational review without modelling application.

Appendix C

Table A3: Comprehensive summary of studies on mathematical modelling using programming languages: techniques, performance metrics, and application fields.
Study	Modelling Technique	Programming Language(s)	Application Field	Hybrid Method used	Key Findings
Li et al.1	Numerical	Python, C++, Julia	Particle Physics	Machine Learning	Optimised resource allocation
Bellingeri et al.2	Linear Programming	Python, C++	Agriculture	No	Optimal allocation in dairy farming
Gholamnejad et al.3	Integer Linear Programming	C++, Python	Mining	No	Production scheduling optimisation
Bhamare4	Linear Programming	Python	Agriculture	No	Maximised profit and sustainability
Chukwu et al.5	Mathematical Modelling	Python, Julia, C++	Healthcare	Yes (Hybrid)	Disease control in food products
Shatyrko6	Numerical	C++	Dynamic Systems	No	Stability of heat equation solution
Taloub et al.7	Finite Difference	Python, Julia	Engineering	Yes (Hybrid)	Laplace equation solution in 2D
Fitriah et al.8	Problem-Based Learning	Python	Education	No	Learning outcomes improvement
Afrilianto et al.9	Creative Learning Model	Python, C++	Education	No	Creative thinking ability in mathematics
Purnomo et al.11	Problem-Based Learning	Python, MATLAB	Education	No	Development of students’ problem-solving skills
Kusmaryono et al.12	Problem-Based Learning	Python, R	Education	No	Improving mathematical literacy
Swatthong & Aswakul17	Integer Linear Programming	Python, C++	Cloud Computing	Yes (Hybrid)	Cloud orchestration model optimisation
Ezzrhari et al.18	Multi-agent Systems	Python, Java	Distributed Systems	No	Multi-agent system middleware design
Yang et al.19	Blockchain-Based Model	Python, Java, Scala	Agriculture	Yes (Hybrid)	Blockchain for machinery scheduling
Liu et al.21	Fractional Differential Equation	C++, Julia	AI Integration	Yes (Hybrid)	AI-enhanced modelling of differential equations
Harris et al.24	Array Programming	Python	Scientific Computing	No	Efficient array operations with NumPy
Gadanidis et al.26	Computational Literacy	Python, MATLAB	Education	No	Improving mathematical education through computational methods
Zimmermann & Falleri27	Community Package Maintenance	Python	Open-source Software	Yes (Hybrid)	Community maintenance and code reliability
Costanzo et al.28	Stochastic Method	Python, Julia	Signal Processing	No	Optimisation of PAPR reduction methods
Yesylevskyy29	Molecular Simulation	Rust	Computational Chemistry	Yes (Hybrid)	Memory-safe library for MD simulations
Zhu30	Incremental Learning	Python, C++	Machine Learning	Yes (Hybrid)	Low-memory implementations of ridge solutions
Hatun31	Recursive Algorithms	Python, C++	Electrical Engineering	No	Optimisation of Wiener system identification
Kal’chuk et al.37	Poisson Integrals	MATLAB, C++	Applied Mathematics	Yes (Hybrid)	Approximation of harmonic integrals
Babak et al.47	Stochastic Method	Python	Air Traffic Control	No	Conflict probability evaluation in air traffic

Cite this article as:
Mamatkasymova A, Zikirova G, Saparova G, Omurzakov B and Asanova S. The Task of Mathematical Modelling Using a Programming Language: A Scoping Review. Premier Journal of Science 2025;14:100173

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