International Journal of Advances in Engineering & Scientific Research
Print ISSN : 2349 –4824
Online ISSN : 2349 –3607
Frequency : Continuous
Current Issue : Volume 13 , Issue 1
2026
Purushottam Bhaskar
Purushottam Bhaskar, Assistant Professor Department of Mathematics, S.P.U. PG College Falna, Pali Rajasthan
DOI : https://doi.org/ 10.5281/zenodo.19235234 Page No : 46-65
Published Online : 2026-03-26
Download Full Article : PDF Check for Updates
ABSTRACT
Aim of the study: This study aims to explore the pivotal role of numerical analysis in solving complex mathematical and engineering problems that are often intractable through analytical methods.
Methodology: This investigation employs a comprehensive review of various numerical techniques, including finite element, finite difference, and boundary element methods, to assess their efficacy in diverse applications such as structural mechanics, fluid dynamics, and electromagnetics.
Findings: The application of these methods consistently yields robust approximations for systems characterized by non-linearities, high dimensionality, or intricate boundary conditions, where exact solutions are mathematically unfeasible. This capability is particularly crucial in contemporary computational mathematics, which extensively leverages algorithms and numerical methods to address multifaceted problems across scientific and engineering domains.
Implications: The increasing complexity of real-world phenomena necessitates advanced computational tools, and numerical analysis provides the foundational framework for developing accurate and efficient simulation methodologies. The evolution of these methodologies has been profoundly influenced by advancements in digital computing, enabling the resolution of large sets of simultaneous arithmetic problems that were previously unmanageable.
Originality/Value: This work contributes a holistic perspective on the integration of numerical analysis within computational engineering, highlighting its indispensable role in bridging theoretical mathematical constructs with practical engineering applications. This foundational aspect underpins the ability to model and predict the behavior of complex systems with increasing precision, thereby driving innovation across numerous industrial sectors, including aerospace, automotive, and biomedical fields.
Keywords: Numerical analysis, computational mathematics, engineering problems, finite element method, finite difference method, boundary element method.
References
Aghaei, A. A., Moghaddam, M. M., & Parand, K. (2024). PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2409.01899
Akhare, D., Luo, T., & Wang, J. (2024). Diffhybrid-Uq: Uncertainty Quantification for Differentiable Hybrid Neural Modeling. https://doi.org/10.2139/ssrn.4712361
Akinsola, V. O. (2023). Numerical Methods: Euler and Runge-Kutta. In IntechOpen eBooks. IntechOpen. https://doi.org/10.5772/intechopen.108533
Alaneme, G. U., Bahrami, A., Iro, U. I., Ganasen, N., Otu, O. N., Udeala, R. C., Ifebude, B. O., & Onwusereaka, E. A. (2023). Numerical Analysis and Parametric Study on Multiple Degrees-of-Freedom Frames. Civil Engineering Journal, 9(7), 1709. https://doi.org/10.28991/cej-2023-09-07-012
Alber, M., Tepole, A. B., Cannon, W. R., De, S., Durá-Bernal, S., Garikipati, K., Karniadakis, G. E., Lytton, W. W., Perdikaris, P., Petzold, L., & Kuhl, E. (2019). Integrating machine learning and multiscale modeling—perspectives, challenges, and opportunities in the biological, biomedical, and behavioral sciences. Npj Digital Medicine, 2(1), 115. https://doi.org/10.1038/s41746-019-0193-y
AL-JAWARY, M., & Nabi, A. (2019). Reliable Iterative Methods for Solving Convective Straight and Radial Fins with Temperature-Dependent Thermal Conductivity Problems. GAZI UNIVERSITY JOURNAL OF SCIENCE, 32(3), 967. https://doi.org/10.35378/gujs.429896
Alotaibi, H., Soutis, C., & Jabbari, M. (2025). FEM and FVM Methods for Design and Manufacturing of Hierarchical Aerospace Composites: A Review. Applied Sciences, 15(16), 8896. https://doi.org/10.3390/app15168896
Amaefule, E. O., Chukwuebuka, A. O., Chukwudi, N. N., Oscar, N. C., Nelson, O. C., Keji, A. H., Ibeamaka, O. M., Sunday, O. I., Don-Ugbaga, C., & Ezeokpube, G. (2022). Finite Element Analysis of Continuous Plates Using a High-Performance Programming Language (MATLAB). Path of Science, 8(8), 3001. https://doi.org/10.22178/pos.84-9
Anshu, K., Kemi, O., Vyacheslav, K., & V., D., Fabio. (2025). Survey of Mathematical Models of and Numerical Methods for Fluid Dynamics Water Engineering. ArXiv.Org. https://doi.org/10.48550/arxiv.2512.16351
Arangarajan, M., R, Revathi., Sivaprasad, R., Alfurhood, B. S., & Mane, A. (2024). Structural Analysis and Finite Element Methods: Modeling and Simulation in Mechanical Engineering. Journal Of Advanced Zoology, 45(1). https://doi.org/10.17762/jaz.v45is1.2884
Arcos, A. L. C. (2022). Finite Element Model Updating for Civil Engineering Structures : Applying convex optimization. Research Portal Denmark, 126. https://local.forskningsportal.dk/local/dki-cgi/ws/cris-link?src=dtu&id=dtu-aa22d803-4d3e-4a90-948f-a799d911675b&ti=Finite%20Element%20Model%20Updating%20for%20Civil%20Engineering%20Structures%20%3A%20Applying%20convex%20optimization
Bala, D. M. (2025). Mathematical Modeling of Fluids: Theory, Equations, and Applications. International Journal of Scientific Research in Science Engineering and Technology, 12(4), 402. https://doi.org/10.32628/ijsrset2512525
Baradi, L., Gosukonda, S., & Babu, B. S. (2024). Turbulent Flow Heat and Mass Transfer of CuO -Water Nano Fluid Through a Square Channel with Heated Inner Triangular Groove. Research Square (Research Square). https://doi.org/10.21203/rs.3.rs-4853467/v1
Barone, M., Hammond, S., Sprague, M., & Womble, D. E. (2015). A2E High Fidelity Modeling: Strategic Planning Meetings. https://doi.org/10.2172/1225833
Bartoldson, B. R., Hu, Y., Saini, A., Cadena, J., Fu, Y., Bao, J., Xu, Z., Ng, B., & Nguyen, P. (2023). Scientific Computing Algorithms to Learn Enhanced Scalable Surrogates for Mesh Physics. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2304.00338
Bhat, P. G. (2020). Survey on Applications Numerical Methods in Engineering. International Journal for Research in Applied Science and Engineering Technology, 8(5), 1602. https://doi.org/10.22214/ijraset.2020.5260
Bocco, A. (2020). A variable precision hardware acceleration for scientific computing. HAL (Le Centre Pour La Communication Scientifique Directe). https://theses.hal.science/tel-03102749
Briol, F., Oates, C. J., Girolami, M., Osborne, M. A., & Sejdinović, D. (2019). Probabilistic Integration: A Role in Statistical Computation? Statistical Science, 34(1). https://doi.org/10.1214/18-sts660
Buryachenko, K., Chugunova, M., & Kolomoitsev, Y. (2025). Editorial: Approximation methods and analytical modeling using partial differential equations. Frontiers in Applied Mathematics and Statistics, 11. https://doi.org/10.3389/fams.2025.1584781
Chandravadia, K., Prakash, P., & Swaminarayan, P. (2023). Enhancing Signature Verification through Neural Network Ensemble. International Journal of Advanced Trends in Computer Science and Engineering, 12(3), 117. https://doi.org/10.30534/ijatcse/2023/031232023
Chen, C., Yang, Y., Yang, X., & Hao, W. (2024). Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2405.14099
Chen, S., Bi, Z., Liu, J., Peng, B., Zhang, S., Pan, X., Xu, J., Wang, J., Chen, K., Yin, C. H., Feng, P., Wen, Y., Wang, T., Li, M., Ren, J., Niu, Q., & Liu, M. (2024). Deep Learning and Machine Learning -- Python Data Structures and Mathematics Fundamental: From Theory to Practice. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2410.19849
Chen, Y., Zhang, L., Xu, Z., Zhou, H., & Ren, Z. (2025). OptMetaOpenFOAM: Large Language Model Driven Chain of Thought for Sensitivity Analysis and Parameter Optimization based on CFD. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2503.01273
Chen, Z. (2005). Finite Element Methods and Their Applications. In Scientific computation. Springer Nature. https://doi.org/10.1007/3-540-28078-2
Chowdhury, I. A. (2024). State-of-the-Art CFD Simulation: A Review of Techniques, Validation Methods, and Application Scenarios. Journal of Recent Trends in Mechanics, 9(2), 45. https://doi.org/10.46610/jortm.2024.v09i02.005
Conti, P., Guo, M., Frangi, A., & Manzoni, A. (2025). Progressive multi-fidelity learning for physical system predictions. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2510.13762
Dai, L., & Foroutan, K. (2024). A semi-analytical and numerical approach for solving 2-D and 6-D nonlinear and complex functionally graded tubular systems. Research Square (Research Square). https://doi.org/10.21203/rs.3.rs-3998430/v1
Dall’Agnol, C., Marchi, C. H., & Moro, D. F. (2024). An Enhanced Hybrid Methodology for Iteration Error Estimation and Reduction in Heat Transfer Modeling. Mathematical Modelling and Engineering Problems, 11(2), 349. https://doi.org/10.18280/mmep.110207
Deeb, A., & Dutykh, D. (2024). Error estimation for numerical approximations of ODEs via composition techniques. Part I: One-step methods. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2409.10548
Dey, S., & Kumar, B. V. R. (2023). Finite element analysis of modified N-S equations coupled with energy transfer for hybrid nanofluid flow in complex domains. Computers & Mathematics with Applications, 150, 37. https://doi.org/10.1016/j.camwa.2023.09.002
Dhere, P. (2018). Introduction to Numerical Computing. Journal of Applied & Computational Mathematics, 7(4). https://doi.org/10.4172/2168-9679.1000423
Dietrich, F., & Schilders, W. H. A. (2025). Scientific machine learning. Mathematische Semesterberichte, 72(2), 89. https://doi.org/10.1007/s00591-025-00399-4
Dimov, I., & Georgieva, R. (2025). Cutting-Edge Stochastic Approach: Efficient Monte Carlo Algorithms with Applications to Sensitivity Analysis. Algorithms, 18(5), 252. https://doi.org/10.3390/a18050252
Doğan, K., Hacıoğlu, E., Gürsoy, F., Ertürk, M., & Milovanović, G. V. (2025). Convergence-Equivalent DF and AR Iterations with Refined Data Dependence: Non-Asymptotic Error Bounds and Robustness in Fixed-Point Computations. Axioms, 14(10), 738. https://doi.org/10.3390/axioms14100738
Foroutan, M., Hashemi, M. S., Rezapour, S., İnç, M., & Habibi, F. (2024). Solving an electrically conducting nanofluid over an impermeable stretching cylinder problem with a spectral reproducing kernel method. Journal of Thermal Analysis and Calorimetry, 149(15), 8747. https://doi.org/10.1007/s10973-024-13336-9
Fronk, C., & Petzold, L. (2024). Training Stiff Neural Ordinary Differential Equations with Explicit Exponential Integration Methods. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2412.01181
Hamayd, H. J. (2025). ANALYSIS AND OPTIMIZATION OF ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS. International Journal of Apllied Mathematics, 38, 178. https://doi.org/10.12732/ijam.v38i8s.555
Haque, Md. N., Akter, R., & Mojumder, M. S. H. (2025). An Efficient Explicit Scheme for Solving the 2D Heat Equation with Stability and Convergence Analysis. Journal of Applied Mathematics and Physics, 13(7), 2234. https://doi.org/10.4236/jamp.2025.137127
Hasan, A. B. M. M., Ali, M. M., Hossain, S., Siam, N. H., Rony, A., & Shohan, A.-A. (2025). A comprehensive review on natural convection in various shaped enclosures by FEM: Engineering applications. South African Journal of Chemical Engineering, 54, 308. https://doi.org/10.1016/j.sajce.2025.08.009
Higham, N. J. (1999). Notes on Accuracy and Stability of Algorithms in Numerical Linear Algebra. In Springer series in computational mathematics (p. 47). Springer Nature. https://doi.org/10.1007/978-3-662-03972-4_2
Hoerold, F., Ivanov, I. R., Dhruv, A., Moses, W. S., Dubey, A., Wahib, M., & Domke, J. (2025). RAPTOR: Practical Numerical Profiling of Scientific Applications. arXiv (Cornell University). https://doi.org/10.1145/3712285.3759810
Hong, S.-P. (2023). Different Numerical Techniques, Modeling and Simulation in Solving Complex Problems. Journal of Machine and Computing, 58. https://doi.org/10.53759/7669/jmc202303007
Hossain, M. A., Menz, P., & Stockie, J. M. (2021). An Open-Access Clicker Question Bank for Numerical Analysis. PRIMUS, 32(8), 858. https://doi.org/10.1080/10511970.2021.1954113
Jacobsen, C. (2024). Enhancing Physical Modeling with Interpretable Physics-Aware Machine Learning. Deep Blue (University of Michigan). https://doi.org/10.7302/23070
Kahan, W. (1966). Numerical Linear Algebra. Canadian Mathematical Bulletin, 9(5), 757. https://doi.org/10.4153/cmb-1966-083-2
Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. (2021). Physics-informed machine learning. Nature Reviews Physics, 3(6), 422. https://doi.org/10.1038/s42254-021-00314-5
Kashi, A., Lü, H., Brewer, W., Rogers, D., Matheson, M. J., Shankar, M., & Wang, F. (2024). Mixed-precision numerics in scientific applications: survey and perspectives. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2412.19322
Katende, R. (2024). A Nonlinear Generalization of the Bauer-Fike Theorem and Novel Iterative Methods for Solving Nonlinear Eigenvalue Problems. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2409.11098
Krishna, A. (2022). A Review on Vibro-Acoustic Analysis of a Launch Vehicle Structure [Review of A Review on Vibro-Acoustic Analysis of a Launch Vehicle Structure]. International Journal for Research in Applied Science and Engineering Technology, 10(6), 4154. International Journal for Research in Applied Science and Engineering Technology (IJRASET). https://doi.org/10.22214/ijraset.2022.44873
Łach, Ł., & Svyetlichnyy, D. (2025). Advances in Numerical Modeling for Heat Transfer and Thermal Management: A Review of Computational Approaches and Environmental Impacts. Energies, 18(5), 1302. https://doi.org/10.3390/en18051302
Le-Duc, T., Nguyen‐Xuan, H., & Lee, J. (2023). A finite-element-informed neural network for parametric simulation in structural mechanics. Finite Elements in Analysis and Design, 217, 103904. https://doi.org/10.1016/j.finel.2022.103904
Lenggana, B. W., Prabowo, A. R., Ubaidillah, U., Imaduddin, F., Surojo, E., Nubli, H., Adiputra, R., Lenggana, B. W., Prabowo, A. R., Ubaidillah, U., Imaduddin, F., Surojo, E., Nubli, H., & Adiputra, R. (2021). Effects of mechanical vibration on designed steel-based plate geometries: behavioral estimation subjected to applied material classes using finite-element method. Curved and Layered Structures, 8(1), 225. https://doi.org/10.1515/cls-2021-0021
Li, A., & Li, Y. (2024). Rational Empirical Interpolation Methods with Applications. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2406.19339
Li, L., Chang, J., Vakanski, A., Wang, Y., Yao, T., & Xian, M. (2024). Uncertainty quantification in multivariable regression for material property prediction with Bayesian neural networks. Scientific Reports, 14(1). https://doi.org/10.1038/s41598-024-61189-x
Li, N., & Dong, H. (2023). Analysis of supply chain finance risk assessment based on numerical analysis algorithm. 3C Empresa Investigación y Pensamiento Crítico, 12(2), 217. https://doi.org/10.17993/3cemp.2023.120252.217-234
Liu, W. K., Li, S., & Park, H. S. (2021). Eighty Years of the Finite Element Method: Birth, Evolution, and Future. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2107.04960
Liu, W. K., Li, S., & Park, H. S. (2022). Eighty Years of the Finite Element Method: Birth, Evolution, and Future. Archives of Computational Methods in Engineering, 29(6), 4431. https://doi.org/10.1007/s11831-022-09740-9
Lopes, I. A. R., Furtado, C., Pereira, S. L., & Furtado, A. T. (2024). Editorial: Envisioning the future of computational methods applied to engineering and sciences. Engineering Computations, 41(6), 1369. https://doi.org/10.1108/ec-07-2024-970
Lubis, T. A. (2025). Penerapan Metode Numerik dalam Penyelesaian Persamaan Diferensial. Pentagon Jurnal Matematika Dan Ilmu Pengetahuan Alam, 3(1), 131. https://doi.org/10.62383/pentagon.v3i1.421
Lustri, C. J., Crew, S. C., & Chapman, S. J. (2023). Exponential Asymptotics using Numerical Rational Approximation in Linear Differential Equations. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2312.07773
Martello, G. (2016). Discretization Analysis in FEM Models. MATEC Web of Conferences, 53, 1063. https://doi.org/10.1051/matecconf/20165301063
Mirzaei, D. (2024). Fundamentals of Computing. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2412.19958
Moradi, H., & Kefayat, E. (2025). Investigation of the Estimation Accuracy of 5 Different Numerical ODE Solvers on 3 Case Studies. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2502.10289
Moradi, H., Kefayat, E., & Hossei, H. (2025). Evaluation of the Efficiency and Comparison of Different Numerical Differentiation Methods on Three Case Studies. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2502.10285
Niazai, S., Rahimzai, A. A., & Atifnigar, H. (2023). Applications of MATLAB in Natural Sciences: A Comprehensive Review [Review of Applications of MATLAB in Natural Sciences: A Comprehensive Review]. European Journal of Theoretical and Applied Sciences, 1(5), 1006. https://doi.org/10.59324/ejtas.2023.1(5).87
Numerical methods for nonlinear engineering models. (2009). Choice Reviews Online, 47(3), 47. https://doi.org/10.5860/choice.47-1444
Ọmọle, E. Ọlaoluwa, Adeyefa, E. O., Apanpa, K. I., Ayodele, V. I., Amoyedo, F. E., & Emadifar, H. (2024). Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems. PLoS ONE, 19(5). https://doi.org/10.1371/journal.pone.0301505
Orhan, F. (2024). Computational Math Modeling and AI Optimization for Better Decision-Making: Applications in Machine Learning. Next Frontier., 8(1), 79. https://doi.org/10.62802/61e8x381
Ortega-Quintana, F. A. (2025). Numerical Methods and Computational Models with Python in Heat Transfer. In IntechOpen eBooks. IntechOpen. https://doi.org/10.5772/intechopen.1009917
Pasupuleti, M. K. (2024). Advanced Computational Techniques: Mathematical Modeling and Numerical Simulation Tools (p. 55). https://doi.org/10.62311/nesx/932106
Patel, R. B. (2023). Computational Mathematics: Solving Complex Problems with the Latest Techniques. International Journal of Computer Applications, 184(46), 37. https://doi.org/10.5120/ijca2023922574
Ping, W., & Zou, Q. (2024). Analysis of any order Runge-Kutta Spectral Volume Schemes for 1D Hyperbolic Equations. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2409.13485
Plevris, V., Georgioudakis, M., & Kioumarsi, M. (2025). Computational Methods in Structural Engineering: Current Advances and Future Perspectives. Computation, 13(9), 224. https://doi.org/10.3390/computation13090224
Plevris, V., & Tsiatas, G. (2018). Computational Structural Engineering: Past Achievements and Future Challenges. Frontiers in Built Environment, 4. https://doi.org/10.3389/fbuil.2018.00021
Prasetiyo, A. B., Prastowo, R., Sekarjati, K. A., Susiana, A., Ardiyat, I. N., Prabowo, F. Y., Assagaf, I. P. A., Rohi, J. R., & Ito, Y. (2022). Finite Element Analysis (FEA) of blade weed design using Ansys workbench. SINERGI, 26(3), 371. https://doi.org/10.22441/sinergi.2022.3.012
Quarteroni, A., Gervasio, P., & Regazzoni, F. (2025). Combining physics-based and data-driven models: advancing the frontiers of research with Scientific Machine Learning. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2501.18708
Quarteroni, A., Sacco, R., & Saleri, F. (2006). Numerical Mathematics. In Texts in applied mathematics. Springer Nature. https://doi.org/10.1007/b98885
Qureshi, S., Argyros, I. K., Soomro, A., Gdawiec, K., Shaikh, A. A., & Hınçal, E. (2023). A new optimal root-finding iterative algorithm: local and semilocal analysis with polynomiography. Numerical Algorithms, 95(4), 1715. https://doi.org/10.1007/s11075-023-01625-7
Rastogi, Y. (2023). Extremal Growth of Multiple Toeplitz Operators and Implications for Numerical Stability of Approximation Schemes. Research Square (Research Square). https://doi.org/10.21203/rs.3.rs-3399905/v1
Razzaq, A. S., Hnait, H. S. H., Mohammed, S., Jamil, O. B., Alkareem, S. R., & Rasool, S. M. (2025). Error Analysis in Numerical Techniques using Euler’s Method with a Practical Application. Journal of University of Babylon for Pure and Applied Sciences, 33, 41. https://doi.org/10.29196/jubpas.v33i.5928
Rincon-Tabares, J.-S., Aristizábal, M., Balcer, M., Montoya, A., Millwater, H., & Restrepo, D. (2024). Efficient Sensitivity Analysis of the Thermal Profile in Powder Bed Fusion of Metals using HYPercomplex Automatic Differentiation Finite Element Method. Additive Manufacturing, 94, 104488. https://doi.org/10.1016/j.addma.2024.104488
Rommel, J. B. (2022). From Prescriptive to Predictive: an Interdisciplinary Perspective on the Future of Computational Chemistry. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2103.02933
Sahani, S. K. (2024). An In-Depth Stability and Convergence Analysis of the Runge-Kutta 4th Order Method for Nonlinear Ordinary Differential Equations. Panamerican Mathematical Journal., 34(2), 300. https://doi.org/10.52783/pmj.v34.i2.5583
Sala, R. (2021). Computational Rational Engineering and Development: Synergies and Opportunities. In Lecture notes in networks and systems (p. 744). Springer International Publishing. https://doi.org/10.1007/978-3-030-82193-7_50
San, O., Rasheed, A., & Kvamsdal, T. (2021). Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution. GAMM-Mitteilungen, 44(2). https://doi.org/10.1002/gamm.202100007
San, O., Rasheed, A., & Kvamsdal, T. (2022). Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2103.14629
Santos, R. D. C. dos, & Sales, J. H. de O. (2023). Mathematical analysis for simulation of incompressible fluid flow. The Journal of Engineering and Exact Sciences, 9(12), 17392. https://doi.org/10.18540/jcecvl9iss12pp17392
Santos, R. D. C. dos, & Sales, J. H. de O. (2024). Advancements in mathematical modeling and numerical simulation: Immersed boundary method for Navier-Stokes equations applied to stationary and rotating isothermal cylinders. The Journal of Engineering and Exact Sciences, 10(8), 18894. https://doi.org/10.18540/jcecvl10iss8pp18894
Sapienza, F., Bolíbar, J., Schäfer, F., Groenke, B., Pal, A., Boussange, V., Heimbach, P., Hooker, G., Pérez, F., Persson, P., & Rackauckas, C. (2024). Differentiable Programming for Differential Equations: A Review [Review of Differentiable Programming for Differential Equations: A Review]. arXiv (Cornell University). Cornell University. https://doi.org/10.48550/arxiv.2406.09699
Shachar, A. (2024). Introduction to Algogens. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2403.01426
Shahane, S., Chammas, S., Bezgin, D. A., Buhendwa, A. B., Schmidt, S. J., Adams, N. A., Bryngelson, S. H., Chen, Y., Wang, Q., Sha, F., & Zepeda-Núñez, L. (2024). Rational-WENO: A lightweight, physically-consistent three-point weighted essentially non-oscillatory scheme. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2409.09217
Shahoo, S. T., Renu Singh, Priyanka Tiwari, Lovely Nigam, Pooja. (2025). Analysis And Implementation of Euler Method for Solving Ordinary Differential Equation. Zenodo (CERN European Organization for Nuclear Research). https://doi.org/10.5281/zenodo.15148232
Shallal, M. A., & Jumaa, B. F. (2016). Numerical Solutions Based on Finite Difference Techniques for Two Dimensional Advection-Diffusion Equation. British Journal of Mathematics & Computer Science, 16(2), 1. https://doi.org/10.9734/bjmcs/2016/25464
Sharapov, D. (2024). Building accurate numerical models. E3S Web of Conferences, 583, 7012. https://doi.org/10.1051/e3sconf/202458307012
Sharma, H., Novák, L., & Shields, M. D. (2024). Physics-constrained polynomial chaos expansion for scientific machine learning and uncertainty quantification. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2402.15115
Shen, G. (2025). Mathematical Models and Numerical Simulations Based on High Precision Multiscale Algorithms. Procedia Computer Science, 261, 118. https://doi.org/10.1016/j.procs.2025.04.179
Shylaja, G., Naidu, V. K., Venkatesh, B., & Mallikarjunaiah, S. M. (2025). Finite element modeling of V-notched thermoelastic strain-limiting solids containing inclusions. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2507.09300
Șimon-Marinică, A. B., Vlasin, N. I., Manea, F., & Florea, G.-D. (2021). Finite element method to solve engineering problems using ansys. MATEC Web of Conferences, 342, 1015. https://doi.org/10.1051/matecconf/202134201015
Singh, R., Menezes, A. A. A., Irfan, R., & Ramsundar, B. (2024). Open source Differentiable ODE Solving Infrastructure. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2411.19882
Soofastaei, A. (2022). Numerical Simulation - Advanced Techniques for Science and Engineering. In IntechOpen eBooks. IntechOpen. https://doi.org/10.5772/intechopen.100786
Soofastaei, A. (2023). Introductory Chapter: Numerical Simulation. In IntechOpen eBooks. IntechOpen. https://doi.org/10.5772/intechopen.112097
Srinvias, T. A. S., Donald, A. D., Sameena, M., Rekha, K., & Srihith, I. D. (2023). Unlocking the Power of Matlab: A Comprehensive Survey. International Journal of Advanced Research in Science Communication and Technology, 20. https://doi.org/10.48175/ijarsct-9005
Srivastava, N. (2018). Optimization of Convergence of Mixed Finite Element Approximations Analysis. Türk Bilgisayar ve Matematik Eğitimi Dergisi, 9(2), 692. https://doi.org/10.17762/turcomat.v9i2.13871
Thacker, B. H., Doebling, S. W., Hemez, F., Anderson, M. C., Pepin, J., & Rodriguez, E. A. (2004). Concepts of Model Verification and Validation. https://doi.org/10.2172/835920
Turab, A., Montoyo, A., & Nescolarde‐Selva, J. A. (2024). Computational and analytical analysis of integral-differential equations for modeling avoidance learning behavior. Journal of Applied Mathematics and Computing. https://doi.org/10.1007/s12190-024-02130-3
Tyrus, W., Andrew, P., Christopher, D., Ian, G., Charuka, W., & Pradeep, R. (2025). Physics-Informed Neural Network Frameworks for the Analysis of Engineering and Biological Dynamical Systems Governed by Ordinary Differential Equations. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2511.00043
Vedral, J., Rupp, A., & Kuzmin, D. (2024). Strongly consistent low-dissipation WENO schemes for finite elements. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2407.04646
Vella, C. (2024). Solveurs efficaces et préservant la structure par décomposition propre généralisée pour la dynamique des solides [Ludwig-Maximilians-Universität München]. In Munich Personal RePEc Archive (Ludwig Maximilian University of Munich). http://www.theses.fr/2024ULILN032/document
Vu‐Bac, N., Lahmer, T., Zhuang, X., Nguyen‐Thoi, T., & Rabczuk, T. (2016). A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 100, 19. https://doi.org/10.1016/j.advengsoft.2016.06.005
Wakil, S. M. (2019). Development and Validation of a Finite Element Programme for Determining Stress and Displacement in Truss. International Journal for Research in Applied Science and Engineering Technology, 7(8), 811. https://doi.org/10.22214/ijraset.2019.8119
Welch-Phillips, A., Gibbons, D., Ahern, D. P., & Butler, J. S. (2020). What Is Finite Element Analysis? Clinical Spine Surgery A Spine Publication, 33(8), 323. https://doi.org/10.1097/bsd.0000000000001050
Wheeler, M. F., Anciaux–Sedrakian, A., & Tran, Q. H. (2019). Numerical methods and HPC. Oil & Gas Science and Technology – Revue d’IFP Energies Nouvelles, 74. https://doi.org/10.2516/ogst/2019036
Workineh, Y., Mekonnen, H., & Belew, B. (2024). Numerical methods for solving second-order initial value problems of ordinary differential equations with Euler and Runge-Kutta fourth-order methods. Frontiers in Applied Mathematics and Statistics, 10. https://doi.org/10.3389/fams.2024.1360628
Yan, S., Jingrun, C., & Keke, W. (2025). A Hybrid Discontinuous Galerkin Neural Network Method for Solving Hyperbolic Conservation Laws with Temporal Progressive Learning. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2508.16032
Yang, Y., Yan, M., Zhang, Z., Chen, X., Hào, Ð. N., & Weixiong, C. (2025). Steady-state and transient thermal stress analysis using a polygonal finite element method. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2501.03908
Zhang, E., Kahana, A., Turkel, E., Ranade, R., Pathak, J., & Karniadakis, G. E. (2022). Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2208.13273
Zheng, B. (2023). Ordinary Differential Equation and Its Application. Highlights in Science Engineering and Technology, 72, 645. https://doi.org/10.54097/rnnev212
Zhu, Y., Zabaras, N., Koutsourelakis, P., & Perdikaris, P. (2019). Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. Journal of Computational Physics, 394, 56. https://doi.org/10.1016/j.jcp.2019.05.024
Cite this Article as: Purushottam B (2026), “Numerical Analysis Concept and Use in Computational Mathematical and Engineering Problem”, International Journal of Advances in Engineering & Scientific Research, Volume 13, Issue 1, 2026, pp 46-65. DOI : https://doi.org/ 10.5281/zenodo.19235234
References
Aghaei, A. A., Moghaddam, M. M., & Parand, K. (2024). PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2409.01899
Akhare, D., Luo, T., & Wang, J. (2024). Diffhybrid-Uq: Uncertainty Quantification for Differentiable Hybrid Neural Modeling. https://doi.org/10.2139/ssrn.4712361
Akinsola, V. O. (2023). Numerical Methods: Euler and Runge-Kutta. In IntechOpen eBooks. IntechOpen. https://doi.org/10.5772/intechopen.108533
Alaneme, G. U., Bahrami, A., Iro, U. I., Ganasen, N., Otu, O. N., Udeala, R. C., Ifebude, B. O., & Onwusereaka, E. A. (2023). Numerical Analysis and Parametric Study on Multiple Degrees-of-Freedom Frames. Civil Engineering Journal, 9(7), 1709. https://doi.org/10.28991/cej-2023-09-07-012
Alber, M., Tepole, A. B., Cannon, W. R., De, S., Durá-Bernal, S., Garikipati, K., Karniadakis, G. E., Lytton, W. W., Perdikaris, P., Petzold, L., & Kuhl, E. (2019). Integrating machine learning and multiscale modeling—perspectives, challenges, and opportunities in the biological, biomedical, and behavioral sciences. Npj Digital Medicine, 2(1), 115. https://doi.org/10.1038/s41746-019-0193-y
AL-JAWARY, M., & Nabi, A. (2019). Reliable Iterative Methods for Solving Convective Straight and Radial Fins with Temperature-Dependent Thermal Conductivity Problems. GAZI UNIVERSITY JOURNAL OF SCIENCE, 32(3), 967. https://doi.org/10.35378/gujs.429896
Alotaibi, H., Soutis, C., & Jabbari, M. (2025). FEM and FVM Methods for Design and Manufacturing of Hierarchical Aerospace Composites: A Review. Applied Sciences, 15(16), 8896. https://doi.org/10.3390/app15168896
Amaefule, E. O., Chukwuebuka, A. O., Chukwudi, N. N., Oscar, N. C., Nelson, O. C., Keji, A. H., Ibeamaka, O. M., Sunday, O. I., Don-Ugbaga, C., & Ezeokpube, G. (2022). Finite Element Analysis of Continuous Plates Using a High-Performance Programming Language (MATLAB). Path of Science, 8(8), 3001. https://doi.org/10.22178/pos.84-9
Anshu, K., Kemi, O., Vyacheslav, K., & V., D., Fabio. (2025). Survey of Mathematical Models of and Numerical Methods for Fluid Dynamics Water Engineering. ArXiv.Org. https://doi.org/10.48550/arxiv.2512.16351
Arangarajan, M., R, Revathi., Sivaprasad, R., Alfurhood, B. S., & Mane, A. (2024). Structural Analysis and Finite Element Methods: Modeling and Simulation in Mechanical Engineering. Journal Of Advanced Zoology, 45(1). https://doi.org/10.17762/jaz.v45is1.2884
Arcos, A. L. C. (2022). Finite Element Model Updating for Civil Engineering Structures : Applying convex optimization. Research Portal Denmark, 126. https://local.forskningsportal.dk/local/dki-cgi/ws/cris-link?src=dtu&id=dtu-aa22d803-4d3e-4a90-948f-a799d911675b&ti=Finite%20Element%20Model%20Updating%20for%20Civil%20Engineering%20Structures%20%3A%20Applying%20convex%20optimization
Bala, D. M. (2025). Mathematical Modeling of Fluids: Theory, Equations, and Applications. International Journal of Scientific Research in Science Engineering and Technology, 12(4), 402. https://doi.org/10.32628/ijsrset2512525
Baradi, L., Gosukonda, S., & Babu, B. S. (2024). Turbulent Flow Heat and Mass Transfer of CuO -Water Nano Fluid Through a Square Channel with Heated Inner Triangular Groove. Research Square (Research Square). https://doi.org/10.21203/rs.3.rs-4853467/v1
Barone, M., Hammond, S., Sprague, M., & Womble, D. E. (2015). A2E High Fidelity Modeling: Strategic Planning Meetings. https://doi.org/10.2172/1225833
Bartoldson, B. R., Hu, Y., Saini, A., Cadena, J., Fu, Y., Bao, J., Xu, Z., Ng, B., & Nguyen, P. (2023). Scientific Computing Algorithms to Learn Enhanced Scalable Surrogates for Mesh Physics. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2304.00338
Bhat, P. G. (2020). Survey on Applications Numerical Methods in Engineering. International Journal for Research in Applied Science and Engineering Technology, 8(5), 1602. https://doi.org/10.22214/ijraset.2020.5260
Bocco, A. (2020). A variable precision hardware acceleration for scientific computing. HAL (Le Centre Pour La Communication Scientifique Directe). https://theses.hal.science/tel-03102749
Briol, F., Oates, C. J., Girolami, M., Osborne, M. A., & Sejdinović, D. (2019). Probabilistic Integration: A Role in Statistical Computation? Statistical Science, 34(1). https://doi.org/10.1214/18-sts660
Buryachenko, K., Chugunova, M., & Kolomoitsev, Y. (2025). Editorial: Approximation methods and analytical modeling using partial differential equations. Frontiers in Applied Mathematics and Statistics, 11. https://doi.org/10.3389/fams.2025.1584781
Chandravadia, K., Prakash, P., & Swaminarayan, P. (2023). Enhancing Signature Verification through Neural Network Ensemble. International Journal of Advanced Trends in Computer Science and Engineering, 12(3), 117. https://doi.org/10.30534/ijatcse/2023/031232023
Chen, C., Yang, Y., Yang, X., & Hao, W. (2024). Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2405.14099
Chen, S., Bi, Z., Liu, J., Peng, B., Zhang, S., Pan, X., Xu, J., Wang, J., Chen, K., Yin, C. H., Feng, P., Wen, Y., Wang, T., Li, M., Ren, J., Niu, Q., & Liu, M. (2024). Deep Learning and Machine Learning -- Python Data Structures and Mathematics Fundamental: From Theory to Practice. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2410.19849
Chen, Y., Zhang, L., Xu, Z., Zhou, H., & Ren, Z. (2025). OptMetaOpenFOAM: Large Language Model Driven Chain of Thought for Sensitivity Analysis and Parameter Optimization based on CFD. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2503.01273
Chen, Z. (2005). Finite Element Methods and Their Applications. In Scientific computation. Springer Nature. https://doi.org/10.1007/3-540-28078-2
Chowdhury, I. A. (2024). State-of-the-Art CFD Simulation: A Review of Techniques, Validation Methods, and Application Scenarios. Journal of Recent Trends in Mechanics, 9(2), 45. https://doi.org/10.46610/jortm.2024.v09i02.005
Conti, P., Guo, M., Frangi, A., & Manzoni, A. (2025). Progressive multi-fidelity learning for physical system predictions. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2510.13762
Dai, L., & Foroutan, K. (2024). A semi-analytical and numerical approach for solving 2-D and 6-D nonlinear and complex functionally graded tubular systems. Research Square (Research Square). https://doi.org/10.21203/rs.3.rs-3998430/v1
Dall’Agnol, C., Marchi, C. H., & Moro, D. F. (2024). An Enhanced Hybrid Methodology for Iteration Error Estimation and Reduction in Heat Transfer Modeling. Mathematical Modelling and Engineering Problems, 11(2), 349. https://doi.org/10.18280/mmep.110207
Deeb, A., & Dutykh, D. (2024). Error estimation for numerical approximations of ODEs via composition techniques. Part I: One-step methods. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2409.10548
Dey, S., & Kumar, B. V. R. (2023). Finite element analysis of modified N-S equations coupled with energy transfer for hybrid nanofluid flow in complex domains. Computers & Mathematics with Applications, 150, 37. https://doi.org/10.1016/j.camwa.2023.09.002
Dhere, P. (2018). Introduction to Numerical Computing. Journal of Applied & Computational Mathematics, 7(4). https://doi.org/10.4172/2168-9679.1000423
Dietrich, F., & Schilders, W. H. A. (2025). Scientific machine learning. Mathematische Semesterberichte, 72(2), 89. https://doi.org/10.1007/s00591-025-00399-4
Dimov, I., & Georgieva, R. (2025). Cutting-Edge Stochastic Approach: Efficient Monte Carlo Algorithms with Applications to Sensitivity Analysis. Algorithms, 18(5), 252. https://doi.org/10.3390/a18050252
Doğan, K., Hacıoğlu, E., Gürsoy, F., Ertürk, M., & Milovanović, G. V. (2025). Convergence-Equivalent DF and AR Iterations with Refined Data Dependence: Non-Asymptotic Error Bounds and Robustness in Fixed-Point Computations. Axioms, 14(10), 738. https://doi.org/10.3390/axioms14100738
Foroutan, M., Hashemi, M. S., Rezapour, S., İnç, M., & Habibi, F. (2024). Solving an electrically conducting nanofluid over an impermeable stretching cylinder problem with a spectral reproducing kernel method. Journal of Thermal Analysis and Calorimetry, 149(15), 8747. https://doi.org/10.1007/s10973-024-13336-9
Fronk, C., & Petzold, L. (2024). Training Stiff Neural Ordinary Differential Equations with Explicit Exponential Integration Methods. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2412.01181
Hamayd, H. J. (2025). ANALYSIS AND OPTIMIZATION OF ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS. International Journal of Apllied Mathematics, 38, 178. https://doi.org/10.12732/ijam.v38i8s.555
Haque, Md. N., Akter, R., & Mojumder, M. S. H. (2025). An Efficient Explicit Scheme for Solving the 2D Heat Equation with Stability and Convergence Analysis. Journal of Applied Mathematics and Physics, 13(7), 2234. https://doi.org/10.4236/jamp.2025.137127
Hasan, A. B. M. M., Ali, M. M., Hossain, S., Siam, N. H., Rony, A., & Shohan, A.-A. (2025). A comprehensive review on natural convection in various shaped enclosures by FEM: Engineering applications. South African Journal of Chemical Engineering, 54, 308. https://doi.org/10.1016/j.sajce.2025.08.009
Higham, N. J. (1999). Notes on Accuracy and Stability of Algorithms in Numerical Linear Algebra. In Springer series in computational mathematics (p. 47). Springer Nature. https://doi.org/10.1007/978-3-662-03972-4_2
Hoerold, F., Ivanov, I. R., Dhruv, A., Moses, W. S., Dubey, A., Wahib, M., & Domke, J. (2025). RAPTOR: Practical Numerical Profiling of Scientific Applications. arXiv (Cornell University). https://doi.org/10.1145/3712285.3759810
Hong, S.-P. (2023). Different Numerical Techniques, Modeling and Simulation in Solving Complex Problems. Journal of Machine and Computing, 58. https://doi.org/10.53759/7669/jmc202303007
Hossain, M. A., Menz, P., & Stockie, J. M. (2021). An Open-Access Clicker Question Bank for Numerical Analysis. PRIMUS, 32(8), 858. https://doi.org/10.1080/10511970.2021.1954113
Jacobsen, C. (2024). Enhancing Physical Modeling with Interpretable Physics-Aware Machine Learning. Deep Blue (University of Michigan). https://doi.org/10.7302/23070
Kahan, W. (1966). Numerical Linear Algebra. Canadian Mathematical Bulletin, 9(5), 757. https://doi.org/10.4153/cmb-1966-083-2
Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. (2021). Physics-informed machine learning. Nature Reviews Physics, 3(6), 422. https://doi.org/10.1038/s42254-021-00314-5
Kashi, A., Lü, H., Brewer, W., Rogers, D., Matheson, M. J., Shankar, M., & Wang, F. (2024). Mixed-precision numerics in scientific applications: survey and perspectives. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2412.19322
Katende, R. (2024). A Nonlinear Generalization of the Bauer-Fike Theorem and Novel Iterative Methods for Solving Nonlinear Eigenvalue Problems. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2409.11098
Krishna, A. (2022). A Review on Vibro-Acoustic Analysis of a Launch Vehicle Structure [Review of A Review on Vibro-Acoustic Analysis of a Launch Vehicle Structure]. International Journal for Research in Applied Science and Engineering Technology, 10(6), 4154. International Journal for Research in Applied Science and Engineering Technology (IJRASET). https://doi.org/10.22214/ijraset.2022.44873
Łach, Ł., & Svyetlichnyy, D. (2025). Advances in Numerical Modeling for Heat Transfer and Thermal Management: A Review of Computational Approaches and Environmental Impacts. Energies, 18(5), 1302. https://doi.org/10.3390/en18051302
Le-Duc, T., Nguyen‐Xuan, H., & Lee, J. (2023). A finite-element-informed neural network for parametric simulation in structural mechanics. Finite Elements in Analysis and Design, 217, 103904. https://doi.org/10.1016/j.finel.2022.103904
Lenggana, B. W., Prabowo, A. R., Ubaidillah, U., Imaduddin, F., Surojo, E., Nubli, H., Adiputra, R., Lenggana, B. W., Prabowo, A. R., Ubaidillah, U., Imaduddin, F., Surojo, E., Nubli, H., & Adiputra, R. (2021). Effects of mechanical vibration on designed steel-based plate geometries: behavioral estimation subjected to applied material classes using finite-element method. Curved and Layered Structures, 8(1), 225. https://doi.org/10.1515/cls-2021-0021
Li, A., & Li, Y. (2024). Rational Empirical Interpolation Methods with Applications. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2406.19339
Li, L., Chang, J., Vakanski, A., Wang, Y., Yao, T., & Xian, M. (2024). Uncertainty quantification in multivariable regression for material property prediction with Bayesian neural networks. Scientific Reports, 14(1). https://doi.org/10.1038/s41598-024-61189-x
Li, N., & Dong, H. (2023). Analysis of supply chain finance risk assessment based on numerical analysis algorithm. 3C Empresa Investigación y Pensamiento Crítico, 12(2), 217. https://doi.org/10.17993/3cemp.2023.120252.217-234
Liu, W. K., Li, S., & Park, H. S. (2021). Eighty Years of the Finite Element Method: Birth, Evolution, and Future. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2107.04960
Liu, W. K., Li, S., & Park, H. S. (2022). Eighty Years of the Finite Element Method: Birth, Evolution, and Future. Archives of Computational Methods in Engineering, 29(6), 4431. https://doi.org/10.1007/s11831-022-09740-9
Lopes, I. A. R., Furtado, C., Pereira, S. L., & Furtado, A. T. (2024). Editorial: Envisioning the future of computational methods applied to engineering and sciences. Engineering Computations, 41(6), 1369. https://doi.org/10.1108/ec-07-2024-970
Lubis, T. A. (2025). Penerapan Metode Numerik dalam Penyelesaian Persamaan Diferensial. Pentagon Jurnal Matematika Dan Ilmu Pengetahuan Alam, 3(1), 131. https://doi.org/10.62383/pentagon.v3i1.421
Lustri, C. J., Crew, S. C., & Chapman, S. J. (2023). Exponential Asymptotics using Numerical Rational Approximation in Linear Differential Equations. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2312.07773
Martello, G. (2016). Discretization Analysis in FEM Models. MATEC Web of Conferences, 53, 1063. https://doi.org/10.1051/matecconf/20165301063
Mirzaei, D. (2024). Fundamentals of Computing. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2412.19958
Moradi, H., & Kefayat, E. (2025). Investigation of the Estimation Accuracy of 5 Different Numerical ODE Solvers on 3 Case Studies. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2502.10289
Moradi, H., Kefayat, E., & Hossei, H. (2025). Evaluation of the Efficiency and Comparison of Different Numerical Differentiation Methods on Three Case Studies. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2502.10285
Niazai, S., Rahimzai, A. A., & Atifnigar, H. (2023). Applications of MATLAB in Natural Sciences: A Comprehensive Review [Review of Applications of MATLAB in Natural Sciences: A Comprehensive Review]. European Journal of Theoretical and Applied Sciences, 1(5), 1006. https://doi.org/10.59324/ejtas.2023.1(5).87
Numerical methods for nonlinear engineering models. (2009). Choice Reviews Online, 47(3), 47. https://doi.org/10.5860/choice.47-1444
Ọmọle, E. Ọlaoluwa, Adeyefa, E. O., Apanpa, K. I., Ayodele, V. I., Amoyedo, F. E., & Emadifar, H. (2024). Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems. PLoS ONE, 19(5). https://doi.org/10.1371/journal.pone.0301505
Orhan, F. (2024). Computational Math Modeling and AI Optimization for Better Decision-Making: Applications in Machine Learning. Next Frontier., 8(1), 79. https://doi.org/10.62802/61e8x381
Ortega-Quintana, F. A. (2025). Numerical Methods and Computational Models with Python in Heat Transfer. In IntechOpen eBooks. IntechOpen. https://doi.org/10.5772/intechopen.1009917
Pasupuleti, M. K. (2024). Advanced Computational Techniques: Mathematical Modeling and Numerical Simulation Tools (p. 55). https://doi.org/10.62311/nesx/932106
Patel, R. B. (2023). Computational Mathematics: Solving Complex Problems with the Latest Techniques. International Journal of Computer Applications, 184(46), 37. https://doi.org/10.5120/ijca2023922574
Ping, W., & Zou, Q. (2024). Analysis of any order Runge-Kutta Spectral Volume Schemes for 1D Hyperbolic Equations. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2409.13485
Plevris, V., Georgioudakis, M., & Kioumarsi, M. (2025). Computational Methods in Structural Engineering: Current Advances and Future Perspectives. Computation, 13(9), 224. https://doi.org/10.3390/computation13090224
Plevris, V., & Tsiatas, G. (2018). Computational Structural Engineering: Past Achievements and Future Challenges. Frontiers in Built Environment, 4. https://doi.org/10.3389/fbuil.2018.00021
Prasetiyo, A. B., Prastowo, R., Sekarjati, K. A., Susiana, A., Ardiyat, I. N., Prabowo, F. Y., Assagaf, I. P. A., Rohi, J. R., & Ito, Y. (2022). Finite Element Analysis (FEA) of blade weed design using Ansys workbench. SINERGI, 26(3), 371. https://doi.org/10.22441/sinergi.2022.3.012
Quarteroni, A., Gervasio, P., & Regazzoni, F. (2025). Combining physics-based and data-driven models: advancing the frontiers of research with Scientific Machine Learning. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2501.18708
Quarteroni, A., Sacco, R., & Saleri, F. (2006). Numerical Mathematics. In Texts in applied mathematics. Springer Nature. https://doi.org/10.1007/b98885
Qureshi, S., Argyros, I. K., Soomro, A., Gdawiec, K., Shaikh, A. A., & Hınçal, E. (2023). A new optimal root-finding iterative algorithm: local and semilocal analysis with polynomiography. Numerical Algorithms, 95(4), 1715. https://doi.org/10.1007/s11075-023-01625-7
Rastogi, Y. (2023). Extremal Growth of Multiple Toeplitz Operators and Implications for Numerical Stability of Approximation Schemes. Research Square (Research Square). https://doi.org/10.21203/rs.3.rs-3399905/v1
Razzaq, A. S., Hnait, H. S. H., Mohammed, S., Jamil, O. B., Alkareem, S. R., & Rasool, S. M. (2025). Error Analysis in Numerical Techniques using Euler’s Method with a Practical Application. Journal of University of Babylon for Pure and Applied Sciences, 33, 41. https://doi.org/10.29196/jubpas.v33i.5928
Rincon-Tabares, J.-S., Aristizábal, M., Balcer, M., Montoya, A., Millwater, H., & Restrepo, D. (2024). Efficient Sensitivity Analysis of the Thermal Profile in Powder Bed Fusion of Metals using HYPercomplex Automatic Differentiation Finite Element Method. Additive Manufacturing, 94, 104488. https://doi.org/10.1016/j.addma.2024.104488
Rommel, J. B. (2022). From Prescriptive to Predictive: an Interdisciplinary Perspective on the Future of Computational Chemistry. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2103.02933
Sahani, S. K. (2024). An In-Depth Stability and Convergence Analysis of the Runge-Kutta 4th Order Method for Nonlinear Ordinary Differential Equations. Panamerican Mathematical Journal., 34(2), 300. https://doi.org/10.52783/pmj.v34.i2.5583
Sala, R. (2021). Computational Rational Engineering and Development: Synergies and Opportunities. In Lecture notes in networks and systems (p. 744). Springer International Publishing. https://doi.org/10.1007/978-3-030-82193-7_50
San, O., Rasheed, A., & Kvamsdal, T. (2021). Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution. GAMM-Mitteilungen, 44(2). https://doi.org/10.1002/gamm.202100007
San, O., Rasheed, A., & Kvamsdal, T. (2022). Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2103.14629
Santos, R. D. C. dos, & Sales, J. H. de O. (2023). Mathematical analysis for simulation of incompressible fluid flow. The Journal of Engineering and Exact Sciences, 9(12), 17392. https://doi.org/10.18540/jcecvl9iss12pp17392
Santos, R. D. C. dos, & Sales, J. H. de O. (2024). Advancements in mathematical modeling and numerical simulation: Immersed boundary method for Navier-Stokes equations applied to stationary and rotating isothermal cylinders. The Journal of Engineering and Exact Sciences, 10(8), 18894. https://doi.org/10.18540/jcecvl10iss8pp18894
Sapienza, F., Bolíbar, J., Schäfer, F., Groenke, B., Pal, A., Boussange, V., Heimbach, P., Hooker, G., Pérez, F., Persson, P., & Rackauckas, C. (2024). Differentiable Programming for Differential Equations: A Review [Review of Differentiable Programming for Differential Equations: A Review]. arXiv (Cornell University). Cornell University. https://doi.org/10.48550/arxiv.2406.09699
Shachar, A. (2024). Introduction to Algogens. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2403.01426
Shahane, S., Chammas, S., Bezgin, D. A., Buhendwa, A. B., Schmidt, S. J., Adams, N. A., Bryngelson, S. H., Chen, Y., Wang, Q., Sha, F., & Zepeda-Núñez, L. (2024). Rational-WENO: A lightweight, physically-consistent three-point weighted essentially non-oscillatory scheme. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2409.09217
Shahoo, S. T., Renu Singh, Priyanka Tiwari, Lovely Nigam, Pooja. (2025). Analysis And Implementation of Euler Method for Solving Ordinary Differential Equation. Zenodo (CERN European Organization for Nuclear Research). https://doi.org/10.5281/zenodo.15148232
Shallal, M. A., & Jumaa, B. F. (2016). Numerical Solutions Based on Finite Difference Techniques for Two Dimensional Advection-Diffusion Equation. British Journal of Mathematics & Computer Science, 16(2), 1. https://doi.org/10.9734/bjmcs/2016/25464
Sharapov, D. (2024). Building accurate numerical models. E3S Web of Conferences, 583, 7012. https://doi.org/10.1051/e3sconf/202458307012
Sharma, H., Novák, L., & Shields, M. D. (2024). Physics-constrained polynomial chaos expansion for scientific machine learning and uncertainty quantification. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2402.15115
Shen, G. (2025). Mathematical Models and Numerical Simulations Based on High Precision Multiscale Algorithms. Procedia Computer Science, 261, 118. https://doi.org/10.1016/j.procs.2025.04.179
Shylaja, G., Naidu, V. K., Venkatesh, B., & Mallikarjunaiah, S. M. (2025). Finite element modeling of V-notched thermoelastic strain-limiting solids containing inclusions. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2507.09300
Șimon-Marinică, A. B., Vlasin, N. I., Manea, F., & Florea, G.-D. (2021). Finite element method to solve engineering problems using ansys. MATEC Web of Conferences, 342, 1015. https://doi.org/10.1051/matecconf/202134201015
Singh, R., Menezes, A. A. A., Irfan, R., & Ramsundar, B. (2024). Open source Differentiable ODE Solving Infrastructure. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2411.19882
Soofastaei, A. (2022). Numerical Simulation - Advanced Techniques for Science and Engineering. In IntechOpen eBooks. IntechOpen. https://doi.org/10.5772/intechopen.100786
Soofastaei, A. (2023). Introductory Chapter: Numerical Simulation. In IntechOpen eBooks. IntechOpen. https://doi.org/10.5772/intechopen.112097
Srinvias, T. A. S., Donald, A. D., Sameena, M., Rekha, K., & Srihith, I. D. (2023). Unlocking the Power of Matlab: A Comprehensive Survey. International Journal of Advanced Research in Science Communication and Technology, 20. https://doi.org/10.48175/ijarsct-9005
Srivastava, N. (2018). Optimization of Convergence of Mixed Finite Element Approximations Analysis. Türk Bilgisayar ve Matematik Eğitimi Dergisi, 9(2), 692. https://doi.org/10.17762/turcomat.v9i2.13871
Thacker, B. H., Doebling, S. W., Hemez, F., Anderson, M. C., Pepin, J., & Rodriguez, E. A. (2004). Concepts of Model Verification and Validation. https://doi.org/10.2172/835920
Turab, A., Montoyo, A., & Nescolarde‐Selva, J. A. (2024). Computational and analytical analysis of integral-differential equations for modeling avoidance learning behavior. Journal of Applied Mathematics and Computing. https://doi.org/10.1007/s12190-024-02130-3
Tyrus, W., Andrew, P., Christopher, D., Ian, G., Charuka, W., & Pradeep, R. (2025). Physics-Informed Neural Network Frameworks for the Analysis of Engineering and Biological Dynamical Systems Governed by Ordinary Differential Equations. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2511.00043
Vedral, J., Rupp, A., & Kuzmin, D. (2024). Strongly consistent low-dissipation WENO schemes for finite elements. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2407.04646
Vella, C. (2024). Solveurs efficaces et préservant la structure par décomposition propre généralisée pour la dynamique des solides [Ludwig-Maximilians-Universität München]. In Munich Personal RePEc Archive (Ludwig Maximilian University of Munich). http://www.theses.fr/2024ULILN032/document
Vu‐Bac, N., Lahmer, T., Zhuang, X., Nguyen‐Thoi, T., & Rabczuk, T. (2016). A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 100, 19. https://doi.org/10.1016/j.advengsoft.2016.06.005
Wakil, S. M. (2019). Development and Validation of a Finite Element Programme for Determining Stress and Displacement in Truss. International Journal for Research in Applied Science and Engineering Technology, 7(8), 811. https://doi.org/10.22214/ijraset.2019.8119
Welch-Phillips, A., Gibbons, D., Ahern, D. P., & Butler, J. S. (2020). What Is Finite Element Analysis? Clinical Spine Surgery A Spine Publication, 33(8), 323. https://doi.org/10.1097/bsd.0000000000001050
Wheeler, M. F., Anciaux–Sedrakian, A., & Tran, Q. H. (2019). Numerical methods and HPC. Oil & Gas Science and Technology – Revue d’IFP Energies Nouvelles, 74. https://doi.org/10.2516/ogst/2019036
Workineh, Y., Mekonnen, H., & Belew, B. (2024). Numerical methods for solving second-order initial value problems of ordinary differential equations with Euler and Runge-Kutta fourth-order methods. Frontiers in Applied Mathematics and Statistics, 10. https://doi.org/10.3389/fams.2024.1360628
Yan, S., Jingrun, C., & Keke, W. (2025). A Hybrid Discontinuous Galerkin Neural Network Method for Solving Hyperbolic Conservation Laws with Temporal Progressive Learning. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2508.16032
Yang, Y., Yan, M., Zhang, Z., Chen, X., Hào, Ð. N., & Weixiong, C. (2025). Steady-state and transient thermal stress analysis using a polygonal finite element method. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2501.03908
Zhang, E., Kahana, A., Turkel, E., Ranade, R., Pathak, J., & Karniadakis, G. E. (2022). Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2208.13273
Zheng, B. (2023). Ordinary Differential Equation and Its Application. Highlights in Science Engineering and Technology, 72, 645. https://doi.org/10.54097/rnnev212
Zhu, Y., Zabaras, N., Koutsourelakis, P., & Perdikaris, P. (2019). Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. Journal of Computational Physics, 394, 56. https://doi.org/10.1016/j.jcp.2019.05.024
