A Robust Numerical Method For The Singularly Perturbed Integro-Differential Equations

Year 2025, Early View, 1 - 1

Derya Arslan , Ercan Celık

https://doi.org/10.35378/gujs.1498203

Abstract

In the present study, NIM is given to approximate the solution of SPFIDEs. As a first step, the FDM on a uniform mesh is used, followed by the TM for integrals. After these calculations, the difference equation is obtained in the form of a system of equations. This system of equations is solved with the TA. Finally, example applications are made, showing the accuracy and economics of the presented method.

Keywords

Singularly Perturbed Problem(SPP), Fredholm Integro-Differential Equation(FIDE), Numerical Integration Method(NIM), Finite difference Method(FDM), Thomas Algorithm(TA)

References

• [1] Geçmen, M. Z., Çelik, E., “Numerical solution of Volterra–Fredholm integral equations with Hosoya polynomials”, Mathematical Methods in the Applied Sciences, 44(14): 11166-11173, (2021).
• [2] Jerri, A., “Introduction to Integral Equations with Applications”, Wiley, New York, (1999).
• [3] Kılıç, S. Ş. Ş., Çelik, E., Özdemir, E., “The Solution of linear Volterra integral equation of the first kind with ZZ- transform”, Turkish Journal of Science, 6 (3): 127-133, (2021).
• [4] Kythe, K., Puri, P., “Computational Methods for Linear Integral Equations”, Birkhauser, Boston, (2002).
• [5] Pandey, P. K., “Numerical solution of linear Fredholm integro-differential equations by non-standard finite difference method”, Applications and Applied Mathematics, 10(2): 1019-1026, (2015).
• [6] Polyanin, A.D., Manzhirov, A.V., “Handbook of Integral Equations”, Chapman and Hall/CRC, Boca Raton, (2008).
• [7] Alao, S., Akinboro, F. S., Akinpelu, FO., Oderinu, R. A., “Numerical solution of integro-differential equation using Adomian decomposition and variational iteration methods”, IOSR Journal of Mathematics, 10(4): 18-22, (2014).
• [8] Arqub, O. A., Al-Smadi, M., Shawagfeh N., “Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method”, Applied Mathematics and Computation, 219(17): 8938-8948, (2013).
• [9] Chen, J., He M., Huang Y., “A fast multiscale Galerkin method for solving second order linear Fredholm integro-differential equation with Dirichlet boundary conditions”, Journal of Computational and Applied Mathematics, 364(1), (2020).
• [10] Fathy, M., El-Gamel M., El-Azab, M. S., “Legendre–Galerkin method for the linear Fredholm integro-differential equations”, Applied Mathematics and Computation, 243: 789-800, (2014).
• [11] Jalilian, R., Tahernezhad T., “Exponential spline method for approximation solution of Fredholm integro-diferential equation”, International Journal of Computer Mathematics, 97(4): 79-80, (2020).
• [12] Maleknejad, K., Mahmoudi, Y., “Numerical solution of linear Fredholm integral equation by using hybrid Taylor and block-pulse functions”, Applied Mathematics and Computation, 149(2): 799–806, (2004).
• [13] Amiraliyev, G. M., Durmaz, M. E., Kudu, M., “Fitted second order numerical method for a singularly perturbed Fredholm integro-differential equation”, Bulletin of the Belgian Mathematical Society, 27(1): 71-88, (2020).
• [14] Çimen, E, Çakır M., “A uniform numerical method for solving singularly perturbed Fredholm integro-differential problem”, Computational & Applied Mathematics, 40(42): 1-14, (2021).
• [15] Durmaz, M. E., Amiraliyev G. M., “A robust numerical method for a singularly perturbed Fredholm integro-differential equation”, Mediterranean Journal of Mathematics, 18(24): 1-17, (2021).
• [16] Angulo, O., López-Marcos, J. C., López-Marcos M. A., “A second-order method for the numerical integration of a size-structured cell population model”, Abstract and Applied Analysis, Article ID 549168, 8, (2015).
• [17] Gemechis, F., Reddy, N. Y., “Numerical integration of a class of singularly perturbed delay differential equations with small shift”, International Journal of Differential Equations, Article ID 572723:12, (2012).
• [18] Reddy, N. Y., “A Numerical integration method for solving singular perturbation problems”, Applied Mathematics and Computation, 37(2): 83-95, (1990).
• [19] Soujanya, G., Phnaeendra, K., “Numerical intergration method for singular-singularly perturbed two- point boundary value problems”, Procedia Engineering, 127: 545-552, (2015).
• [20] Arslan, D., Çelik E., “An approximate solution of singularly perturbed problem on uniform mesh”, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 14(1): 74-80, (2024).
• [21] Arslan, D., “A robust numerical approach for singularly perturbed problem with integral boundary condition”, Gazi university Journal of Science, 36(4): 1647-1656, (2023).
• [22] Farrell, P. A., Hegarty A. H., Miller J. J. H., O’Riordan E., Shishkin, GI., “Robust Computational Techniques for Boundary Layers”, Chapman-Hall/CRC, New York, (2000).
• [23] Miller, J. J. H., O’Riordan, E., Shishkin, G. I., “Fitted numerical methods for singular perturbation problems”, World Scientific, Singapore, 48-69, (1996).
• [24] Roos, H. G., Stynes, M., Tobiska L., “Numerical methods for singularly perturbed differential equations”, Convection-Diffusion and Flow Problems, Springer Verlag, Berlin, (1996).
• [25] Andargie, A., Reddy, Y. N., “Numerical integration method for singular perturbation problems with mixed boundary conditions”, Journal of Applied Mathematics and Informatics, 26(5-6): 1273-1287, (2018).
• [26] Arslan, D., “A numerical solution for singularly perturbed multi-point boundary value problems with the numerical ıntegration method”, BEU Journal of Science, 9(1): 157-167,(2020).
• [27] Ranjan, R., Prasad, H. S., “An efficient method of numerical integration for a class of singularly perturbed two point boundary value problems”, Wseas Transactions on Mathematics, 17: 265-273, (2018).
• [28] Amiraliyev, G. M., Amirali, I., “Nümerik Analiz Teori ve Uygulamaları”, Seçkin Yayıncılık, Ankara, Türkiye, (2018).