Abstract
İntegro-diferansiyel denklemler mekanik, fizik, kimya, biyofizik, astronomi, ekonomi teorisi ve nüfus dinamiği gibi çalışma alanlarında karşımıza çıkmaktadır. Nadir durumlarda diferansiyel ve/veya integral denklemlerin çözüm yöntemleri integro-diferansiyel denklemlere genelleştirilebilir; ancak genel olarak sayısal yöntemlerin uygulanması gerekir. Son yıllarda integro-diferansiyel denklemlere uygulanabilen çok sayıda yöntem geliştirilmiştir. Bu çalışma, bu yeni yöntemleri, ağırlıklı kalıntı yöntemlerinden biri olan klasik nokta kollokasyon yöntemi ile karşılaştırmayı amaçlamaktadır. Yöntem, literatürden seçilen doğrusal ve doğrusal olmayan integro-diferansiyel denklemlerden oluşan test problemlerine uygulanmış ve iyi sonuçlar verdiği görülmüştür.
Keywords
Collocation yöntemi doğrusal olmayan integro-diferansiyel denklemler Volterra integro-diferansiyel denklemi Fredholm integro-diferansiyel denklemleri sayısal çözüm.
References
- Abdi, A., Berrut, J.-P., & Hosseini, S.A. (2022). Explicit methods based on barycentric rational interpolants for solving non-stiff Volterra integral equations. Applied Numerical Mathematics, 174, 127-141. https://doi.org/10.1016/j.apnum.2022.01.004
- Ahmadinia, M., Afshariarjmand, H., & Salehi, M. (2023). Numerical solution of multi-dimensional Itô Volterra integral equations by the second kind Chebyshev wavelets and parallel computing process. Applied Mathematics and Computation, 450. https://doi.org/10.1016/j.amc.2023.127988
- Al-Saar, F., & Ghadle, K. (2021). Solving nonlinear Fredholm integro-differential equations via modifications of some numerical methods. Advances in the Theory of Nonlinear Analysis and its Applications, 5(2), 260-276. https://doi.org/10.31197/atnaa.872432
- Al-Towaiq, M., & Kasasbeh, A., (2017). Modified Algorithm for Solving Linear Integro-Differential Equations of the Second Kind. American Journal of Computational Mathematics, 7(2), 157-165. https://doi.org/10.4236/ajcm.2017.72014
- Avudainayagam, A., & Vani, C. (2000). Wavelet–Galerkin method for integro–differential equations. Applied Numerical Mathematics, 32(3), 247-254. https://doi.org/10.1016/S0168-9274(99)00026-4
- Boonklurb, R., Duangpan, A., & Gugaew, P. (2020). Numerical solution of direct and inverse problems for time-dependent volterra ıntegro-differential equation using finite ıntegration method with shifted chebyshev polynomial. Symmetry, 12(4), 497. https://doi.org/10.3390/sym12040497
- Cimen, E., & Enterili, K. (2020). Fredholm İntegro Diferansiyel Denklemin Sayısal Çözümü için Alternatif Bir Yöntem. Erzincan University Journal of Science and Technology, 13(1), 46-53. https://doi.org/10.18185/erzifbed.633899
- Çakır, M., & Güneş, B. (2022). A new difference method for the singularly perturbed Volterra-Fredholm integro-differential equations on a Shishkin mesh. Hacettepe Journal of Mathematics and Statistics, 51(3), 787-799. https://doi.org/10.15672/hujms.950075
- Dzhumabaev, D.S. (2016). On one approach to solve the linear boundary value problems for Fredholm integro-differential equations. Journal of Computational and Applied Mathematics, 294, 342-357. https://doi.org/10.1016/j.cam.2015.08.023
- Jaradat, H., Alsayyed, O., & Al-Shara, S. (2008). Numerical Solution of Linear Integro-Differential Equations. Journal of Mathematics and Statistics, 4(4), 250-254. https://doi.org/10.3844/jmssp.2008.250.254
- Lakshmikantham, V., & Rama Mohana Rao, M. (1995). Theory of Integro-Differential Equations (1st ed.). Lausanne, Switzerland: Gordon & Breach Science Publishers.
- Olayiwola, M. O., & Kareem, K. (2022). A New Decomposition Method for Integro-Differential Equations. Cumhuriyet Science Journal, 43(2), 283-288. https://doi.org/10.17776/csj.986019
- Rabiei, F., Abd Hamid, F., Md Lazim, N., Ismail, F., & Abdul Majid, Z. (2019). Numerical Solution of Volterra Integro-Differential Equations Using Improved Runge-Kutta Methods. Applied Mechanics and Materials, 892, 193–199. http://dx.doi.org/10.4028/www.scientific.net/AMM.892.193
- Rahmani, L., Rahimi, B., & Mordad, M. (2011). Numerical Solution of Volterra-Fredholm Integro-Differential Equation by Block Pulse Functions and Operational Matrices. General Mathematics Notes, 4(2), 37-48. https://www.kurims.kyoto-u.ac.jp/EMIS/journals/GMN/yahoo_site_admin/assets/docs/4_GMN-482-V4N2.160165143.pdf
- Sakran, M.R.A. (2019). Numerical solutions of integral and integro-differential equations using Chebyshev polynomials of the third kind. Applied Mathematics and Computation, 351, 66-82. https://doi.org/10.1016/j.amc.2019.01.030
- Sepehrian, B., & Razzaghi, M. (2004). Single-term Walsh series method for the Volterra integro-differential equations. Engineering Analysis with Boundary Elements, 28(11), 1315-1319. https://doi.org/10.1016/j.enganabound.2004.05.001
- Islam, S. U., Aziz I., & Fayyaz, M. (2013). A new approach for numerical solution of integro-differential equations via Haar wavelets. International Journal of Computer Mathematics, 90(9), 1971-1989. https://doi.org/10.1080/00207160.2013.770481
- Wen, J., & Huang, C. (2024). Multistep Runge–Kutta methods for Volterra integro-differential equations. Journal of Computational and Applied Mathematics, 436. https://doi.org/10.1016/j.cam.2023.115384
- Xu, L. (2007). Variational Iteration Method for Solving Integral Equations. Computers & Mathematics with Applications, 54(7-8), 1071-1078. https://doi.org/10.1016/j.camwa.2006.12.053
- Zarebnia, M. (2010). Sinc numerical solution for the Volterra integro-differential equation. Communications in Nonlinear Science and Numerical Simulation, 15(3), 700-706. https://doi.org/10.1016/j.cnsns.2009.04.021
Abstract
Integro-differential equations are encountered in such fields of study as mechanics, physics, chemistry, biophysics, astronomy, economic theory, and population dynamics. In rare cases the solution methods for differential and/or integral equations can be generalized to integro-differential equations; but in general, numerical methods have to be applied. Recent years have seen the development of a large number of methods applicable to integro-differential equations. The present study aims to compare these newer methods with the classical method of point collocation, which is one of the weighted residual methods. The method was applied to test problems chosen from the literature, both linear and nonlinear integro-differential equations, and was seen to give good results.
Keywords
Collocation method nonlinear integro-differential equations Volterra integro-differential equation Fredholm integro-differential equations numerical solution.
Thanks
The authors would like to thank Prof.Dr. Erol Uzal for his suggestions and contributions to the study.
References
- Abdi, A., Berrut, J.-P., & Hosseini, S.A. (2022). Explicit methods based on barycentric rational interpolants for solving non-stiff Volterra integral equations. Applied Numerical Mathematics, 174, 127-141. https://doi.org/10.1016/j.apnum.2022.01.004
- Ahmadinia, M., Afshariarjmand, H., & Salehi, M. (2023). Numerical solution of multi-dimensional Itô Volterra integral equations by the second kind Chebyshev wavelets and parallel computing process. Applied Mathematics and Computation, 450. https://doi.org/10.1016/j.amc.2023.127988
- Al-Saar, F., & Ghadle, K. (2021). Solving nonlinear Fredholm integro-differential equations via modifications of some numerical methods. Advances in the Theory of Nonlinear Analysis and its Applications, 5(2), 260-276. https://doi.org/10.31197/atnaa.872432
- Al-Towaiq, M., & Kasasbeh, A., (2017). Modified Algorithm for Solving Linear Integro-Differential Equations of the Second Kind. American Journal of Computational Mathematics, 7(2), 157-165. https://doi.org/10.4236/ajcm.2017.72014
- Avudainayagam, A., & Vani, C. (2000). Wavelet–Galerkin method for integro–differential equations. Applied Numerical Mathematics, 32(3), 247-254. https://doi.org/10.1016/S0168-9274(99)00026-4
- Boonklurb, R., Duangpan, A., & Gugaew, P. (2020). Numerical solution of direct and inverse problems for time-dependent volterra ıntegro-differential equation using finite ıntegration method with shifted chebyshev polynomial. Symmetry, 12(4), 497. https://doi.org/10.3390/sym12040497
- Cimen, E., & Enterili, K. (2020). Fredholm İntegro Diferansiyel Denklemin Sayısal Çözümü için Alternatif Bir Yöntem. Erzincan University Journal of Science and Technology, 13(1), 46-53. https://doi.org/10.18185/erzifbed.633899
- Çakır, M., & Güneş, B. (2022). A new difference method for the singularly perturbed Volterra-Fredholm integro-differential equations on a Shishkin mesh. Hacettepe Journal of Mathematics and Statistics, 51(3), 787-799. https://doi.org/10.15672/hujms.950075
- Dzhumabaev, D.S. (2016). On one approach to solve the linear boundary value problems for Fredholm integro-differential equations. Journal of Computational and Applied Mathematics, 294, 342-357. https://doi.org/10.1016/j.cam.2015.08.023
- Jaradat, H., Alsayyed, O., & Al-Shara, S. (2008). Numerical Solution of Linear Integro-Differential Equations. Journal of Mathematics and Statistics, 4(4), 250-254. https://doi.org/10.3844/jmssp.2008.250.254
- Lakshmikantham, V., & Rama Mohana Rao, M. (1995). Theory of Integro-Differential Equations (1st ed.). Lausanne, Switzerland: Gordon & Breach Science Publishers.
- Olayiwola, M. O., & Kareem, K. (2022). A New Decomposition Method for Integro-Differential Equations. Cumhuriyet Science Journal, 43(2), 283-288. https://doi.org/10.17776/csj.986019
- Rabiei, F., Abd Hamid, F., Md Lazim, N., Ismail, F., & Abdul Majid, Z. (2019). Numerical Solution of Volterra Integro-Differential Equations Using Improved Runge-Kutta Methods. Applied Mechanics and Materials, 892, 193–199. http://dx.doi.org/10.4028/www.scientific.net/AMM.892.193
- Rahmani, L., Rahimi, B., & Mordad, M. (2011). Numerical Solution of Volterra-Fredholm Integro-Differential Equation by Block Pulse Functions and Operational Matrices. General Mathematics Notes, 4(2), 37-48. https://www.kurims.kyoto-u.ac.jp/EMIS/journals/GMN/yahoo_site_admin/assets/docs/4_GMN-482-V4N2.160165143.pdf
- Sakran, M.R.A. (2019). Numerical solutions of integral and integro-differential equations using Chebyshev polynomials of the third kind. Applied Mathematics and Computation, 351, 66-82. https://doi.org/10.1016/j.amc.2019.01.030
- Sepehrian, B., & Razzaghi, M. (2004). Single-term Walsh series method for the Volterra integro-differential equations. Engineering Analysis with Boundary Elements, 28(11), 1315-1319. https://doi.org/10.1016/j.enganabound.2004.05.001
- Islam, S. U., Aziz I., & Fayyaz, M. (2013). A new approach for numerical solution of integro-differential equations via Haar wavelets. International Journal of Computer Mathematics, 90(9), 1971-1989. https://doi.org/10.1080/00207160.2013.770481
- Wen, J., & Huang, C. (2024). Multistep Runge–Kutta methods for Volterra integro-differential equations. Journal of Computational and Applied Mathematics, 436. https://doi.org/10.1016/j.cam.2023.115384
- Xu, L. (2007). Variational Iteration Method for Solving Integral Equations. Computers & Mathematics with Applications, 54(7-8), 1071-1078. https://doi.org/10.1016/j.camwa.2006.12.053
- Zarebnia, M. (2010). Sinc numerical solution for the Volterra integro-differential equation. Communications in Nonlinear Science and Numerical Simulation, 15(3), 700-706. https://doi.org/10.1016/j.cnsns.2009.04.021
Details
| Primary Language | English |
|---|---|
| Subjects | Numerical Methods in Mechanical Engineering |
| Journal Section | Mechanical Engineering |
| Authors | |
| Publication Date | December 3, 2023 |
| Submission Date | August 21, 2023 |
| Published in Issue | Year 2023 |