Spline quasi-interpolation numerical methods for integro-differential equations with weakly singular kernels
Abstract
In this work, we introduce a numerical approach that utilizes spline quasi-interpolation operators over a bounded interval. This method is designed to provide a numerical solution for a class of Fredholm integro-differential equations with weakly singular kernels. We outline the computational components involved in determining the approximate solution and provide theoretical findings regarding the convergence rate. This convergence rate is analyzed in relation to both the degree of the quasi-interpolant and the grading exponent of the graded grid partition. Finally, we present numerical experiments that validate the theoretical findings.
Keyword : quasi-interpolation operators, numerical methods, redholm integro-differential equations, weakly singular kernel, graded grids
How to Cite
Saou, A., Sbibih, D., Tahrichi, M., & Barrera, D. (2024). Spline quasi-interpolation numerical methods for integro-differential equations with weakly singular kernels. Mathematical Modelling and Analysis, 29(3), 442–459. https://doi.org/10.3846/mma.2024.18832
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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B. Tair, H. Guebbai, S. Segni and M. Ghiat. An approximation solution of linear Fredholm integro-differential equation using collocation and Kantorovich methods. Journal of Applied Mathematics and Computing, 68(5):3505–3525, 2021. https://doi.org/10.1007/s12190-021-01654-2
B. Tair, H. Guebbai, S. Segni and M. Ghiat. Solving linear Fredholm integro-differential equation by Nystro¨m method. Journal of Applied Mathematics and Computational Mechanics, 20(3):53–64, 2021. https://doi.org/10.17512/jamcm.2021.3.05
G. Vainikko. Multidimensional Weakly Singular Integral Equations. Lecture Notes in Mathematics. Springer Berlin Heidelberg, 1993. https://doi.org/10.1007/BFb0088979
S. Yalc¸inba¸s and M. Sezer. A Taylor collocation method for the approximate solution of general linear Fredholm-Volterra integro-difference equations with mixed argument. Applied Mathematics and Computation, 175(1):675–690, 2006. https://doi.org/10.1016/j.amc.2005.07.038
X. Zhang and H. Du. A generalized collocation method in reproducing kernel space for solving a weakly singular Fredholm integrodifferential equations. Applied Numerical Mathematics, 156:158–173, 2020. https://doi.org/10.1016/j.apnum.2020.04.019
A.T. Assanova and Sh.N. Nurmukanbet. A solution to a boundary-value problem for integro-differential equations with weakly singular kernels. Russian Mathematics, 65(11):1–13, 2021. https://doi.org/10.3103/s1066369x21110013
A.T. Assanova and S.N. Nurmukanbet. A solvability of a problem for a Fredholm integro-differential equation with weakly singular kernel. Lobachevskii Journal of Mathematics, 43(1):182–191, 2022. https://doi.org/10.1134/s1995080222040047
C. de Boor. A Practical Guide to Splines. Applied Mathematical Sciences. Springer New York, 2001.
G. Deng, Y. Yang and E. Tohidi. High accurate pseudo-spectral Galerkin scheme for pantograph type Volterra integro-differential equations with singular kernels. Applied Mathematics and Computation, 396:125866, 2021. https://doi.org/10.1016/j.amc.2020.125866
S.M. El-Sayed, D. Kaya and S. Zarea. The decomposition method applied to solve high-order linear Volterra-Fredholm integro-differential equations. International Journal of Nonlinear Sciences and Numerical Simulation, 5(2):105–112, 2004. https://doi.org/10.1515/ijnsns.2004.5.2.105
M. Ganesh and I.H. Sloan. Optimal order spline methods for nonlinear differential and integro-differential equations. Applied Numerical Mathematics, 29(4):445–478, 1999. https://doi.org/10.1016/s0168-9274(98)00067-1
I. Hashim. Adomian decomposition method for solving BVPs for fourth-order integro-differential equations. Journal of Computational and Applied Mathematics, 193(2):658–664, 2006. https://doi.org/10.1016/j.cam.2005.05.034
H. Hawsar, M.S. Hari Mohan, H. Mudhafar, O.M. Pshtiwan, Y.A. Musawa and B. Dumitru. Novel algorithms to approximate the solution of nonlinear integrodifferential equations of volterra-fredholm integro type. AIMS Mathematics, 8(6):14572–14591, 2023. https://doi.org/10.3934/math.2023745
M.J. Ibáñez, A. Lamnii, H. Mraoui and D. Sbibih. Construction of spherical spline quasi-interpolants based on blossoming. Journal of Computational and Applied Mathematics, 234(1):131–145, 2010. https://doi.org/10.1016/j.cam.2009.12.010
H. Jaradat, O. Alsayyed and S. Al-Shara. Numerical solution of linear integrodifferential equations. J. Math. Statist., 4:250–254, 2008.
T. Lyche and K. Morken. Spline methods draft. Department of Informatics, Center of Mathematics for Applications, University of Oslo, Oslo, 2008.
S. Nas, S. Yalcinbas and M. Sezer. A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations. International Journal of Mathematical Education in Science and Technology, 31(2):213–225, 2000. https://doi.org/10.1080/0020739x.2022.12131593
A. Pedas and E. Tamme. Spline collocation method for integro-differential equations with weakly singular kernels. Journal of Computational and Applied Mathematics, 197(1):253–269, 2006. https://doi.org/10.1016/j.cam.2005.07.035
A. Saadatmandi and M. Dehghan. Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients. Computers & Mathematics with Applications, 59(8):2996–3004, 2010. https://doi.org/10.1016/j.camwa.2010.02.018
P. Sablonnière. Quadratic spline quasi-interpolants on bounded domains of Rd, d = 1,2,3. Spline and radial functions, 61:229–246, 2003
.
P. Sablonnière. Univariate spline quasi-interpolants and applications to numerical analysis. arXiv:math/0504022, 2005.
X. Shao, L. Yang and A. Guo. A feedforward neural network based on Legendre polynomial for solving linear Fredholm integro-differential equations. International Journal of Computer Mathematics, 100(7):1480–1499, 2023. https://doi.org/10.1080/00207160.2023.2191746
B. Tair, H. Guebbai, S. Segni and M. Ghiat. An approximation solution of linear Fredholm integro-differential equation using collocation and Kantorovich methods. Journal of Applied Mathematics and Computing, 68(5):3505–3525, 2021. https://doi.org/10.1007/s12190-021-01654-2
B. Tair, H. Guebbai, S. Segni and M. Ghiat. Solving linear Fredholm integro-differential equation by Nystro¨m method. Journal of Applied Mathematics and Computational Mechanics, 20(3):53–64, 2021. https://doi.org/10.17512/jamcm.2021.3.05
G. Vainikko. Multidimensional Weakly Singular Integral Equations. Lecture Notes in Mathematics. Springer Berlin Heidelberg, 1993. https://doi.org/10.1007/BFb0088979
S. Yalc¸inba¸s and M. Sezer. A Taylor collocation method for the approximate solution of general linear Fredholm-Volterra integro-difference equations with mixed argument. Applied Mathematics and Computation, 175(1):675–690, 2006. https://doi.org/10.1016/j.amc.2005.07.038
X. Zhang and H. Du. A generalized collocation method in reproducing kernel space for solving a weakly singular Fredholm integrodifferential equations. Applied Numerical Mathematics, 156:158–173, 2020. https://doi.org/10.1016/j.apnum.2020.04.019