Nedjem Eddine RAMDANI, Sandra PINELAS

Solving nonlinear integro-differential equations using numerical method

The aim of this paper is to establish conditions for the existence and uniqueness of the solution of a nonlinear integro-differential equation. Moreover, it is to propose a quadrature method in order to find an approximate solution and establish the convergence of the method. We conclude by providing the algorithm and some numerical simulation to confirm our theoretical results.

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[1] Arikoglu A, Ozkol I. Solutions of integral and integro-differential equation systems by using differential transform method, Computers & Mathematics with Applications 2008; 56 (9): 2411-2417. doi.org/10.1016/j.camwa.2008.05.017
[2] Babolian E, Shahsavaran A. Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets, Journal of Computational and Applied Mathematics 2009; 225: 87-95. doi.org/10.1016/j.cam.2008.07.003
[3] Borzabadi AH, Fard OS. A numerical scheme for a class of nonlinear Fredholm integral equations of the second kind, Journal of Computational and Applied Mathematics 2009; 232: 449-454. doi:10.1016/j.cam.2009.06.038
[4] Borzabadi AH, Kamyad AV, Mehne HH. A different approach for solving the nonlinear Fredholm integral equations of the second kind, Applied Mathematics and Computation 2006; 173 (2): 724-735. doi:10.1016/j.amc.2005.04.008
[5] Chebbah H, Mennouni A, Ramdani NE. Numerical Solution of Generalized Logarithmic Integral Equations of the Second Kind by Projections, Malaysian Journal of Mathematical Sciences 2018; 12 (3): 349-367.
[6] David K, Moler C, Nash S. Numerical methods and software, Englewood Cliffs, Prentice Hall, 1989.
[7] Dobritoiu M. An Application of the Admissibility types in b-Metric Spaces, Transylvanian Journal of Mathematics and Mechanics 2020; 12 (1): 11-16.
[8] Dobritoiu M. A Nonlinear Fredholm Integral Equation, Transylvanian Journal of Mathematics and Mechanics 2009; 1 (1-2): 25-32.
[9] Golberg MA. Numerical Solution of Integral Equations, Mathematical Concepts and Methods in Science and Engineering, Springer, Boston, MA, 1990. doi:10.1007/978-1-4899-2593-0
[10] Haq EU, Khan AS, Ali M. An Efficient Approach for Solving the Linear and Nonlinear Integro-differential Equation, Computational Research Progress in Applied Science & Engineering 2020; 06 (02).
[11] Jerri AJ. Introduction to Integral Equations with Applications, 2 nd edition, Clarkson University, 1986.
[12] Kheybari S, Darvishi MT, Wazwaz AM. A semi analytical algorithm to solve systems of integro-differential equations under mixed boundary conditions, Journal of Computational and Applied Mathematics 2017; 317: 72-89. doi: 10.1016/j.cam.2016.11.029
[13] Kheybari S, Darvishi MT, Wazwaz AM. A semi analytical approach to solve integro-differential equations, Journal of Computational and Applied Mathematics 2017; 317: 17-30. doi: 10.1016/j.cam.2016.11.011
[14] Laurie DP. Calculation of Gauss-Kronrod Quadrature Rules, Mathematics of Computation 1997; 66 (219). doi: 10.1090/S0025-5718-97-00861-2
[15] Mennouni A, Ramdani NE. Collocation method to solve second order Cauchy integro-differential equations, in Differential and Difference Equations with Applications 2018; ICDDEA 2017 proceedings, Editors: Pinelas S, Caraballo T, Kloeden P, Graef JR. (Eds.), springer, ISBN 978-3-319-75647-9. doi: 10.1007/978-3-319-75647-9_25
[16] Mennouni A, Ramdani NE, Zennir K. A New Class of Fredholm Integral Equations of the Second Kind with Non Symmetric Kernel: Solving by Wavelets Method, Boletim da Socieda de Paranaense de Matemática 2021; 6 (39): 67-80. doi:10.5269/bspm.41734
[17] Moradi S, Anjedani MM, Analoei E. On existence and uniqueness of solutions of a nonlinear VolterraFredholm integral equation, International Journal of Nonlinear Analysis and Applications 2015; 6 (1): 62-68. doi:10.22075/ijnaa.2015.179
[18] Sauer T. Numerical Analysis, $2^{nd}$ Edition, Pearson Education, 2006.
[19] Stoer J, Bulirsch R. Introduction to Numerical Analysis, Springer Verlag, New York, 1993.
[20] Tuan T, Hong NT. A class of Fredholm equations and systems of equations related to the Kontorovich-Lebedev and the Fourier integral transforms, Turkish Journal of Mathematics 2020; 44: 643-655. doi:10.3906/mat-2001-24
[21] Wazwaz AM. Nonlinear Fredholm Integro-Differential Equations, Linear and Nonlinear Integral Equations 2011; 517-546. doi: 10.1007/978-3-642-21449-3_16
[22] Zhang P, Hao X. Existence and uniqueness of solutions for a class of nonlinear integro-differential equations on unbounded domains in Banach spaces, Advances in Difference Equations 2018; 247. doi: 10.1186/s13662-018-1681-0