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## Archive of Issues

Algeria; Russia Guelma; Ouargla; Tambov
Year
2020
Volume
30
Issue
2
Pages
176-188
 Section Mathematics Title Difference derivative for an integro-differential nonlinear Volterra equation Author(-s) Guebbai H.a, Lemita S.b, Segni S.a, Merchela W.c Affiliations University 8 Mai 1945a, Ecole Normale Supérieure de Ouarglab, Tambov State Universityc Abstract In this article, we propose a new numerical approximation method to deal with the unique solution of the nonlinear integro-differential Volterra equation. We are interested in a very particular form of this equation, in which the derivative of the sought solution appears under the integral sign in a nonlinear manner. Our vision is based on two different approaches: We use the Nyström method to transform the integral into a finite sum using a numerical integration formula, then we use the numerical backward difference derivative method to approach the derivative of our solution. This collocation between two different methods, the first outcome of the numerical processing of integral equations and the second outcome of the numerical processing of differential equations, gives a new nonlinear system for approaching the solution of our equation. We show that the system has a unique solution and that this numerical solution converges perfectly to our solution. A section is dedicated to numerical tests, in which we show the effectiveness of our new vision compared to two methods based only on numerical integration. Keywords Volterra integro-differential equation, nonlinear equation, fixed point, numerical derivative, Nyström method UDC 517.988 MSC 45D05, 45J99, 65R20 DOI 10.35634/vm200203 Received 17 January 2020 Language English Citation Guebbai H., Lemita S., Segni S., Merchela W. Difference derivative for an integro-differential nonlinear Volterra equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 2, pp. 176-188. References Linz P. Analytical and numerical methods for Volterra equations, Society for Industrial and Applied Mathematics, 1985. https://doi.org/10.1137/1.9781611970852 Dareiotis K. On finite difference schemes for partial integro-differential equations of Lévy type, Journal of Computational and Applied Mathematics, 2020, vol. 368, 112587. https://doi.org/10.1016/j.cam.2019.112587 Behera S., Saha Ray S. An operational matrix based scheme for numerical solutions of nonlinear weakly singular partial integro-differential equations, Applied Mathematics and Computation, 2020, vol. 367, 124771. https://doi.org/10.1016/j.amc.2019.124771 Rajagopal N., Balaji S., Seethalakshmi R., Balaji V.S. A new numerical method for fractional order Volterra integro-differential equations, Ain Shams Engineering Journal, 2020, vol. 11, issue 1, pp. 171-177. https://doi.org/10.1016/j.asej.2019.08.004 Xu D. Analytical and numerical solutions of a class of nonlinear integro-differential equations with $L$$1$ kernels, Nonlinear Analysis: Real World Applications, 2020, vol. 51, 103002. https://doi.org/10.1016/j.nonrwa.2019.103002 Sato T. Sur l'équation intégrale non linéaire de Volterra, Compositio Mathematica, 1953, vol. 11, pp. 271-290. http://www.numdam.org/item/?id=CM_1953__11__271_0 Atkinson K., Han W. Theoretical numerical analysis: A functional analysis framework, New York: Springer, 2009. https://doi.org/10.1007/978-1-4419-0458-4 Brunner H. The numerical treatment of Volterra integro-differential equations with unbounded delay, Journal of Computational and Applied Mathematics, 1989, vol. 28, pp. 5-23. https://doi.org/10.1016/0377-0427(89)90318-X Pachpatte B.G. On higher order Volterra-Fredholm integro-differential equation, Fasciculi Mathematici, 2007, no. 37, pp. 35-48. http://www.math.put.poznan.pl/artykuly/FM37_Pachpatte-wyd02.pdf Guebbai H., Aissaoui M.Z., Debbar I., Khalla B. Analytical and numerical study for an integro-differential nonlinear Volterra equation, Applied Mathematics and Computation, 2014, vol. 229, pp. 367-373. https://doi.org/10.1016/j.amc.2013.12.046 Segni S., Ghiat M., Guebbai H. New approximation method for Volterra nonlinear integro-differential equation, Asian-European Journal of Mathematics, 2019, vol. 12, no. 1, 1950016. https://doi.org/10.1142/S1793557119500165 Ghiat M., Guebbai H. Analytical and numerical study for an integro-differential nonlinear Volterra equation with weakly singular kernel, Computational and Applied Mathematics, 2018, vol. 37, issue 4, pp. 4661-4674. https://doi.org/10.1007/s40314-018-0597-3 Pachpatte B.G. On Fredholm type integrodifferential equation, Tamkang Journal of Mathematics, 2008, vol. 39, no. 1, pp. 85-94. https://doi.org/10.5556/j.tkjm.39.2008.48 Full text