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Algeria; Russia Guelma; Ouargla; Tambov
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.
  1. Linz P. Analytical and numerical methods for Volterra equations, Society for Industrial and Applied Mathematics, 1985.
  2. Dareiotis K. On finite difference schemes for partial integro-differential equations of Lévy type, Journal of Computational and Applied Mathematics, 2020, vol. 368, 112587.
  3. 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.
  4. 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.
  5. 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.
  6. Sato T. Sur l'équation intégrale non linéaire de Volterra, Compositio Mathematica, 1953, vol. 11, pp. 271-290.
  7. Atkinson K., Han W. Theoretical numerical analysis: A functional analysis framework, New York: Springer, 2009.
  8. 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.
  9. Pachpatte B.G. On higher order Volterra-Fredholm integro-differential equation, Fasciculi Mathematici, 2007, no. 37, pp. 35-48.
  10. 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.
  11. 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.
  12. 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.
  13. Pachpatte B.G. On Fredholm type integrodifferential equation, Tamkang Journal of Mathematics, 2008, vol. 39, no. 1, pp. 85-94.
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