Numerical Solutions for Nonlinear Volterra Integral
Equations of the Second Kind with a Domain
Decomposition and Modified Decomposition
Methods

Siddiga Abdalla Osman

doi:https://doi.org/10.14445/22315373/IJMTT-V68I3P504

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 68 | Issue 3 | Year 2022 | Article Id. IJMTT-V68I3P504 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I3P504

Numerical Solutions for Nonlinear Volterra Integral Equations of the Second Kind with a Domain Decomposition and Modified Decomposition Methods

Siddiga Abdalla Osman

Abstract

In this work, a domain decomposition method (ADM) and Modified a domain decomposition method (MADM) is used to solve the nonlinear Volterra integral equations, also some comparison between the two methods is done for the same class of nonlinear equations. We outline some differences between the two methods and show that the Modified decomposition method is more effective than the standard decomposition method . Numerical examples and their solutions are obtained using MATLAB.

Keywords

Volterra integral equations, A domain decomposition method, Modified decomposition method.

References

[1] P. Linz, Analytical and Numerical Methods for Volterra equations. Society for Industrial and Applied Mathematics, (1985).
[2] R.P. Kanwal, Linear Integral Equations Theory and Technique. Boston, (1996).
[3] K.E.Atkinson, The Numerical Solution of Integral Equation of the Second Kind. Cambridge, (1997).
[4] A.M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications. Springer ,(2011).
[5] L. Gabet, The Decomposition Method and Distributions. Computers & Mathematics with Applications, 27(3) (1994) 41-49.
[6] S. N. Venkatarangan, & K. Rajalakshmi, A Modification of a Domain's Solution for Nonlinear Oscillatory Systems. Computers & Mathematics with Applications, 29(6) (1995) 67-73.
[7] A.M. Wazwaz, A Reliable Modification of Decomposition Method, Appl. Math. Comput. 102 (1999) 77–86.
[8] A.M. Wazwaz, A Reliable Treatment for Mixed Volterra–Fredholm Integral Equations, Appl. Math. Comput. 127 (2002) 405–414.
[9] A.M. Wazwaz, The Numerical Solution of Special Fourth-Order Boundary Value Problems by the Modified Decomposition Method, Int.J. Comput. Math. 79(3) (2002) 345–356.
[10] A.M. Wazwaz, A First Course in Integral Equations, World Scientific, (1997).
[11] A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema, Netherlands, (2002).
[12] A.M. Wazwaz, A Reliable Technique for Solving the Weakly Singular Second-Kind Volterra-Type Integral Equations, Appl. Math. Comput. 80 (1996) 287–299.
[13] A.M. Wazwaz, Analytical Approximations and Pade’ approximants for Volterra’s Population Model, Appl. Math. Comput. 100 (1999) 13–25.
[14] A.M. Wazwaz, the Modified Decomposition Method and Pade’ Approximants for Solving Thomas–Fermi equation, Appl. Math.Comput. 105 (1999) 11–19.
[15] A.M. Wazwaz, A new algorithm for calculating A Domain Polynomials for Nonlinear Operators, Appl. Math. Comput. 111 (2000) 53–69.
[16] G. A domain, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, 1994.
[17] G. A domain, A review of the Decomposition Method and Some Recent Results for Nonlinear Equation, Math. Comput. Modell. 13(7) (1992) 17–43.
[18] G. A domian, The Decomposition Method for Nonlinear Dynamical Systems. Journal of mathematical analysis and applications, 120(1) (1986) 370-383.
[19] K. Maleknejad, K. Nouri, &R. Mollapourasl, Existence of solutions for some nonlinear integral equations. Communications in Nonlinear Science and Numerical Simulation, 14(6) (2009) 2559-2564.
[20] L. J. Xie, A New Modification of a Domain Decomposition Method for Volterra Integral Equations of the Second Kind. Journal of Applied Mathematics, (2013).
[21] H. H., Ali, &F. Abdelwahid, Modified A Domain Techniques Applied to Non-Linear Volterra Integral Equations. (2013).
[22] A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999).
[23] R.K. Miller, Nonlinear Volterra Integral Equations, W.A. Benjamin, Menlo Park, CA, (1967).
[24] H. Brunner, On the Numerical Solution of Nonlinear Volterral–Fredholm Integral Equation bBy Collocation Methods, SIAM J. Numer. Anal. 27 (4) (1990) 987–1000.
[25] L. Hacia, On Approximate Solution for Integral Equations of Mixed Type, ZAMM. Z. Angew. Math. Mech. 76 (1996) 415–416.

Citation :

Siddiga Abdalla Osman, "Numerical Solutions for Nonlinear Volterra Integral Equations of the Second Kind with a Domain Decomposition and Modified Decomposition Methods," International Journal of Mathematics Trends and Technology (IJMTT), vol. 68, no. 3, pp. 15-20, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I3P504