A Comparison Some of Numerical and Analytical methods for Solving Nonlinear Volterra Integral Equations with Applications

Fatimah Ali  Othman; Sizar mohammed

doi:10.5269/bspm.78020

Fatimah Ali Othman University of Duhok
Sizar mohammed University of Duhok

Abstract

This paper presents an in-depth investigation into the numerical solutions of nonlinear Volterra integral equations of the second kind (NVIE-II). Two distinct methods are applied for approximating the solutions: combining the Laplace transform and the Adomian Decomposition Method (ADM) using He's polynomials, and the Variational Iteration Method (VIM). The first method involves transforming the integral equation using the Laplace transform, followed by solving it through the Adomian Decomposition Method to produce a series of approximations, which are then compared with the exact solution. The second method, based on VIM, converts the Volterra integral equation into an integro-differential equation and demonstrates the rapid convergence of successive approximations to the exact solution. A thorough comparison is conducted between the exact solution, the approximate results obtained from both methods, and the associated error estimations. The findings offer a detailed evaluation of the effectiveness and accuracy of these methods, providing insight into their practical application for solving nonlinear (NVIE-II).

Downloads

Download data is not yet available.

References

[1] Wazwaz, A.-M. (2011). Linear and nonlinear integral equations: Methods and applications. Springer.
[2] Wazwaz, A.-M. (2015). A first course in integral equations (2nd ed.). World Scientific.
[3] Ahmad, N., & Singh, B. (2022). Numerical solution of Volterra nonlinear integral equation by using the Laplace–Adomian decomposition method. International Journal of Applied Mathematics, 35(1), 39–48.
[4] Ullah, M. A., Uddin, J., & Shah, M. A. (2024). Enhanced Laplace–Adomian decomposition method for nonlinear Volterra integral equation. The Sciencetech, 5(1). https://thesciencetech.com/index.php/tech/article/view/587/434
[5] Al-Saar, F. M., & Ghadle, K. P. (2019). The numerical solutions of linear and nonlinear Volterra integral equations of the second kind using the variational iteration method. Acta Universitatis Matthiae Belii, Series Mathematics, 2019, 32–42. http://actamath.savbb.sk
[6] Mirzaei, S. M. (2011). Homotopy perturbation method and variational iteration method for Volterra integral equations. Journal of Applied Mathematics and Bioinformatics, 1(1), 105–113.
[7] Adomian, G. (1988). A review of the decomposition method in applied mathematics. Journal of Mathematical Analysis and Applications, 135(3), 501–544. https://doi.org/10.1016/0022-247X(88)90170-9
[8] Wazwaz, A.-M. (2010). The combined Laplace transform–Adomian decomposition method for handling nonlinear Volterra integro-differential equations. Applied Mathematics and Computation, 216, 1304–1309. https://doi.org/10.1016/j.amc.2010.02.023
[9] He, J. H. (1999). Variational iteration method – A kind of non-linear analytical technique: Some examples. International Journal of Non-Linear Mechanics, 34, 699–708.
[10] Rani, D., & Mishra, V. (2018). Modification of the Laplace–Adomian decomposition method for solving nonlinear Volterra integral and integro-differential equations based on Newton–Raphson formula. European Journal of Pure and Applied Mathematics, 11(1), 202–214.
[11] Atkinson, K. E. (1997). The numerical solution of integral equations of the second kind. Cambridge University Press.
[12] Saha Ray, S., & Sahu, P. K. (2019). Novel methods for solving linear and nonlinear integral equations. CRC Press.