Solving nonlinear Fredholm integro-differential equations via modifications of some numerical methods

Abstract

This paper presents the modications of the variational iteration method (MVIM), the Laplace Adomian decomposition method (MLADM), and the homotopy perturbation method (MHPM) for solving the nonlinear Fredholm integro-dierential equation of the second kind. In these techniques, a series is established, the summation of which gives the solution of the discussed equation. The conditions ensuring convergence of this series are presented. Some examples to illustrate the investigated methods are presented as well, and the results reveal that the proposed methods are very eective. Moreover, we present the comparison between our proposed methods with the exact solution and some traditional methods during numerical examples. The results show that (MHPM) and (MLADM) lead to an exact solution, whereas (MVIM) leads to limited solutions. Finally, the uniqueness of solutions and the convergence of the proposed methods are also proved.

New Fixed Point Theorem on Triple Controlled Metric Type Spaces with Applications to Volterra–Fredholm Integro-Dynamic Equations

“…Fixed point theory provides a basis in solving existence and uniqueness problems involving all types of differential and integral equations. Many researchers have examined the question of the existence and uniqueness of integrodifferential equations (see [7][8][9][10][11][12][13][14][15][16]). Recently, the authors in [17] discussed a particular kind of integro-dynamic equation.…”

Numerical Method for a Filtration Model Involving a Nonlinear Partial Integro-Differential Equation

“…Recently, much interest has been paid to collocation methods-namely, the Legendre wavelet method [17,19,20], Haar wavelet method [21][22][23], Euler wavelets [24], collocation method by sigmoidal functions [25], variational iteration method [26][27][28][29], homotopy perturbation method [30][31][32], moving least square method [33], Adomian's decomposition method [34] and its combination with the homotopy analysis method [35,36], parametric iteration method and spectral collocation method [37], and differential transform method [38].…”