Numerical investigation of the weakly singular Volterra integro-differential equations using He's Homotopy perturbation method

Abstract

In this paper, we present a reliable algorithm for solving a system of Volterra integro-differential equations (VIDE) using He's Homotopy Perturbation Method (HHPM) [9,10] and spectral methods [12]. This method converts a system of Volterra integro-differential equations to the system of linear algebraic equations. Some illustrative examples have been presented to illustrate the implementation of the algorithm and efficiency of the method.

Numerical investigation of the fuzzy integro-differential equations by He$’$s Homotopy perturbation method

“…In this section, we briefly review the main points of the powerful method, known as the He's homotopy perturbation method [16,19,21]. To illustrate the basic ideas of this method, we consider the following differential equation:…”

“…If we take He's homotopy perturbation method [14][15][16][17][18][19][20][21] getting applied to the wide class problems in physics, biology and chemical reactions. If provides this result in a computable terms of rapid convergent series.…”

“…The specialty is integro-differential equations using fuzzy initial value problems have been taken for research by many mathematicians and physicists. He's homotopy perturbation method plays an important role in both the analysis and numerical result of integro-differential equations [16,19,21]. He's homotopy perturbation method can have a significant impact on what is looked upon a practical approach and on the types of problems that can be answered.…”

“…Recently Abbasbandy [1] found the solution of Kawahara equation, generalized Zakharov equation [S. Abbasbandy, E. Babolian and M. Ashtiani [4]], MHD Falkne-Skan flow [S. Abbasbandy and T. Hayat [5]] and nonlinear boundary value problems [S. Abbasbandy and E. Shivanian [6]]. In this article we developed numerical methods for FIDE to get the discrete solutions via He's Homotopy Perturbation method which was studied by Sekar et al [14][15][16][17][18][19][20][21]. The problems are solved in two methods; one is single-term Haar wavelet series method (STHWS) which is presented previously by Sekar et al [13] and another one is He's Homotopy Perturbation method which has been developed exclusively to deal with fuzzy integrodifferential equations.…”

A Review on Fuzzy Differential Equations

“…A large number of studies conducted on FDEs, for instance see [10]- [16], demonstrate that Hukuhara and Seikkala derivatives, despite being equivalent, are more palatable definitions. However, the research results have revealed that these derivatives suffer from a number of major limitations among which the most serious is that the diameter of the fuzzy function under study needs to be necessarily non-decreasing.…”