Hypersingular integral equations of the first kind: A modified homotopy perturbation method and its application to vibration and active control

Abstract

This paper attempts to propose and investigate a modification of the homotopy perturbation method to study hypersingular integral equations of the first kind. Along with considering this matter, of course, the novel method has been compared with the standard homotopy perturbation method. This method can be conveniently fast to get the exact solutions. The validity and reliability of the proposed scheme are discussed. Different examples are included to prove so. According to the results, we further state that new simple homotopy perturbation method is so efficient and promises the exact solution. The modification of the homotopy perturbation method has been discovered to be the significant ideal tool in dealing with the complicated function-theoretic analytical structures within an analytical method.

Theoretical Analysis of Pre-Steady State Behaviour of Non-Linear Double Intermediate Enzymatic Reaction

“…Effective mathematical modelling serves as a valuable tool for elucidating enzymatic reaction processes. Since direct analytical solutions are unavailable for many non-linear enzymatic reaction equations, employing approximate analytical methods such as the variational iteration method (VIM) [4,5], Akbari Ganji method (AGM) [6,7], differential transform method (DTM) [8,9], homotopy perturbation method (HPM) [10,11], and homotopy analysis method (HAM) [12,13] becomes essential for obtaining analytical solutions. Most realistic models across chemistry, engineering, biology, and physics exhibit nonlinearity, making obtaining analytical solutions for such systems impractical.…”

A Novel Collocation Method for Numerical Solution of Hypersingular Integral Equation with Singular Right-Hand Function

“…In certain cases, there exist some analytical methods for solving HSIEs, but in general, there is no known analytical method. Thus, different numerical methods are introduced and developed for solving HSIEs [8,9,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”

On projection method for numerical solution of hypersingular integral equations of the first kind combined with quadrature methods

“…There exist lots of literature on related applications of semi-analytical and numerical methods for hypersingular integral equation (1) on its special case K(x, s) = 1 and L(x, s) = 0. The applied methods involve projection method [10], polynomial approximation method [5,11], Bernestein polynomials approximation method [6], Spline collocation method [12,13], Chebyshev polynomials approximation methods [14,15], modified Adomian decomposition method [16,17] and new homotopy perturbation method [18].…”