Modified homotopy perturbation method for solving hypersingular integral equations of the first kind

Abstract

Modified homotopy perturbation method (HPM) was used to solve the hypersingular integral equations (HSIEs) of the first kind on the interval [−1,1] with the assumption that the kernel of the hypersingular integral is constant on the diagonal of the domain. Existence of inverse of hypersingular integral operator leads to the convergence of HPM in certain cases. Modified HPM and its norm convergence are obtained in Hilbert space. Comparisons between modified HPM, standard HPM, Bernstein polynomials approach Mandal and Bhattacharya (Appl Math Comput 190:1707−1716, 2007), Chebyshev expansion method Mahiub et al. (Int J Pure Appl Math 69(3):265–274, 2011) and reproducing kernel Chen and Zhou (Appl Math Lett 24:636–641, 2011) are made by solving five examples. Theoretical and practical examples revealed that the modified HPM dominates the standard HPM and others. Finally, it is found that the modified HPM is exact, if the solution of the problem is a product of weights and polynomial functions. For rational solution the absolute error decreases very fast by increasing the number of collocation points.

Homotopy perturbation method and Chebyshev polynomials for solving a class of singular and hypersingular integral equations

“…In Eshkuvatov et al. [13] it is shown that both HPM and MHPM give identical solution. In Table 1 comparisons of three methods are given.…”

“…are homo-topic as the effectiveness of MHPM, the author has shown many examples. In 2016, Eshkuvatov et al [13] were able successfully implemented the modifed HPM together with Chebyshev polynomials for the following HSIEs of the first kind.…”

“…HPM has been used for a wide range of problems; for finding the exact and approximate solutions of nonlinear ordinary differential equations (ODEs) (Ramos [39]), linear integral equations ( [26], [43]), the integro-differential equations ( [15], [10], [25]) and nonlinear integral equations ( [16], [17], [18]). HPM has mainly two types of modifications ( [17], [18]) and in the recent decades, modified homotopy perturbation method (MHPM) has been used to solve many types of integral equations including SIEs and HSIEs ( [42], [13], [45], [31]).…”

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

“…What makes a certain hypersingular integral equation efficient is the extent to which that it could be a significant tool for solving a large class of mixed boundary value problems showing up in mathematical physics; especially, the crack problems of fracture mechanics, or water wave scattering problems concerning obstructions; diffracting electromagnetic waves and also aerodynamics problems might be decreased to hypersingular integral equations rather single or disjoint multiple intervals. Ideally, there is an imperative example of hypersingular integral equations of the first kind which has been exercised in dealing with most problems arising in vibration and active control [10][11][12][13][14][15][16][17] 1 p Z 1 À1 uðxÞ ðx À yÞ 2 dx ¼ fðyÞ; ðÀ1 < y < 1Þ (1) Here, f(y) and u(x) are presented as a known function and an unknown function on the finite interval ðÀ1; 1Þ considering the end points conditions uðAE1Þ ¼ 0. In equation 1 ; ðÀ1 < y < 1Þ…”

“…Mahmoudi 37 actually formed a new modified Adomian decomposition method to solve a class of hypersingular integral equations of the second kind. Also, Eshkuvatov et al 13 have presented a work regarding to the modified homotopy perturbation method for solving hypersingular integral equations of the first kind. In the present paper, we consider the method mentioned in literature 33,37 and then determine a new modified homotopy perturbation method to actually solve hypersingular integral equation of the first kind.…”