Legendre wavelet-based iterative schemes for fourth-order elliptic equations with nonlocal boundary conditions

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ORIGINAL ARTICLE

Legendre wavelet-based iterative schemes for fourth-order elliptic equations with nonlocal boundary conditions K. Harish Kumar1 • V. Antony Vijesh1 Received: 15 October 2017 / Accepted: 29 April 2019  Springer-Verlag London Ltd., part of Springer Nature 2019

Abstract In the literature of wavelets, there is limited report of work done to solve nonlinear differential equations with nonlocal boundary conditions. This work is a new attempt to solve a fourth-order elliptic equation with the use of nonlocal boundary conditions by coupling quasilinearization with Legendre wavelet. Since the previously available approach failed to produce reliable accuracy for certain class of problems, this iterative scheme has been suitably modified to deal with a broader class to obtain an accuracy that is reliable. To show the efficiency of the proposed numerical method, a comparison was performed with some existing methods available in the literature. Keywords Collocation method  Fourth-order elliptic equation  Legendre wavelets  Quasilinearization

1 Introduction Fourth-order elliptic equations with various boundary conditions have found relevance in the study of traveling waves in suspension bridges ([8, 13, 15]) and static deflection of a bending beam ([6, 9, 24]). Only a few published reports were found in the literature that dealt with the fourth-order elliptic equation D2 uðx; yÞ  b0 Duðx; yÞ þ c0 uðx; yÞ ¼ f ðx; y; uðx; yÞÞ; ð1:1Þ where a  x  b and c  y  d with nonlocal boundary Rd Rb conditions uðx; i1 Þ ¼ c a bðx; yÞuðx; yÞdxdy þ gð1Þ ðx; i1 Þ; Rd Rb uði2 ; yÞ ¼ c a bðx; yÞuðx; yÞdxdy þ gð1Þ ði2 ; yÞ, Duðx; i1 Þ ¼ Rd Rb bðx; yÞ Duðx; yÞdxdy  gð0Þ ðx; i1 Þ; Duði2 ; yÞ ¼ Rcd Rab ð0Þ bðx; yÞ Duðx; yÞdxdy  g ði ; yÞ, where i1 ¼ c, d, 2 c a i2 ¼ a; b, b0 and c0 are constants with b0  0 and f(x, y, u), bðx; yÞ and gðlÞ ðx0 Þðl ¼ 0; 1Þ that are continuous functions in their respective domains. To solve this problem

& K. Harish Kumar [email protected] 1

School of Basic Sciences, Indian Institute of Technology Indore, Indore 453552, India

numerically, interesting finite difference iterative schemes are developed in [14, 17]. The present article proposes numerical techniques based on two different iterative schemes using Legendre wavelets for solving (1.1). Numerical schemes based on Legendre wavelets are recently reported in the literature for solving different types of differential equations such as ordinary differential equation [16], delay differential Eq. [20], partial differential equation [12, 22] and q-difference equation [21]. Though there are considerably many methods based on wavelet techniques for different types of differential equations, only a few published works are available for linear second-order elliptic partial differential equations [4, 5] with Dirichlet, Neumann and mixed boundary conditions. In the literature, some methods are reported for integral equations and ordinary integro differential equations [1, 2, 7, 10]. Recently, in