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Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method
Author(s) -
Eshaghi Jafar,
Adibi Hojatollah,
Kazem Saeed
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3788
Subject(s) - mathematics , legendre polynomials , volterra integral equation , mathematical analysis , integral equation , fractional calculus , legendre's equation , singular integral , algebraic equation , nonlinear system , legendre wavelet , associated legendre polynomials , legendre function , kernel (algebra) , singularity , orthogonal polynomials , classical orthogonal polynomials , gegenbauer polynomials , discrete wavelet transform , physics , wavelet transform , quantum mechanics , artificial intelligence , computer science , wavelet , combinatorics
In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.