The Implication of Local Thin Plate Splines for Solving Nonlinear Mixed Integro-Differential Equations Based on the Galerkin Scheme
Year: 2019
Author: Mehdi Dehghan, Pouria Assari, Fatemeh Asadi-Mehregan, Mehdi Dehghan
Numerical Mathematics: Theory, Methods and Applications, Vol. 12 (2019), Iss. 4 : pp. 1066–1092
Abstract
In this article, we investigate the construction of a computational method for solving nonlinear mixed Volterra-Fredholm integro-differential equations of the second kind. The method firstly converts these types of integro-differential equations to a class of nonlinear integral equations and then utilizes the locally supported thin plate splines as a basis in the discrete Galerkin method to estimate the solution. The local thin plate splines are known as a type of the free shape parameter radial basis functions constructed on a small set of nodes in the support domain of any node which establish a stable technique to approximate an unknown function. The presented method in comparison with the method based on the globally supported thin plate splines for solving integral equations is well-conditioned and uses much less computer memory. Moreover, the algorithm of the presented approach is attractive and easy to implement on computers. The numerical method developed in the current paper does not require any cell structures, so it is meshless. Finally, numerical examples are considered to demonstrate the validity and efficiency of the new method.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2018-0077
Numerical Mathematics: Theory, Methods and Applications, Vol. 12 (2019), Iss. 4 : pp. 1066–1092
Published online: 2019-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 27
Keywords: Mixed integro-differential equation nonlinear integral equation discrete Galerkin method local thin plate spline meshless method.