Volume 12, Issue 4
The Implication of Local Thin Plate Splines for Solving Nonlinear Mixed Integro-Differential Equations Based on the Galerkin Scheme

Pouria Assari, Fatemeh Asadi-Mehregan & Mehdi Dehghan

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 1066-1092.

Published online: 2019-06

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  • 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.

  • Keywords

Mixed integro-differential equation, nonlinear integral equation, discrete Galerkin method, local thin plate spline, meshless method.

  • AMS Subject Headings

45G10, 45J05, 65R20

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

passari@basu.ac.ir (Pouria Assari)

mdehghan@aut.ac.ir (Mehdi Dehghan)

  • BibTex
  • RIS
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@Article{NMTMA-12-1066, author = {Assari , Pouria and Asadi-Mehregan , Fatemeh and Dehghan , Mehdi }, title = {The Implication of Local Thin Plate Splines for Solving Nonlinear Mixed Integro-Differential Equations Based on the Galerkin Scheme}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {12}, number = {4}, pages = {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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2018-0077}, url = {http://global-sci.org/intro/article_detail/nmtma/13215.html} }
TY - JOUR T1 - The Implication of Local Thin Plate Splines for Solving Nonlinear Mixed Integro-Differential Equations Based on the Galerkin Scheme AU - Assari , Pouria AU - Asadi-Mehregan , Fatemeh AU - Dehghan , Mehdi JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1066 EP - 1092 PY - 2019 DA - 2019/06 SN - 12 DO - http://dor.org/10.4208/nmtma.OA-2018-0077 UR - https://global-sci.org/intro/nmtma/13215.html KW - Mixed integro-differential equation, nonlinear integral equation, discrete Galerkin method, local thin plate spline, meshless method. AB -

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.

Pouria Assari, Fatemeh Asadi-Mehregan & Mehdi Dehghan. (2019). The Implication of Local Thin Plate Splines for Solving Nonlinear Mixed Integro-Differential Equations Based on the Galerkin Scheme. Numerical Mathematics: Theory, Methods and Applications. 12 (4). 1066-1092. doi:10.4208/nmtma.OA-2018-0077
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