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Local radial basis function scheme for solving a class of fractional integro‐differential equations based on the use of mixed integral equations
Author(s) -
Assari Pouria,
AsadiMehregan Fatemeh
Publication year - 2019
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201800236
Subject(s) - mathematics , collocation method , fractional calculus , basis function , discretization , integral equation , differential equation , radial basis function , projection (relational algebra) , mathematical analysis , ordinary differential equation , computer science , algorithm , machine learning , artificial neural network
Integro‐differential equations with non‐integer order derivatives are an all‐purpose subdivision of fractional calculus. In the current paper, we present a numerical method for solving fractional Volterra‐Fredholm integro‐differential equations of the second kind. To establish the scheme, we first convert these types of integro‐differential equations to mixed integral equations by fractional integrating from both sides of them. Then, the discrete collocation method by combining the locally supported radial basis functions are used to approximate the mentioned integral equations. Since the local method proposed in this paper estimates an unknown function via a small set of data instead of all points in the solution domain, it uses much less computer memory in comparison with the global cases. We apply the nonuniform Gauss‐Legendre integration rule to compute the singular‐fractional integrals appearing in the scheme. As the scheme does not need any background meshes, it can be recognized as a meshless method. Numerical results are included to show the validity and efficiency of the new technique and confirm that the algorithm of the presented approach is attractive and easy to implement on computers.