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Reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions
Author(s) -
Allahviranloo Tofigh,
Sahihi Hussein
Publication year - 2020
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22502
Subject(s) - orthogonalization , mathematics , galerkin method , representer theorem , hilbert space , reproducing kernel hilbert space , kernel (algebra) , partial differential equation , mathematical analysis , convergence (economics) , stability (learning theory) , kernel method , kernel embedding of distributions , finite element method , algorithm , pure mathematics , physics , artificial intelligence , machine learning , computer science , support vector machine , economics , thermodynamics , economic growth
In this study, the parabolic partial differential equations with nonlocal conditions are solved. To this end, we use the reproducing kernel method (RKM) that is obtained from the combining fundamental concepts of the Galerkin method, and the complete system of reproducing kernel Hilbert space that was first introduced by Wang et al. who implemented RKM without Gram–Schmidt orthogonalization process. In this method, first the reproducing kernel spaces and their kernels such that satisfy the nonlocal conditions are constructed, and then the RKM without Gram–Schmidt orthogonalization process on the considered problem is implemented. Moreover, convergence theorem, error analysis theorems, and stability theorem are provided in detail. To show the high accuracy of the present method several numerical examples are solved.