Journal ArticleDOI
Use of radial basis functions for solving the second‐order parabolic equation with nonlocal boundary conditions
Mehdi Dehghan,Mehdi Tatari +1 more
TLDR
In this paper, the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and nonlocal boundary conditions is considered, and the radial basis functions are used for finding an approximation of the solution of the present problem.Abstract:
Nonlocal mathematical models appear in various problems of physics and engineering. In these models the integral term may appear in the boundary conditions. In this paper the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and nonlocal boundary conditions is considered. These kinds of problems have certainly been one of the fastest growing areas in various application fields. The presence of an integral term in a boundary condition can greatly complicate the application of standard numerical techniques. As a well-known class of meshless methods, the radial basis functions are used for finding an approximation of the solution of the present problem. Numerical examples are given at the end of the paper to compare the efficiency of the radial basis functions with famous finite-difference methods. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008read more
Citations
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Journal ArticleDOI
The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics
TL;DR: In this paper, the authors proposed a numerical method for the solution of the time-fractional nonlinear Schrodinger equation in one and two dimensions which appear in quantum mechanics.
Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial Differential Equations
TL;DR: This monograph differs from other recent books on meshfree methods in that it focuses only on the MQ RBF while others have focused on meshless methods in general.
Journal ArticleDOI
The numerical solution of nonlinear high dimensional generalized Benjamin-Bona-Mahony-Burgers equation via the meshless method of radial basis functions
TL;DR: The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations.
Journal ArticleDOI
An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations
TL;DR: In this paper, the authors proposed a numerical method for the solution of time fractional nonlinear sine-Gordon equation that appears extensively in classical lattice dynamics in the continuum media limit and Klein-Gordon equations which arises in physics.
Journal ArticleDOI
Numerical solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via collocation method based on radial basis functions
Kourosh Parand,Jamal Amani Rad +1 more
TL;DR: This method is a combination of collocation method and radial basis functions with the differentiation process, using zeros of the shifted Legendre polynomial as the collocation points for the solution of nonlinear Volterra–Fredholm–Hammerstein integral equations.
References
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Book ChapterDOI
Interpolation of scattered data: Distance matrices and conditionally positive definite functions
TL;DR: In this paper, it was shown that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R Franke, which is a conjecture that was later proved in the present paper.
Journal ArticleDOI
Radial Basis Functions
Martin D. Buhmann,M. D. Buhmann +1 more
TL;DR: This paper gives a selective but up-to-date survey of several recent developments that explains their usefulness from the theoretical point of view and contributes useful new classes of radial basis function.
Journal ArticleDOI
Multivariate interpolation and conditionally positive definite functions. II
W. R. Madych,S. A. Nelson +1 more
TL;DR: In this paper, the Fourier transform was used to analyze the variational framework for multivariate interpolation and obtained error estimates of arbitrarily high order for a class of interpolation methods that includes multiquadrics.
Journal ArticleDOI
Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations
E.J. Kansa,Y.C. Hon +1 more
TL;DR: This paper explores several techniques, each of which improves the conditioning of the coefficient matrix and the solution accuracy, and recommends using what has been learned from the FEM practitioners and combining their methods with what has be learned in RBF simulations to form a flexible, hybrid approach to solve complex multidimensional problems.