Radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions

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Abstract

The development of numerical methods for the solution of partial differential equations (PDEs) with nonlocal boundary conditions is important, since such type of problems arise as mathematical models of various real-world processes. We use radial basis function (RBF) collocation technique for the solution of a multidimensional linear elliptic equation with classical Dirichlet boundary condition and nonlocal integral conditions. RBF-based meshless methods are easily implemented and efficient, especially for multidimensional problems formulated on complexly shaped domains. In this paper, properties of the method are investigated by studying two- and three-dimensional test examples with manufactured solutions. We analyze the influence of the RBF shape parameter and the distribution of the nodes on the accuracy of the method as well as the influence of nonlocal conditions on the conditioning of the collocation matrix.
Original languageEnglish
Pages (from-to)1407-1420
JournalComputers and Mathematics with Applications
Volume67
Issue number7
DOIs
Publication statusPublished - 2014

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Nonlocal Boundary Conditions
Radial Functions
Elliptic Equations
Basis Functions
Linear equation
Nonlocal Conditions
Boundary conditions
Collocation
Integral Condition
Meshless Method
Shape Parameter
Conditioning
Dirichlet Boundary Conditions
Partial differential equations
Numerical methods
Partial differential equation
Numerical Methods
Mathematical Model
Mathematical models
Three-dimensional

Keywords

  • Multidimensional elliptic equation
  • Nonlocal integral condition
  • Meshless method

Cite this

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title = "Radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions",
abstract = "The development of numerical methods for the solution of partial differential equations (PDEs) with nonlocal boundary conditions is important, since such type of problems arise as mathematical models of various real-world processes. We use radial basis function (RBF) collocation technique for the solution of a multidimensional linear elliptic equation with classical Dirichlet boundary condition and nonlocal integral conditions. RBF-based meshless methods are easily implemented and efficient, especially for multidimensional problems formulated on complexly shaped domains. In this paper, properties of the method are investigated by studying two- and three-dimensional test examples with manufactured solutions. We analyze the influence of the RBF shape parameter and the distribution of the nodes on the accuracy of the method as well as the influence of nonlocal conditions on the conditioning of the collocation matrix.",
keywords = "Multidimensional elliptic equation, Nonlocal integral condition, Meshless method",
author = "Svajūnas Sajavičius",
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PY - 2014

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N2 - The development of numerical methods for the solution of partial differential equations (PDEs) with nonlocal boundary conditions is important, since such type of problems arise as mathematical models of various real-world processes. We use radial basis function (RBF) collocation technique for the solution of a multidimensional linear elliptic equation with classical Dirichlet boundary condition and nonlocal integral conditions. RBF-based meshless methods are easily implemented and efficient, especially for multidimensional problems formulated on complexly shaped domains. In this paper, properties of the method are investigated by studying two- and three-dimensional test examples with manufactured solutions. We analyze the influence of the RBF shape parameter and the distribution of the nodes on the accuracy of the method as well as the influence of nonlocal conditions on the conditioning of the collocation matrix.

AB - The development of numerical methods for the solution of partial differential equations (PDEs) with nonlocal boundary conditions is important, since such type of problems arise as mathematical models of various real-world processes. We use radial basis function (RBF) collocation technique for the solution of a multidimensional linear elliptic equation with classical Dirichlet boundary condition and nonlocal integral conditions. RBF-based meshless methods are easily implemented and efficient, especially for multidimensional problems formulated on complexly shaped domains. In this paper, properties of the method are investigated by studying two- and three-dimensional test examples with manufactured solutions. We analyze the influence of the RBF shape parameter and the distribution of the nodes on the accuracy of the method as well as the influence of nonlocal conditions on the conditioning of the collocation matrix.

KW - Multidimensional elliptic equation

KW - Nonlocal integral condition

KW - Meshless method

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DO - 10.1016/j.camwa.2014.01.014

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JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

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