Solution of partial differential equations by a global radial basis function-based differential quadrature method. (English) Zbl 1081.65546
Summary: The conventional differential quadrature (DQ) method is limited in its application to regular regions by using functional values along a mesh line to approximate derivatives. In this work, we extend the idea of DQ method to a general case. In other words, any spatial derivative is approximated by a linear weighted sum of all the functional values in the whole physical domain. The weighting coefficients in the new approach are determined by the radial basis functions (RBFs). The proposed method combines the advantages of the conventional DQ method and the RBFs. Since the method directly approximates the derivative, it can be consistently well applied to linear and nonlinear problems. It also remains mesh free feature of RBFs. Numerical examples of linear and nonlinear cases showed that RBF-based DQ method has a potential to become an efficient approach for solving partial differential equations.
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
Keywords:
differential quadrature method; radial basis function; Poisson equation; nonlinear elliptic equation; numerical examplesReferences:
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