Galerkin RBF for Integro-Differential Equations

Two methods based on the Galerkin method with Radial basis Functions (RBF) as bases are applied to solve integro-differential equations (IDEs). In the first approach, direct Galerkin RBF method, the unknown function of the IDE is approximated by RBFs and then the derivatives of it are replaced by the derivatives of RBFs. In the second one, indirect Galerkin RBF method, the derivative of the unknown function is approximated by RBFs and then lower order derivatives and unknown function itself are computed by integrating of RBFs. Therefore the Galerkin method is applied to compute these coefficients. Double integrals that appeared in the process, can be reduced to single integrals by using a formula of iterated integrals. In complicated cases, single integrals approximated by Legendre-Gauss-Lobatto quadrature. Illustrative examples are included to demonstrate the validity and applicability of the presented techniques. A comparison of applying these methods shows the efficiency and high accuracy of the indirect Galerkin RBF method rather than direct Galerkin RBF method.