On the numerical solution of a parabolic Fredholm integro-differential equation by the RBF method

This paper presents the numerical solution of an initial boundary value problem for a parabolic Fredholm integro-differential equation (FIDE) in bounded 2D and 3D spatial domains. To reduce the dimensionality of the problem, we employ the Laguerre transformation and Rothe’s method, with both first- and second-order time discretization approximations. As a result, the time-dependent problem is transformed into a recurrent sequence of boundary value problems for elliptic FIDEs. The radial basis function (RBF) method is then applied, where each stationary solution is approximated as a linear combination of radial basis functions centered at specific points, along with polynomial basis functions. The placement of these center points is outlined for both two-dimensional and three-dimensional regions. Collocation at center points generates a sequence of linear systems with integral coefficients. To compute these coefficients numerically, parameterization is performed, and Gauss–Legendre and trapezoidal quadratures are used. The shape parameter of the RBFs is optimized through a real-coded genetic algorithm. Numerical results in both two-dimensional and three-dimensional domains confirm the effectiveness and applicability of the proposed approaches.