Approximation Solution of Fractional Partial Differential Equations by Neural Networks

Neural networks with radial basis functions method are used to solve a class of initial boundary value of fractional partial differential equations with variable coefficients on a finite domain. It takes the case where a left-handed or right-handed fractional spatial derivative may be present in the partial differential equations. Convergence of this method will be discussed in the paper. A numerical example using neural networks RBF method for a two-sided fractional PDE also will be presented and compared with other methods. 1. Introduction In this paper, I will use neural network method to solve the fractional partial differential equation (FPDE) of the form: On a finite domain , . Here, I consider the case , where the parameter is the fractional order (fractor) of the spatial derivative. The function is source/sink term [1]. The functions and may be interpreted as transport-related coefficients. We also assume an initial condition for and zero Dirichlet boundary conditions. For the case , the addition of classical advective term on the right-hand side of (1.1) does not impact the analysis performed in this paper and has been omitted to simplify the notation. The left-hand and right-hand fractional derivatives in (1.1) are the Riemann-Liouville fractional derivatives of order [5] defined by where is an integer such that . If is an integer, then the above definitions give the standard integer derivatives, that is When , and setting , (1.1) becomes the following classical parabolic PDE: Similarly, when and setting , (1.1) reduces to the following classical hyperbolic PDE: The case represents a superdiffusive process, where particles diffuse faster than the classical model (1.4) predicts. For some applications to physics and hydrology, see [2–4]. I also note that the left-handed fractional derivative of at a point depends on all function values to the left of the point , that is, this derivative is a weighted average of such function values. Similarly, the right-handed fractional derivative of at a point depends on all function values to the right of this point. In general, left-handed and right-handed derivatives are not equal unless is an even integer, in which case these derivatives become localized and equal. When is an odd integer, these derivatives become localized and opposite in sign. For more details on fractional derivative concepts and definitions, see [1, 3, 5, 6]. Reference [7] provides a more detailed treatment of the right-handed fractional derivatives as well as a substantial treatment of the left-handed fractional derivatives. Published