A fractional B-spline collocation method for the numerical solution of fractional predator-prey models

We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost

An unstructured mesh control volume method for two-dimensional space fractional diffusion equations with variable coefficients on convex domains

In this paper, we propose a novel unstructured mesh control volume method to deal with the space fractional derivative on arbitrarily shaped convex domains, which to the best of our knowledge is a new contribution to the literature. Firstly, we present the finite volume scheme for the two-dimensional space fractional diffusion equation with variable coefficients and provide the full implementation details for the case where the background interpolation mesh is based on triangular elements. Secondly, we explore the property of the stiffness matrix generated by the integral of space fractional derivative. We find that the stiffness matrix is sparse and not regular. Therefore, we choose a suitable sparse storage format for the stiffness matrix and develop a fast iterative method to solve the linear system, which is more efficient than using the Gaussian elimination method. Finally, we present several examples to verify our method, in which we make a comparison of our method with the finite element method for solving a Riesz space fractional diffusion equation on a circular domain. The numerical results demonstrate that our method can reduce CPU time significantly while retaining the same accuracy and approximation property as the finite element method. The numerical results also illustrate that our method is effective and reliable and can be applied to problems on arbitrarily shaped convex domains.Comment: 18 pages, 5 figures, 9 table

Variable-Order Fractional Partial Differential Equations: Analysis, Approximation and Inverse Problem

Variable-order fractional partial differential equations provide a competitive means in modeling challenging phenomena such as the anomalous diffusion and the memory effects and thus attract widely attentions. However, variable-order fractional models exhibit salient features compared with their constant-order counterparts and introduce mathematical and numerical difficulties that are not common in the context of integer-order and constant-order fractional partial differential equations. This dissertation intends to carry out a comprehensive investigation on the mathematical analysis and numerical approximations to variable-order fractional derivative problems, including variable-order time-fractional, space-fractional, and space-time fractional partial differential equations, as well as the corresponding inverse problems. Novel techniques are developed to accommodate the impact of the variable fractional order and the proposed mathematical and numerical methods provide potential tools to analyze and compute the variable-order fractional problems