On the analysis of mixed-index time fractional differential equation systems

In this paper we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left hand side. We prove a theorem on the solution of the linear system of equations, which collapses to the well-known Mittag-Leffler solution in the case the indices are the same, and also generalises the solution of the so-called linear sequential class of time fractional problems. We also investigate the asymptotic stability properties of this class of problems using Laplace transforms and show how Laplace transforms can be used to write solutions as linear combinations of generalised Mittag-Leffler functions in some cases. Finally we illustrate our results with some numerical simulations.Comment: 21 pages, 6 figures (some are made up of sub-figures - there are 15 figures or sub-figures

A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations

In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using tensor product cubic spline basis functions defined on a background rectangular (interpolation) mesh, which leads to high spatial accuracy and straightforward implementation, and establishes a solid base for extending the computational framework to three-dimensional problems. A semi-implicit time-stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem. A key finding of our study is that the newly proposed mesh-free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero. Both methods produce a large constrained least-squares problem that must be solved at each time step in the advancement of the solution. We have found that regularization yields a relatively well-conditioned system that can be solved accurately using QR factorization. An extensive numerical investigation is performed to illustrate the effectiveness of the present methods, including the application of the new method to a coupled system of time-fractional partial differential equations having different fractional indices in different (irregularly shaped) regions of the solution domain

Alikhanov Legendre–Galerkin spectral method for the coupled nonlinear time-space fractional Ginzburg–Landau complex system

A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg-Landau system is proposed and analyzed. The Alikhanov L2-1 sigma difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Gronwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims