The principal aim of this paper is to analyze and implement two numerical algorithms for solving two kinds of space fractional linear advection-dispersion problems. The proposed numerical solutions are spectral and they are built on assuming the approximate solutions to be certain double shifted Tchebyshev basis. The two typical collocation and Petrov-Galerkin spectral methods are applied to obtain the desired numerical solutions. The special feature of the two proposed methods is that their applications enable one to reduce, through integration, the fractional problem under investigation into linear systems of algebraic equations, which can be efficiently solved via any suitable solver. The convergence and error analysis of the double shifted Tchebyshev basis are carefully investigated, aiming to illustrate the correctness and feasibility of the proposed double expansion. Finally, the efficiency, applicability, and high accuracy of the suggested algorithms are demonstrated by presenting some numerical examples accompanied with comparisons with some other existing techniques discussed in the literature.

While the approximate solutions of one-dimensional nonlinear Volterra–Fredholm integral equations with smooth kernels are now well understood, no systematic studies of the numerical solutions of their multi-dimensional counterparts exist. In this paper, we provide an efficient numerical approach for the multi-dimensional nonlinear Volterra–Fredholm integral equations based on the multi-variate Legendre-collocation approach. Spectral collocation methods for multi-dimensional nonlinear integral equations are known to cause major difficulties from a convergence analysis point of view. Consequently, rigorous error estimates are provided in the weighted Sobolev space showing the exponential decay of the numerical errors. The existence and uniqueness of the numerical solution are established. Numerical experiments are provided to support the theoretical convergence analysis. The results indicate that our spectral collocation method is more flexible with better accuracy than the existing ones.

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This manuscript is devoted to implementing spectral numerical solutions to two kinds of fractional space-time advectiondispersion problems governed by certain constraints conditions. The collocation and tau spectral methods are utilized for obtaining the proposed spectral solutions. A double Legendre expansion is proposed as an approximate solution. The main idea of the algorithm is basically depend on converting the equation with its constraints conditions into linear or nonlinear systems of algebraic equations which can be efficiently solved with the aid of suitable numerical solvers. Some illustrative examples are displayed aiming to confirm robustness, efficiency and accuracy of the proposed spectral solutions.