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A geometrically convergent pseudo-spectral method for multi-dimensional two-sided space fractional partial differential equations
Shina Daniel Oloniiju,Sicelo P Goqo,Precious Sibanda
Keywords:Shifted Chebyshev polynomials, Chebyshev-Gauss-Lobatto quadrature, two-sided space-time fractional partial differentials, fractional calculus, convergence analysis
Abstract:
      In this study, we present a geometrically convergent numerical method for solving multidimensional two-sided space time fractional differential equations. The approach allows for the representation of solutions of differential equations in terms of the shifted Chebyshev polynomials. The expansions are evaluated at the shifted Chebyshev--Gauss--Lobatto nodes. We present the left-sided integration matrix and the left-- and right-sided differentiation matrices based on Caputo fractional operators. The performance of the method is demonstrated using selected two-sided space fractional partial differential equations in one and two dimensions. The numerical results obtained show that the method is accurate and computationally efficient. A theoretical analysis of the convergence of the method is presented, where it is shown that the numerical solution converges for a sufficiently large number of grid points provided the solution is continuously differentiable.
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