Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems

By M. A. Zaky. Applied Numerical Mathematics, 2019

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Abstract Tempered fractional-order models open up new possibilities for robust mathematical modeling of complex multi-scale problems and anomalous transport phenomena. The purpose of this paper is twofold. First, we study existence, uniqueness, and structural stability of solutions to nonlinear tempered fractional differential equations involving the Caputo tempered fractional derivative with generalized boundary conditions. Second, we develop and analyze a singularity preserving spectral-collocation method for the numerical solution of such equations. We derive rigorous error estimates under the L ω θ − 1 , 0 2 - and L ∞ -norms. The most remarkable feature of the method is its capability to achieve spectral convergence for the solution with limited regularity. The results confirm that the method is best suited to discretize tempered fractional differential equations as they naturally take the singular behavior of the solution into account.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

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