Convolution quadrature and discretized operational calculus I.
Numerische Mathematik
Fractionally coupled solutions of the diffusion equation
Applied Mathematics and Computation - Special issue: Advanced special functions and related topics in differential equations, third Melfi workshop, proceedings of the Melfi school on advanced topics in mathematics and physics
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A system of fractional evolution equations results from employing the tool of the Fractional Calculus and following the method used by Dirac to obtain his well-known equation from Klein-Gordon's one. It represents a possible interpolation between Dirac and diffusion and wave equations in one space dimension. In this paper some analytical properties typical of the general solution of this system of equations are obtained and necessary stability bounds for a numerical scheme approximating such equations are found, through the classical discrete Von Neumann-type analysis. The non-local property of the time fractional differential operator leads to discretizations in terms of series. Here, the analytical methods, usually employed in the study of the stability of discrete schemes when dealing with integer order differential equations, have been adapted to the complexity of the real order case.