2012 **Abstract:** Solutions of boundary value problems for a diffusion equation of fractional and variable order in differential and difference settings are studied. It is shown that the method of the energy inequalities is applicable to obtaining a priori estimates for these problems exactly as in the classical case. The credibility of the obtained results is verified by performing numerical calculations for a test problem.

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“…Those Lyapunov functions are quadratic (positive definite) and are accompanied by a new Lemma that allows proving that their fractional derivatives are negative semidefinite or definite so it can be used along with the fractional extension of the Lyapunov direct method to determine stability or asymptotical stability. A similar result can be found in Corollary 1 of [8] which is a consequence of an expression, introduced in that paper, for the Caputo derivative of the product of two functions.…”

confidence: 69%

“…Those Lyapunov functions are quadratic (positive definite) and are accompanied by a new Lemma that allows proving that their fractional derivatives are negative semidefinite or definite so it can be used along with the fractional extension of the Lyapunov direct method to determine stability or asymptotical stability. A similar result can be found in Corollary 1 of [8] which is a consequence of an expression, introduced in that paper, for the Caputo derivative of the product of two functions.…”

confidence: 69%

“…The following Lemma, which generalizes Lemma 1 in [7] and Corollary 1 in [8], will subsequently allow us to extend Lyapunov type results for nonlinear and time-varying fractional order systems.…”

confidence: 96%