Abstract:
In the present paper, the Mainardi problem related to the study of slow diffusion (subdiffusion) processes described by equations with a fractional time derivative and the second derivative with respect to the spatial variable is generalized to the case of equations in a Banach space with the generator of a strongly continuous cosine function. The well-posed solvability of the corresponding Cauchy problem with the Caputo derivative is proved. The type of solution is indicated. As example of the Mainardi scalar equation is used to show wide possibilities and naturalness of our method and its accuracy.
Keywords:fractional integro-differentiation, $C_{0}$-semigroups and cosine functions, well-posed problem.
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