Cauchy problem for an essentially loaded fractional diffusion equation
DOI:
https://doi.org/10.31489/2026m2/136-148Keywords:
loaded differential equation, Cauchy problem, essentially loaded equation, fractional diffusion equation, non-uniqueness, moving load, Wright function, Mittag-Leffler functionAbstract
In this paper, we solve the Cauchy problem for a loaded fractional diffusion equation in an infinite strip. The loaded term is defined as the trace of the fractional derivative of the desired solution on a continuous curve lying inside the domain. We consider all three cases of possible distribution of the order of differentiation in the loaded term (µ) and the order of the time-fractional derivative in the principal differential part of the equation (α). In the first case considered (α>µ), the problem under study is reduced to an integral equation. In the second case (α =µ), we obtain a functional equation. In the third case (α <µ), we are dealing with a differential equation. We show that the condition α>µ ensures the unique solvability of the problem under consideration. In the case of an essentially loaded equation (α≤µ), the problem may lose both uniqueness and solvability. In particular, it is shown that if α<µ, then the problem under consideration ceases to be uniquely solvable, and the corresponding homogeneous problem has infinitely many nontrivial solutions. Moreover, in this case, the solvability requires additional conditions that narrow the set of admissible input data.
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