Abstract
In the three-dimensional open rectangular domain, the problem of the identification of the redefinition function for a partial differential equation with Gerasimov–Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the 0<α≤1 order. A positive parameter is present in the mixed derivatives. The solution of this fractional differential equation is studied in the class of regular functions. The Fourier series method is used, and a countable system of ordinary fractional differential equations with degeneration is obtained. The presentation for the redefinition function is obtained using a given additional condition. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proven.
Highlights
When the boundary of a domain of a physical process is impossible to study, a nonlocal condition of integral form can be obtained as additional information sufficient for the unique solvability of the problem
For the case of the 0 < α ≤ 1 order, we study the regular one value solvability of the inverse boundary value problem for the Gerasimov–Caputo-type fractional partial differential equation with degeneration
We find the pair of functions {U (t, x, y); φ ( x, y)}, the first of which satisfies the partial differential Equation (1), nonlocal integral condition (2), and boundary value conditions
Summary
When the boundary of a domain of a physical process is impossible to study, a nonlocal condition of integral form can be obtained as additional information sufficient for the unique solvability of the problem. In [25], an inverse problem to determine the right-hand side for a mixed type integro-differential equation with fractional order Gerasimov–Caputo operators is considered. In [27], the solvability of the nonlocal boundary problem for a mixed-type differential equation with a fractional-order operator and degeneration is studied. The Gerasimov–Caputo α-order fractional derivative for the function η (t) is defined by the following formula α. For the case of the 0 < α ≤ 1 order, we study the regular one value solvability of the inverse boundary value problem for the Gerasimov–Caputo-type fractional partial differential equation with degeneration. Cauchy Problem for a Fractional Ordinary Differential Equation with Degeneration It is well-known that the two-parametric Mittag–Leffler function is defined as (see, for example, [33]).
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