Inverse Problem for a Nonlinear Pseudiparabolic Differential Equation with Final Condition and Gerasimov–Caputo Operator

Abstract

In this paper in rectangle domain an inverse problem for a fractional analogue of
the pseudoparabolic differential operator with mixed conditions, degeneration and identification
source is considered. Fractional operator is the Gerasimov–Caputo type and the solution of the
nonlinear differential equation with two spatial variables is studied in the class of generalized
functions. The nonlinear Fourier series method is used and by the aid of Kilbas–Saigo function
a nonlinear countable system of functional integral equation is obtained. In the proof of unique
solvability of the countable system is applied the method of successive approximations in combination with the method of compressing mapping. We use the Cauchy–Schwarz inequality and
the Bessel inequality in proving the absolute and uniform convergence of the obtained Fourier
series. Then we derive the desire redefinition function also in the form of Fourier series.