On a Mixed Problem for Hilfer Type Fractional Differential Equation with Degeneration

In three-dimensional domain the single-value solvability of a mixed problem for a Hilfer type nonlinear partial differential equation of the even order with small positive parameters in mixed derivatives is considered. The regular solution of this fractional differential equation is studied in the case $$0<\alpha\leq 1$$ . The Fourier series method is used and a countable system of nonlinear ordinary differential equations is obtained. The ordinary differential equation is integrated and the obtained constants are found by the aid of given initial value conditions. The countable system of nonlinear functional-integral equations is solved by the method of successive approximations. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proved.