On a boundary value problem for a mixed type fractional differential equations with parameters

Abstract

In this paper, we consider a boundary value problem for a mixed type partial differential equation with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The mixed differential equation depends from another positive small parameter in mixed derivatives. The considering mixed type differential equation brings to a spectral problem for a second order differential equation with respect to the second variable. Regarding the first variable, this equation is an ordinary fractional differential equation in the positive part of the considering segment, and is a second-order ordinary differential equation with spectral parameter in the negative part of this segment. Using the spectral method of separation of variables, the solution of the boundary value problem is constructed in the form of a Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. It is proved the stability of solution with respect to boundary function and with respect to small positive parameter given in mixed derivatives. For irregular values of the spectral parameter, an infinite number of solutions in the form of a Fourier series are constructed. 1. Problem statement In a rectangular domain Ω = {(t, x) : −a < t < b, 0 < x < l} we consider the fractional partial differential equation of mixed type 0 =  ( D α, γ − ν D α, γ ∂2 ∂ x2 − ∂2 ∂ x2 ) U (t, x), (t, x) ∈ Ω 1, ( ∂ 2 ∂ t 2 − ν ∂ 4 ∂ t 2 ∂ x 2 − ω2 ∂ 2 ∂ x 2 ) U (t, x), (t, x) ∈ Ω 2, (1.1) where Ω 1 = {(t, x) : 0 < t < b, 0 < x < l}, Ω 2 = {(t, x) : −a < t < 0, 0 < x < l}, ν is positive parameter, ω is positive spectral parameter, a, b are positive real numbers, D γ = Jγ−α 0+ d dt J1−γ 0+ , 0 < α ≤ γ ≤ 1 2010 Mathematics Subject Classification. 35M12, 35J25, 35L20, 30E20, 45E05.