Inverse problem for $2b$-order differential equation with a time-fractional derivative

Abstract

We study the inverse problem for a differential equation of order $2b$ with the Riemann-Liouville fractional derivative of order $\beta\in (0,1)$ in time and given Schwartz type distributions in the right-hand sides of the equation and the initial condition. The problem is to find the pair of functions $(u, g)$: a generalized solution $u$ to the Cauchy problem for such equation and the time dependent multiplier $g$ in the right-hand side of the equation. As an additional condition, we use an analog of the integral condition $$(u(\cdot,t),\varphi_0(\cdot))=F(t), \;\;\; t\in [0,T],$$ where the symbol $(u(\cdot,t),\varphi_0(\cdot))$ stands for the value of an unknown distribution $u$ on the given test function $\varphi_0$ for every $t\in [0,T]$, $F$ is a given continuous function.
 We prove a theorem for the existence and uniqueness of a generalized solution of the Cauchy problem, obtain its representation using the Green's vector-function. The proof of the theorem is based on the properties of conjugate Green's operators of the Cauchy problem on spaces of the Schwartz type test functions and on the structure of the Schwartz type distributions.
 We establish sufficient conditions for a unique solvability of the inverse problem and find a representation of anunknown function $g$ by means of a solution of a certain Volterra integral equation of the second kind with an integrable kernel.

Inverse problem with two unknown time-dependent functions for $2b$-order differential equation with fractional derivative

“…By assumptions, r, g ∈ C[0, T] for each u ∈ S ′ γ,(a),C ( Q). Repeating the proof of Theorem 1 from [16] we get that u ∈ S ′ γ,(a),C ( Q) is the solution of the Cauchy problem (1), (2) if and only if it is the solution of the equation 7) in ( 8) we get (6) for the solution of the problem (1)-( 3).…”

“…The conditions of classical solvability of the Cauchy and boundary value problems to equations with a time fractional derivative were obtained, for example, in [2][3][4][5][6][7][8]. The inverse boundary value problems to a time fractional diffusion equation with different unknown functions or parameters were investigated, for example, in [9][10][11][12][13][14][15][16][17][18][19]. Most papers were devoted to inverse problems with an unknown right-hand sides, mainly under regular data.…”

“…The inverse boundary value problem with an unknown only young coefficient in the case b = 1, one over-determination condition of kind (3) and regular data was studied in [14], and in [15] in the case of given distributions with compact supports in the right-hand sides. Source inverse problem for a 2b-order equation with a time-fractional derivative was studied in [16]. Inverse problems for a time-fractional differential equations with a time-integral over-determination conditions were investigated, for example, in [17][18][19].…”

“…In this paper, the result [16] is generalized to the case when, in addition to the solution of the Cauchy problem, we look for two unknown, time-dependent functions (part of the righthand side and the lowest coefficient in the equation).…”

Inverse problem for semilinear Eidelman type equation

“…Note, that the problems of determination of a parameter in the right-hand side function of the parabolic equations were studied in [7]- [12], of the semilinear ultraparabolic equations in [13,14,15]. The authors used the methods of the integral equations, regularization and the Shauder principle [7,8,10], the methods of finite difference approximations, numerical and iterative methods [11,12], the method of successive approximations [13]- [15].…”