Inverse problem for fractional order pseudo-parabolic equation with involution

Abstract

In this paper, we consider an inverse problem on recovering the right-hand side of a fractional pseudo-parabolic equation with an involution operator. The major obstacle for considering the inverse problems is related with the well-posedness of the problem. Inverse problems are often ill-posed. For example, the inverse heat equation, deducing a previous distribution of temperature from final data, is not well-posed since the solution is highly sensitive to variations in the final data.The advantage of this paper is two-fold. On the one hand, we investigate the solvability of the direct problem and prove the solvability to this problem. On the other hand, we study the inverse problem based on this direct problem and prove the solvability results in this problem, too.First, we investigate the Cauchy problem for the time-fractional pseudo-parabolic equation with the involution operator, and secondly, we consider the inverse problem on recovering the right-hand side from an overdetermined final condition and prove that it is solvable.To achieve our goals, we use methods corresponding to the different areas of mathematics such as the theory of partial differential equations, mathematical physics, and functional analysis. In particular, we use the ℒ-Fourier analysis method to establish the existence and uniqueness of solutions to this problem on the Sobolev space.The classical and generalized solutions of the inverse problem are studied.

A hybrid regularization method for identifying the source term and the initial value simultaneously for fractional pseudo-parabolic equations with involution *

“…It is accessible to prove that L is a self-adjoint operator. For all |ε| < 1, ε ∈ R, nonlocal problem (1.3) has the folllowing eigenvalues, see [10] for the solution procedure…”

“…In [9], Ruzhansky et al used L-Fourier method to obtain the uniqueness and stability of the solution of fractional involution pseudo-parabolic equations in an abstract set of Hilbert Spaces. In [10], Serikbaev used L-Fourier method to obtain the classical and generalized solutions of fractional involution inverse problems on Sobolev space, and proved the existence and uniqueness.…”

Gas Dynamics type Burgers equation with convolutional nonlinearity

“…In [7,8,9,10], the Fourier method is widely used to solve several types of problems of evolution equations for positive operators. In [6], convolution operators and self-adjoint positive operators on an abstract Hilbert space are considered in more detail.…”