On a class of inverse problems for a heat equation with involution perturbation

Abstract

A class of inverse problems for a heat equation with involution perturbation is considered using four different boundary conditions, namely, Dirichlet, Neumann, periodic and anti-periodic boundary conditions. Proved theorems on existence and uniqueness of solutions to these problems are presented. Solutions are obtained in the form of series expansion using a set of appropriate orthogonal basis for each problem. Convergence of the obtained solutions is also discussed.

Initial-boundary value problems for a time-fractional differential equation with involution perturbation

“…It is worth mentioning here that for the special case of α = 1, the above obtained results are in a full agreement with the ones obtained in [2].…”

“…Our aim is to prove the existence and uniqueness of a regular classical solution of the IBVP (2.1)-(2.3). By a regular solution we mean u(x, t) ∈ C(Ω) ∩ C 1,2 t,x (Ω). In particular, we look for a solution in the form of a series expansion using orthogonal basis which can be obtained by considering the corresponding homogeneous equation and using the method of separation of variables.…”

“…The above two inverse problems are a generalization of two problems considered in [2] for the classical heat equation, i.e., for the case α = 1.…”

Global existence and blow-up of solutions of the time-fractional space-involution reaction-diffusion equation

“…Furthermore, for the equations containing transformation of the spatial variable in the diffusion term, we can cite Cabada and Tojo [8], where an example that describes a concrete situation in physics is given. Note that, the direct and inverse problems for diffusion and fractional diffusion equations with involutions were studied in [4,5,12,13].…”

Inverse source problems for a wave equation with involution

“…Concerning the inverse problems for partial differential equations with involutions, some recent works have been implemented in [6][7][8][9][10][11].…”