A variant of quasi‐reversibility method for a class of heat equations with involution perturbation

Abstract

The paper is devoted to investigating a Cauchy problem governed by nonclassical heat equation with involution. The problem is severely ill‐posed in the sense of Hadamard by violating the continuous dependence upon the input Cauchy data. Therefore, in order to obtain a stable solution, we shall use a modified Pseudo‐Parabolic Regularization Method. The main idea is to add a correction term by introducing a third‐order derivation operator to formulate a sequence of well‐posed problems that depend on a regularization parameter ε. Further, we show that the approximate problems are well posed, and we prove some convergence results.

Solvability of mixed problems for the wave equation with reflection of the argument

“…The questions of solvability of direct, inverse problems for differential equations with involution in the case of constant coefficients are discussed in the works of many authors. 11,12,[14][15][16][17][18][31][32][33][34][35][36] For example, in Turmetov and Kadirkulov, 31 solvability of a boundary value problem for a nonlocal analog of a mixed parabolic-hyperbolic equation of fractional order with involution is studied. Previous works [13][14][15][16][17][18]32 are devoted to the study of inverse problems for a fractional parabolic equation with involution.…”

“…Inverse problems for a degenerate parabolic equation of fractional order with involution were considered in Kirane et al 11 The questions of solvability of mixed problems for the heat and wave equations with involution were studied in previous studies. 12,35 However, in all the papers listed, equations with involution with constant coefficients are studied. As for equations of the type (1) with variable coefficients, mixed problems have not yet been studied.…”

“…In Ruzhansky et al, 13 a class of inverse problems for restoring the right‐hand side of a parabolic equation for a large class of positive self‐adjoint operators with discrete spectrum is considered. Inverse problems for a degenerate parabolic equation of fractional order with involution were considered in Kirane et al 11 The questions of solvability of mixed problems for the heat and wave equations with involution were studied in previous studies 12,35 . However, in all the papers listed, equations with involution with constant coefficients are studied.…”

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

“…In [7], AI-Salti and Kirane used local and nonlocal boundary conditions to study the direct and inverse initial boundary value problems of time-fractional heat equations with involution perturbations. In [8], Sassane et al used an improved pseudo-parabolic regularization method to solve a class of heat equations with involution perturbation. At present, the problem of fractional pseudo-parabolic equations with involution has become the subject of extensive research in various mathematical models.…”

The inverse problem for the heat equation with reflection of the argument and with a complex coefficient

“…In this paper, we consider the equation of the type u t (x, t)u xx (x, t) + αu xx (-x, t) + q(x)u(x, t) = f (x), (x, t) ∈ , (1.1) (see, for example, [4][5][6][7][8][9][10][11][12] and references therein). In [4][5][6][7], inverse problems for equations with involution with constant coefficients were considered by the method of separation of variables. In [9], inverse problems were studied for a parabolic equation containing an arbitrary linear positive self-conjugate operator with discrete spectrum.…”