Modified Quasi Boundary Value method for inverse source biparabolic

Abstract

In this study, we study an inverse source problem of the bi-parabolic equation. The problem is severely non-well-posed in the sense of Hadamard, the problem is called well-posed if it satisfies three conditions, such as the existence, the uniqueness, and the stability of the solution. If one of the these properties is not satisfied, the problem is called is non well-posed (ill-posed). According to our research experience, the stability properties of the sought solution are most often violated. Therefore, a regularization method is required. Here, we apply a Modified Quasi Boundary Method to deal with the inverse source problem. Base on this method, we give a regularized solution and we show that the regularized solution satisfies the conditions of the well-posed problem in the sense of Hadarmad. In addition, we present the estimation between the regularized solution and the sought solution by using a priori regularization parameter choice rule.

Regularization of Inverse Initial Problem for Conformable Pseudo-Parabolic Equation with Inhomogeneous Term

“…The number of works on the regularized problem with input data in L 2 is quite abundant. The results of this study can be found in the following documents, attached to the regularization methods: the Tikhonov method, see [14,15], the Fractional Tikhonov method, see [16], the fractional Landweber method, see [17,18], the Quasi Boundary method, see [19], the truncation method, see [20], and their references.…”

Note on the convergence of fractional conformable diffusion equation with linear source term

“…The above equation has various applications in areas such as the harmonic oscillator, the damped oscillator and the forced oscillator (see [3]), electrical circuits (see [5]), chaotic systems in dynamics (see [6]), projectile motion (see [7]). Our paper is one of the braches of directions about fractional PDEs, see [15,16,17,18,20].…”

Identifying inverse source for diffusion equation with conformable time derivative by Fractional Tikhonov method

“…There are a lot of research results for an inverse source problem of a time-fractional diffusion equation. To do that, during the past decades, a lot of technical developments by mathematicians around the world: Quasi-Reversibility method, see [19], Quasi-Boundary Value method, which readers can see in [20,21], the Landweber iterative method (see [22,23]), the Fractional Landweber method (see [24]), a Tikhonov regularization method (see [25]), a Fourier truncation method (see [26]). However, the object of this topic is to restore the source function F(x) of the problem (1) by the Fractional Tikhonov method.…”