Backward heat equation with time dependent variable coefficient

Abstract

Backward heat equation with time dependent variable coefficient is severely ill‐posed in the sense of Hadamard, so we need regularization. In this paper, we consider Backward heat equation with time dependent variable coefficient, and by small perturbing, we obtain an approximation problem. We show this approximation problem is well‐posed with small parameter. Also, we show this approximation system converges to the original problem when parameter goes to zero. Here, we use modified‐quasi boundary value method to regularize this problem. Copyright © 2016 John Wiley & Sons, Ltd.

Stochastic fractional diffusion equations containing finite and infinite delays with multiplicative noise

“…Initially, let us mention about the classical diffusion equations, the paramount importance of the Caputo-Fabrizio operator, and some related studies on time fractional diffusion equations in the deterministic case. It should be noted that if the fractional derivative CF D β t is replaced by the integer order derivative ∂ t then the equations we consider turn to be the primitive diffusion models (also called typical heat equations and classical parabolic equations), which are traditional and have been much studied previously due to their theoretical interest and essential applications in various fields of science such as heat transfer and image processing [3,28,35,37]. Regarding the fractional derivative CF D β t , the presence of this derivative plays the role of modeling several practical phenomena in physics, control systems, biology, fluid dynamics and material science [5,6,7,24].…”

Regularization of the backward stochastic heat conduction problem

“…The deterministic model of Problem (1), i.e., when f (t) Ẇ is omitted, commonly known as the backward heat conduction problem (BHCP) has been extensively studied in the literature over the last few decades, see e.g. [2,3,4,5,9,12,13,15,16] to mention only a few. The BHCP arises in several practical areas such as heat transfer and image processing, [1,9].…”

“…[2,3,4,5,9,12,13,15,16] to mention only a few. The BHCP arises in several practical areas such as heat transfer and image processing, [1,9]. The problem is well-known to be severely ill-posed in the sense that a solution corresponding to the data ξ does not always exist, and in the case of existence, it does not depend continuously on the given data.…”

On regularization and error estimates for the backward heat conduction problem with time-dependent thermal diffusivity factor

“…By using a truncation regularization method, the one dimensional case of the BHCP with the time-dependent diffusion coefficient has been formulated in [30][31][32]. A modified quasi-reversibility method for the n-dimensional BHCP has been also developed in [33]. The most error estimates for the BHCP presented in the literature are of the Hölder type which is not more suitable to measure with adequate accuracy.…”