On the identification of a nonlinear term in a reaction–diffusion equation

Abstract

Reaction-diffusion equations are one of the most common partial differential equations used to model physical phenomenon. They arise as the combination of two physical processes: a driving force f (u) that depends on the state variable u and a diffusive mechanism that spreads this effect over a spatial domain. The canonical form is u t − △u = f (u). Application areas include chemical processes, heat flow models and population dynamics. As part of the model building, assumptions are made about the form of f (u) and these inevitably contain various physical constants. The direct or forwards problem for such equations is now very well-developed and understood, especially when the diffusive mechanism is governed by Brownian motion resulting in an equation of parabolic type.However, our interest lies in the inverse problem of recovering the reaction term f (u) not just at the level of determining a few parameters in a known functional form, but recovering the complete functional form itself. To achieve this we set up the usual paradigm for the parabolic equation where u is subject to both given initial and boundary data, then prescribe overposed data consisting of the solution at a later time T . For example, in the case of a population model this amounts to census data at a fixed time. Our approach will be two-fold. First we will transform the inverse problem into an equivalent nonlinear mapping from which we seek a fixed point. We will be able to prove important features of this map such as a self-mapping property and give conditions under which it is contractive. Second, we consider the direct map from f through the partial differential operator to the overposed data. We will investigate Newton schemes for this case.Classical, Brownian motion diffusion is not the only version and in recent decades various anomalous processes have been used to generalize this case. Amongst the most popular is one that replaces the usual time derivative by a subdiffusion process based on a fractional derivative of order α ≤ 1. We will also include this model in our analysis. The final section of the paper will show numerical reconstructions that demonstrate the viability of the suggested approaches. This will also include the dependence of the inverse problem on both T

Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay

“…In the formulations of inverse problems for partial differential equations additional information about the solution on a part of the boundary is often used (see, for example, [21][22][23][24][25][26][27][28][29][30]). For example, in formulation of so-called inverse backward problem (or retrospective inverse problems) it is required to find a solution at the initial time from a known solution at the final time [31][32][33].…”

A New Forward–Backward Algorithm with Line Searchand Inertial Techniques for Convex Minimization Problems with Applications

“…The problem of finding an unknown reaction term of such an equation can be formulated as a coefficient inverse problem (CIP) for a partial differential equation (PDE). Numerical methods for solving CIPs for PDEs, as well as their applications to various subjects have been widely studied and analyzed; for more comprehensive information on this topic, see [1][2][3][4][5]. To obtain a globally convergent method for solving CIPs for PDEs, many authors have employed the convexification technique, which reformulates CIPs as convex minimization problems; for a more in-depth development and discussion, we refer to [6][7][8].…”

“…where α n is a suitable step size, converges to a solution of (1). Due to its simplicity, this method has been intensively studied and improved by many researchers; see [10,16,18] and the references therein for more information.…”

“…Recently, inspired by [20], Kankam et al [9] proposed a new line search technique and a new algorithm to solve (1). They conducted some experiments on signal processing and found that their method performed better than that of Cruz and Nghia [20].…”

Final value problem for nonlinear time fractional reaction–diffusion equation with discrete data

“…Until now, there are some interesting papers on inverse problem of fractional diffsuion. We can list some well-known results, for example, J. Jia et al [39], J. Liu et al [40], some papers of M. Yamamoto and his group see [43,44,47,46,45], B. Kaltenbacher et al [37,38,50,51], W. Rundell et al [48,49], J. Janno see [41,42], etc. However, to the best of our knowledge, there is no result concerning the backward problem for (2) with random noise.…”