Reconstruction of the time-dependent source in thermal grooving by surface diffusion

In hot polycrystalline materials, when a vertical flat grain boundary meets a horizontal surface the grain boundary forms a groove in the surface. Mathematically modelling features of such thermal grooving mechanism is therefore very important in characterizing polycrystalline materials composed of tiny grains intersecting an external free surface. With this aim in mind, we formulate and investigate a novel inverse problem of reconstructing the unknown time-dependent source term entering the fourth-order parabolic equation of thermal grooving by surface diffusion from a given integral observation. We formulate and prove in Theorems 2.3–2.7 that this linear inverse problem is well-posed. However, in practice, the ideal regularity of data under which the inverse source problem is stable is never satisfied due to the inherent non-smoothness of the measurement. Consequently, this leads to the inverse problem with raw data becoming ill-posed. In order to obtain accurate and stable solutions, we develop and compare two numerical methods, namely, a time-discrete method and an optimization method. We obtain error estimates and convergence rates for the time-discrete method. For the optimization method, an objective functional, which is proved to be Fréchet differentiable, is introduced and the conjugate gradient method (CGM), regularized by the discrepancy principle, is developed to compute the minimizer yielding the source term. The results of two numerical tests illustrate the performance of the two methods for both exact and noisy measured data

Inverse Problems of Determining Coefficients of the Fractional Partial Differential Equations

When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model, for example, the orders of the fractional derivative or the source term, are often unknown, which requires one to discuss inverse problems to identify these physical quantities from some additional information that can be observed or measured practically. This chapter investigates several kinds of inverse coefficient problems for the fractional diffusion equation

К ЧИСЛЕННОМУ РЕШЕНИЮ ОБРАТНЫХ ЗАДАЧ ДЛЯ ЛИНЕЙНОГО ПАРАБОЛИЧЕСКОГО УРАВНЕНИЯ

В статье рассматриваются обратные задачи для линейного параболического уравнения с неизвестными коэффициентами в правой части. К данным задачам, в частности, приводятся краевые задачи, с нелокальными условиями. Отдельно рассмотрены случаи, когда идентифицируемые коэффициенты зависят либо только от временной переменной, либо только от пространственной координаты. Предлагается методика численного решения задач с использованием метода прямых. Приводятся результаты численных экспериментов, проведенных на тестовых задача

On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation

This paper is devoted to the analysis of some uniqueness properties of a classical reaction-diffusion equation of Fisher-KPP type, coming from population dynamics in heterogeneous environments. We work in a one-dimensional interval $(a,b)$ and we assume a nonlinear term of the form $u \, (\mu(x)-\gamma u)$ where $\mu$ belongs to a fixed subset of $C^{0}([a,b])$. We prove that the knowledge of $u$ at $t=0$ and of $u$, $u_x$ at a single point $x_0$ and for small times $t\in (0,\varepsilon)$ is sufficient to completely determine the couple $(u(t,x),\mu(x))$ provided $\gamma$ is known. Additionally, if $u_{xx}(t,x_0)$ is also measured for $t\in (0,\varepsilon)$, the triplet $(u(t,x),\mu(x),\gamma)$ is also completely determined. Those analytical results are completed with numerical simulations which show that, in practice, measurements of $u$ and $u_x$ at a single point $x_0$ (and for $t\in (0,\varepsilon)$) are sufficient to obtain a good approximation of the coefficient $\mu(x).$ These numerical simulations also show that the measurement of the derivative $u_x$ is essential in order to accurately determine $\mu(x)$

Inverse Problems of Determining Sources of the Fractional Partial Differential Equations

In this chapter, we mainly review theoretical results on inverse source problems for diffusion equations with the Caputo time-fractional derivatives of order $\alpha\in(0,1)$. Our survey covers the following types of inverse problems: 1. determination of time-dependent functions in interior source terms 2. determination of space-dependent functions in interior source terms 3. determination of time-dependent functions appearing in boundary condition