On the Cauchy problem for the fast diffusion equation

For , the author studies the existence of a kind of weak solution to the Cauchy problem

     

A problem with wells for the steady diffusion equation

This paper deals with a problem with wells for which nonlocal boundary conditions are given. It is shown that the problem is equivalent to a mixed problem without wells. For this formulation, an error estimate of a mixed finite element method in the 2D case is studied.     

Cauchy problem for fast diffusion equation with localized reaction

This paper studies the Cauchy problem for the fast diffusion equation with a localized reaction. We establish the Fujita type theorem to the problem, and then obtain the diffusion-independent blow-up rate for the non-global solutions. Moreover, we prove that the blow-up set for the problem consists of a single point under large initial data. These conclusions are quite different from those for the slow diffusion case.     

Mixed problem for a nonlinear ultraparabolic equation that generalizes the diffusion equation with inertia

We consider a mixed problem for a nonlinear ultraparabolic equation that is a nonlinear generalization of the diffusion equation with inertia and the special cases of which are the Fokker-Planck equation and the Kolmogorov equation. Conditions for the existence and uniqueness of a solution of this problem are established. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 9, pp. 1192–1210, September, 2006.     

Monotone method for initial value problem for fractional diffusion equation

Using the method of upper and lower solutions and its associated monotone iterative, consider the existence and uniqueness of solution of an initial value problem for the nonlinear fractional diffusion equation.     

A backward problem for the time‐fractional diffusion equation

In this paper, we are concerned with the backward problem of reconstructing the initial condition of a time‐fractional diffusion equation from interior measurements. We establish uniqueness results and provide stability analysis. Our method is based on the eigenfunction expansion of the forward solution and the Tikhonov regularization to tackle the ill‐posedness issue of the underlying inverse problem. Some numerical examples are included to illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Three-scale asymptotics for a diffusion problem coupled with the wave equation

In this article, we present an asymptotic analysis of waves of elastic stress in an infinite solid whose boundary is subject to a rapid thermal load. The problem under consideration couples the wave equation and the heat equation, and the asymptotic approximation of the solution requires three-scaled variables. The asymptotic approximation is supplied with a rigorous remainder estimate and is illustrated numerically.     

Tikhonov regularization method for a backward problem for the time-fractional diffusion equation

This paper is devoted to solve a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain by the Tikhonov regularization method. Based on the eigenfunction expansion of the solution, the backward problem for searching the initial data is changed to solve a Fredholm integral equation of the first kind. The conditional stability for the backward problem is obtained. We use the Tikhonov regularization method to deal with the integral equation and obtain the series expression of solution. Furthermore, the convergence rates for the Tikhonov regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Two numerical examples in one-dimensional and two-dimensional cases respectively are investigated. Numerical results show that the proposed method is effective and stable.     

Inverse problem for the diffusion equation in the case of spherical symmetry

The initial boundary value problem for the diffusion equation is considered in the case of spherical symmetry and an unknown initial condition. Additional information used for determining the unknown initial condition is an external volume potential whose density is the Laplace operator applied to the solution of the initial boundary value problem. The uniqueness of the solution of the inverse problem is studied depending on the parameters entering into the boundary conditions. It is shown that the solution of the inverse problem is either unique or not unique up to a one-dimensional linear subspace.     

The diffusion equation for the concentration tensor

The purpose of this paper is to show what changes the tensor nature of the concentration introduces in the solution of an equation in comparison with the solution for a scalar quantity. The subsequent analysis of the latter may serve as the basis for experimental determination of the relations between the different components of the concentration tensor (or the chemical potential tensor).Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 36–39.     

The fractional Landweber method for identifying the space source term problem for time-space fractional diffusion equation

Numerical Algorithms - This paper is devoted to solve an inverse problem for identifying the source term of a time-fractional nonhomogeneous diffusion equation with a fractional Laplacian in a...     

Tikhonov regularization method for a backward problem for the inhomogeneous time-fractional diffusion equation

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogs and they are used to model anomalous diffusion, especially in physics. In this paper, we study a backward problem for an inhomogeneous time-fractional diffusion equation with variable coefficients in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The backward problem is ill-posed and we propose a regularizing scheme by using Tikhonov regularization method. We also prove the convergence rate for the regularized solution by using an a priori regularization parameter choice rule. Numerical examples illustrate applicability and high accuracy of the proposed method.     

有限分形介质中分数阶反应扩散方程及其解析解

建立了有限分形介质中具有吸附效应的分数阶反应扩散积分方程.利用Lap lace变换、广义有限H ankel变换及其相应的逆变换得到了以M ittag-Leffler函数为主要形式的解析解,并研究了解的渐近性态.     

Inverse source problem for a diffusion equation involving the fractional spectral Laplacian

In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach.     

Well-posedness of the Dirichlet problem for the non-linear diffusion equation in non-smooth domains

We investigate the Dirichlet problem for the parablic equation

0, \end{displaymath}">

in a non-smooth domain . In a recent paper [U.G. Abdulla, J. Math. Anal. Appl., 260, 2 (2001), 384-403] existence and boundary regularity results were established. In this paper we present uniqueness and comparison theorems and results on the continuous dependence of the solution on the initial-boundary data. In particular, we prove -contraction estimation in general non-smooth domains.

     

Nonexistence results for a fractional problem arising in thermal diffusion in fractal media

We consider the Cauchy problem for the fractional differential equation

D^1+αu+D^βu=Δu+h(t,x)|u|^p