A Nonlocal Diffusion Equation whose Solutions Develop a Free Boundary

Let J:\mathbbR ? \mathbbRJ:\mathbb{R} \to \mathbb{R} be a nonnegative, smooth compactly supported function such that ò_\mathbbR J(r)dr = 1. \int_\mathbb{R} {J(r)dr = 1.} We consider the nonlocal diffusion problem

Similarity Solutions of a Non-linear Diffusion Equation

In several physical contexts the equations for the dispersionof a buoyant contaminant can be approximated by the Erdogan-Chatwin(1967) equation {dot}c = {dot}_y{[D_o + ({dot}_yc)²D₂]{dot}_yc}. Here it is shown that in the limit of strong non-linearity (i.e.D_o = 0) there are similarity solutions for a concentration jumpand for a finite discharge. A stability analysis for the latterproblem involves a new family of orthogonal polynomials Y_n(z)where (1 – z⁴)Y – 6z³Y + n(n + 5)z² Y_n = 0 and the degree n is restricted to the values 0, 1, 4, 5, 8,9,.... A numerical solution of the Erdogan-Chatwin equationis given which describes the transition between the non-linearand linear (Gaussian) similarity solutions.     

非线性奇异扩散方程的第二初边值问题

讨论了非线性奇异扩散方程的第二初边值问题，证明了存在唯一的光滑解，且解关于初值是连续依赖的．同时还简洁地得到解的渐近性质：ｔ^ｌ｜ｕ－ｕ_０｜→０，０＜ｌ＜１/２．     

Recovering the Initial Distribution for a Time-Fractional Diffusion Equation

We consider the inverse problem of reconstructing the initial condition of a one-dimensional time-fractional diffusion equation from measurements collected at a single interior location over a finite time-interval. The method relies on the eigenfunction expansion of the forward solution in conjunction with a Tikhonov regularization scheme to control the instability inherent in the problem. We show that the inverse problem has a unique solution provided exact data is given, and prove stability results regarding the regularized solution. Numerical realization of the method and illustrations using a finite-element discretization are given at the end of this paper.     

The Small-time Evolution of a Non-linear Diffusion Equation

The small-time evolution of the equation . t 0, – -$$\infty $$ < y < $$\infty $$ is consideredfor initial conditions appropriate in physical terms to botha concentration jump and a finite discharge. It is shown thatthe strong non-linear similarity solution is not uniformly validfor all y but is singular along an unknown moving boundary inthe (y, t) plane which has to be determined as part of the solution.The nature of this singularity and its resolution is investigatedby the introduction of a transition region between the strongnon-linear and linear regimes.     

Iterative Identification of the Diffusion Coefficient in an Initial Boundary Value Problem for the Subdiffusion Equation

Journal of Applied and Industrial Mathematics - We propose an iterative solution method for an implicit finite-difference analog of the inverse problem of identifying the diffusion coefficient in...     

Asymptotic Solution of a Model Non-linear Convective Diffusion Equation

In this paper we consider the large-time solution of the equation for initial data with compact support. With m = 4 and n = 3the equation models the flow of a thin viscous sheet on an inclinedbed while for n m > 1 it has application in porous mediaflow under gravity. The equation can also be regarded as ananalogue of Burgers equation in non-linear diffusion. It isknown that two moving boundaries exist along which certain boundaryconditions are required to hold. The paper extends earlier workand determines the analytic behaviour of the moving boundariesexist along which certain boundary conditions are required tohold. The paper extends earlier work and determines the analyticbehaviour of the moving boundaries together with the structureof the solution which at large times is shown to depend cruciallyon the location in (n, m) parameter space.     

具有吸附和非线性边值条件的P-Laplace方程

研究了一类具吸附和非线性边值条件的 P- L aplace方程弱解的存在唯一性 .     

KdV-KSV方程的初边值问题

本文证明了KdV－KSV方程的周期边值问题和Cauchy问题广义解和古典解的整体存在性、正则性及唯一性．     

Initial Boundary Value Problem for a Damped Nonlinear Hyperbolic Equation

In the paper, the existence and uniqueness of the generalized global solution and the classical global solution of the initial boundary value problems for the nonlinear hyperbolic equation u_{tt} + k_1u_{xxxx} + k_2u_{xxxxt} + g(u_{xx})_{xx} = f(x, t) are proved by Galerkin method and the sufficient conditions of blow-up of solution in finite time are given.     

奇异扩散方程的非线性边值问题

本文讨论具有非线性边界条件的奇异扩散方程混合边值问题整体光滑解的存在性,唯一性和关于初值的连续依赖性。     

Initial Boundary Value Problem for the KdV Equation on a Semiaxis with Homogeneous Boundary Conditions

We consider the Korteweg–de Vries equation on the semiaxis with zero boundary conditions at x = 0 and arbitrary smooth decreasing initial data. We show that the problem can be effectively integrated by the inverse scattering transform method if the associated linear equation has no discrete spectrum. Under these assumptions, we prove the global solvability of the problem.     

具有振动边值的扩散方程Dirichlet问题

本文以上（下）连续函数作为扩散方程u_t=1/2Δu+cu 在Ｄ内的Ｄｉｒｉｃｈｌｅｔ问题边值函数，讨论了振动边值的Ｄｉｒｉｃｈｌｅｔ问题，并用概率方法证明解的存在性、唯一性和稳定性，把古典Ｄｉｒｉｃｈｌｅｔ问题边值条件减弱到最一般情形     

Boundary Integral Solution of the Time-Fractional Diffusion Equation

Here we consider initial boundary value problem for the time–fractional diffusion equation by using the single layer potential representation for the solution. We derive the equivalent boundary integral equation. We will show that the single layer potential admits the usual jump relations and discuss the mapping properties of the single layer operator in the anisotropic Sobolev spaces. Our main theorem is that the single layer operator is coercive in an anisotropic Sobolev space. Based on the coercivity and continuity of the single layer operator we finally show the bijectivity of the operator in a certain range of anisotropic Sobolev spaces.      

Inverse Problem with Free Boundary for Heat Equation

We establish conditions for the existence and uniqueness of a solution of the inverse problem for a one-dimensional heat equation with unknown time-dependent leading coefficient in the case where a part of the boundary of the domain is unknown.     

Fractional Diffusion Equation Degenerating in the Initial Hyperplane

Ukrainian Mathematical Journal - We consider a fractional extension of the parabolic equation degenerating in the initial hyperplane. For this equation, we construct and investigate the fundamental...     

Boundary Integral Equation Solution of Non-linear Plane Potential Problems

This paper presents the boundary integral equation (BIE) formulation,and numerical solution procedure for two-dimensional problemsgoverned by Laplac's equation and subject to non-linear boundaryconditions. The introduction of non-linear terms constitutesa fundamental extension of the BIE method, as previous applicationshave been restricted entirely to linear problems. Furthermore,non-linearities necessitate the use of iterative solution techniqueswhich present the conceptual disadvantage that a solution isnot guaranteed. However, such difficulties were not encounteredwith the Newton—Raphson method employed in this study.The various features of the BIE technique are illustrated bythe application to a physical problem which is of significancein heat exchanger design.