On a nonlinear renewal equation with diffusion

In this paper, we consider a nonlinear age structured McKendrick–von Foerster population model with diffusion term. Here we prove existence and uniqueness of the solution of the equation. We consider a particular type of nonlinearity in the renewal term and prove Generalized Relative Entropy type inequality. Longtime behavior of the solution has been addressed for both linear and nonlinear versions of the equation. In linear case, we prove that the solution converges to the first eigenfunction with an exponential rate. In nonlinear case, we have considered a particular type of nonlinearity that is present in the mortality term in which we can predict the longtime behavior. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotic behavior for a class of the renewal nonlinear equation with diffusion

In this paper, we consider nonlinear age‐structured equation with diffusion under nonlocal boundary condition and non‐negative initial data. More precisely, we prove that under some assumptions on the nonlinear term in a model of McKendrick–Von Foerster with diffusion in age, solutions exist and converge (long‐time convergence) towards a stationary solution. In the first part, we use classical analysis tools to prove the existence, uniqueness, and the positivity of the solution. In the second part, using comparison principle, we prove the convergence of this solution towards the stationary solution. Copyright © 2012 John Wiley & Sons, Ltd.     

The minimal solution of a Markov renewal equation

Given a killed Markov process, one can use a procedure of Ikedaet al. to revive the process at the killing times. The revived process is again a Markov process and its transition function is the minimal solution of a Markov renewal equation. In this paper we will calculate such solutions for a class of revived processes.     

A very singular solution for the slow diffusion equation with nonlinear convection

We here investigate an existence and uniqueness of the nontrivial, nonnegative solution of a nonlinear ordinary differential equation:

(fm^)″+βrf^′+αf+σ(fq^)′=0     

A general solution of the diffusion equation for semiinfinite geometries

The solution of the time-dependent diffusion equation in a semiinfinite planar, cylindrical, or spherical geometry with common initial and asymptotic boundary conditions is considered. It is shown that this boundary value problem may be described by a single equation which involves only a first order spatial derivative and a half order time derivative. The replacement is exact in the planar and spherical geometry cases but approximate in the cylindrical case. This replacement permits the solution of the original boundary value problem to be written for any boundary condition at the origin. It also leads to a simple relationship between the boundary flux and the boundary intensive variable, which does not require a calculation of the intensive variable at all positions and times.     

Composite spectral method for solution of the diffusion equation with specification of energy

Many physical subjects are modeled by nonclassical parabolic boundary value problems with nonlocal boundary conditions replacing the classic boundary conditions. In this article, we introduce a new numerical method for solving the one‐dimensional parabolic equation with nonlocal boundary conditions. The approximate proposed method is based upon the composite spectral functions. The properties of composite spectral functions consisting of terms of orthogonal functions are presented and are utilized to reduce the problem to some algebraic equations. The method is easy to implement and yields very accurate result. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

The explicit solution of a diffusion equation with singularity

We give the explicit solution of the Cauchy problem for the diffusion equation with a singular term:

where . We construct the solution on the basis of a generalization of the Fourier transform. We next show that the solution is expressed by an analytic semigroup, and examine smoothness of and continuity of .

     

The Decay of mass for a nonlinear fractional reaction–diffusion equation

We study the large time behavior of non‐negative solutions to the nonlinear fractional reaction–diffusion equation ?_tu = ? t^σ( ? Δ)^{α ∕ 2}u ? h(t)u^p (α ∈ (0,2]) posed on and supplemented with an integrable initial condition, where σ ≥ 0, p > 1, and h : [0, ∞ ) → [0, ∞ ). Defining the mass , under certain conditions on the function h, we show that the asymptotic behavior of the mass can be classified along two cases as follows:

• if , then there exists M _∞ ∈ (0, ∞ ) such that ;
• if , then .

Copyright © 2014 John Wiley & Sons, Ltd.     

On the renewal of a solution of the Dirichlet boundary-value problem for a biharmonic equation

We study the problem of renewal of a solution of the Dirichlet boundary-value problem for a biharmonic equation on the basis of the known information about the boundary function. The obtained estimates of renewal error are unimprovable in certain cases. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1147–1151, August, 1998.     

一类非线性扩散方程解的blow up速率估计

利用F riedm an-M cleod方法和变动尺度方法研究了一类具有非线性边界条件的非线性扩散方程解的b low up问题,证明了解在有限时间b low up,并且得到了b low up速率估计.     

Numerical solution of a fast diffusion equation

In this paper, the authors consider the first boundary value problem for the nonlinear reaction diffusion equation: in , a smooth bounded domain in with the zero lateral boundary condition and with a positive initial condition, (fast diffusion problem), and . Sufficient conditions on the initial data are obtained for the solution to vanish or become infinite in a finite time. A scheme for the discretization in time of this problem is proposed. The numerical scheme preserves the essential properties of the initial problem; namely existence of an extinction or a blow-up time, for which estimates have been obtained. The convergence of the method is also proved.

     

Numerical solution of a diffusion equation with a reproducing nonlinearity

Summary We study the Faedo-Galerkin approximations of the Burger Equation, which we write as an operator equation of the typeu _t+Au+Nu=0 in a Hilbert spaceH. We show that the nonlinear operatorN is finitely reproducing relative to the orthonormal sequence {u _i} generated byAu=u and study the numerical behavior of the approximations.

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Zusammenfassung Wir untersuchen die Faedo-Galerkin Approximationen der Burger Gleichung, die wir als Operatorgleichung des Typesu _t+Au+Nu=0, in einem Hilbertraum auffassen. Wir zeigen, daß der nichtlineare OperatorN, endlich reproduzierend bezüglich einer Orthonormalfolgeu _i ist, die durchAu=u erzeugt wird und untersuchen das numerische Verhalten der Approximationen.

     

Homogeneous renewal equation

Let F be a non-arithmetic distribution on the line , and W be the class of bounded functions w without discontinuity of the second kind such that

.In this paper, we show that the solution of the homogeneous renewal equation w = w F in the class W is a constant-function.     

The uniqueness of the solution to the diffusion equation with a damping term

Consider a non-linear diffusion equation with a damping term. If the diffusion coefficient is positive, then the solutions are not unique generally. However, if the diffusion coefficient degenerates, the situation may change. In this paper, not only the existence of the weak solution is established, but also the uniqueness of the weak solutions is proved, even the boundary value condition is not imposed. The conclusions imply that, on the boundary, the degeneracy of diffusion coefficient can eliminate the action from the damping term.     

On the numerical solution of the spherically symmetric diffusion equation

Summary A new stability proof for a difference approximation to the spherically symmetric diffusion equation is given.     

Numerical solution of a nonlinear reaction diffusion equation

In this paper, the authors propose a numerical method to compute the solution of a Cauchy problem with blow-up of the solution. The problem is split in two parts: a hyperbolic problem which is solved by using Hopf and Lax formula and a parabolic problem solved by a backward linearized Euler method in time and a finite element method in space. It is proved that the numerical solution blows up in a finite time as the exact solution and the support of the approximation of a self-similar solution remains bounded. The convergence of the scheme is obtained.