# Heat Equation (heat + equation)

Distribution by Scientific Domains

## Selected Abstracts

### Ricci flows and infinite dimensional algebras

FORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 6-7 2004
I. Bakas
The renormalization group equations of two-dimensional sigma models describe geometric deformations of their target space when the world-sheet length changes scale from the ultra-violet to the infra-red. These equations, which are also known in the mathematics literature as Ricci flows, are analyzed for the particular case of two-dimensional target spaces, where they are found to admit a systematic description as Toda system. Their zero curvature formulation is made possible with the aid of a novel infinite dimensional Lie algebra, which has anti-symmetric Cartan kernel and exhibits exponential growth. The general solution is obtained in closed form using Bäcklund transformations, and special examples include the sausage model and the decay process of conical singularities to the plane. Thus, Ricci flows provide a non-linear generalization of the heat equation in two dimensions with the same dissipative properties. Various applications to dynamical problems of string theory are also briefly discussed. Finally, we outline generalizations to higher dimensional target spaces that exhibit sufficient number of Killing symmetries. [source]

### Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right-hand side in Lp (p,1)

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2009
Pierre-Étienne Druet
Abstract Accurate modelling of heat transfer in high-temperature situations requires accounting for the effect of heat radiation. In complex industrial applications involving dissipative heating, we hardly can expect from the mathematical theory that the heat sources will be in a better space than L1. In this paper, we focus on a stationary heat equation with nonlocal boundary conditions and Lp right-hand side, with p,1 being arbitrary. Thanks to new coercivity results, we are able to produce energy estimates that involve only the Lp norm of the heat sources and to prove the existence of weak solutions. Copyright © 2008 John Wiley & Sons, Ltd. [source]

### Asymptotic analysis and estimates of blow-up time for the radial symmetric semilinear heat equation in the open-spectrum case

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2007
N. I. Kavallaris
Abstract We estimate the blow-up time for the reaction diffusion equation ut=,u+ ,f(u), for the radial symmetric case, where f is a positive, increasing and convex function growing fast enough at infinity. Here ,>,*, where ,* is the ,extremal' (critical) value for ,, such that there exists an ,extremal' weak but not a classical steady-state solution at ,=,* with ,w(,, ,),,,, as 0<,,,*,. Estimates of the blow-up time are obtained by using comparison methods. Also an asymptotic analysis is applied when f(s)=es, for ,,,*,1, regarding the form of the solution during blow-up and an asymptotic estimate of blow-up time is obtained. Finally, some numerical results are also presented. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### Evolution completeness of separable solutions of non-linear diffusion equations in bounded domains

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2004
V. A. Galaktionov
Abstract As a basic example, we consider the porous medium equation (m > 1) (1) where , , ,N is a bounded domain with the smooth boundary ,,, and initial data . It is well-known from the 1970s that the PME admits separable solutions , where each ,k , 0 satisfies a non-linear elliptic equation . Existence of at least a countable subset , = {,k} of such non-linear eigenfunctions follows from the Lusternik,Schnirel'man variational theory from the 1930s. The first similarity pattern t,1/(m,1),0(x), where ,0 > 0 in ,, is known to be asymptotically stable as t , , and attracts all nontrivial solutions with u0 , 0 (Aronson and Peletier, 1981). We show that if , is discrete, then it is evolutionary complete, i.e. describes the asymptotics of arbitrary global solutions of the PME. For m = 1 (the heat equation), the evolution completeness follows from the completeness-closure of the orthonormal subset , = {,k} of eigenfunctions of the Laplacian , in L2. The analysis applies to the perturbed PME and to the p -Laplacian equations of second and higher order. Copyright © 2004 John Wiley & Sons, Ltd. [source]

### Local solutions to a model of piezoelectric materials

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2004
Kamel Hamdache
Abstract A local existence theorem is proved for a non-linear coupled system modelling the electromechanical motion of a one-dimensional piezoelectric body with domain switching. The system is composed by a heat equation describing the behaviour of the number of electric dipoles and by a wave equation governing the dynamic of the electric displacement. The main coupling in the system appears in the time-dependent velocity of the waves depending on the number of electric dipoles. The proof of the result relies on a time decay estimate satisfied by the number of electric dipoles and an uniform estimate of the solution of the regularized wave equation. Copyright © 2004 John Wiley & Sons, Ltd. [source]

### Unsaturated incompressible flows in adsorbing porous media

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2003
A. Fasano
We study a free boundary problem modelling the penetration of a liquid through a porous material in the presence of absorbing granules. The geometry is one dimensional. The early stage of penetration is considered, when the flow is unsaturated. Since the hydraulic conductivity depends both on saturation and on porosity and the latter change due to the absorption, the main coefficient in the flow equation depends on the free boundary and on the history of the process. Some results have been obtained in Fasano (Math. Meth. Appl. Sci. 1999; 22:605) for a simplified version of the model. Here existence and uniqueness are proved in a class of weighted Hölder spaces in a more general situation. A basic tool are the estimates on a non-standard linear boundary value problem for the heat equation in an initially degenerate domain (Rend. Mat. Acc. Lincei 2002; 13:23). Copyright © 2003 John Wiley & Sons, Ltd. [source]

### L1 Decay estimates for dissipative wave equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2001
Albert Milani
Let u and v be, respectively, the solutions to the Cauchy problems for the dissipative wave equation $$u_{tt}+u_t-\Delta u=0$$\nopagenumbers\end(1) and the heat equation $$v_t-\Delta v=0$$\nopagenumbers\end(2) We show that, as $t\rightarrow+\infty$\nopagenumbers\end, the norms $\|\partial_t^k\,D_x^\alpha u(\,\cdot\,,t)\|_{L^1({\rm R}^n)}$\nopagenumbers\end and $\|\partial_t^k\,D_x^\alpha v(\,\cdot\,,t)\|_{L^1({\rm R}^n)}$\nopagenumbers\end decay to 0 with the same polynomial rate. This result, which is well known for decay rates in $L^p({\rm R}^n)$\nopagenumbers\end with $2\leq p\leq+\infty$\nopagenumbers\end, provides another illustration of the asymptotically parabolic nature of the hyperbolic equation (1). Copyright © 2001 John Wiley & Sons, Ltd. [source]

### Approximate identities in variable Lp spaces

MATHEMATISCHE NACHRICHTEN, Issue 3 2007
D. Cruz-Uribe SFO
Abstract We give conditions for the convergence of approximate identities, both pointwise and in norm, in variable Lp spaces. We unify and extend results due to Diening [8], Samko [18] and Sharapudinov [19]. As applications, we give criteria for smooth functions to be dense in the variable Sobolev spaces, and we give solutions of the Laplace equation and the heat equation with boundary values in the variable Lp spaces. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Compact difference schemes for heat equation with Neumann boundary conditions

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2009
Zhi-Zhong Sun
Abstract In this article, two recent proposed compact schemes for the heat conduction problem with Neumann boundary conditions are analyzed. The first difference scheme was proposed by Zhao, Dai, and Niu (Numer Methods Partial Differential Eq 23, (2007), 949,959). The unconditional stability and convergence are proved by the energy methods. The convergence order is O(,2 + h2.5) in a discrete maximum norm. Numerical examples demonstrate that the convergence order of the scheme can not exceeds O(,2 + h3). An improved compact scheme is presented, by which the approximate values at the boundary points can be obtained directly. The second scheme was given by Liao, Zhu, and Khaliq (Methods Partial Differential Eq 22, (2006), 600,616). The unconditional stability and convergence are also shown. By the way, it is reported how to avoid computing the values at the fictitious points. Some numerical examples are presented to show the theoretical results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source]

### Numerical studies of a nonlinear heat equation with square root reaction term

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2009
Ron Buckmire
Abstract Interest in calculating numerical solutions of a highly nonlinear parabolic partial differential equation with fractional power diffusion and dissipative terms motivated our investigation of a heat equation having a square root nonlinear reaction term. The original equation occurs in the study of plasma behavior in fusion physics. We begin by examining the numerical behavior of the ordinary differential equation obtained by dropping the diffusion term. The results from this simpler case are then used to construct nonstandard finite difference schemes for the partial differential equation. A variety of numerical results are obtained and analyzed, along with a comparison to the numerics of both standard and several nonstandard schemes. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source]

### A high-order finite difference method for 1D nonhomogeneous heat equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2009
Yuan Lin
Abstract In this article a sixth-order approximation method (in both temporal and spatial variables) for solving nonhomogeneous heat equations is proposed. We first develop a sixth-order finite difference approximation scheme for a two-point boundary value problem, and then heat equation is approximated by a system of ODEs defined on spatial grid points. The ODE system is discretized to a Sylvester matrix equation via boundary value method. The obtained algebraic system is solved by a modified Bartels-Stewart method. The proposed approach is unconditionally stable. Numerical results are provided to illustrate the accuracy and efficiency of our approximation method along with comparisons with those generated by the standard second-order Crank-Nicolson scheme as well as Sun-Zhang's recent fourth-order method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source]

### A stabilized Hermite spectral method for second-order differential equations in unbounded domains

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2007
Heping Ma
Abstract A stabilized Hermite spectral method, which uses the Hermite polynomials as trial functions, is presented for the heat equation and the generalized Burgers equation in unbounded domains. In order to overcome instability that may occur in direct Hermite spectral methods, a time-dependent scaling factor is employed in the Hermite expansions. The stability of the scheme is examined and optimal error estimates are derived. Numerical experiments are given to confirm the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source]

### Anisotropic a posteriori error estimate for an optimal control problem governed by the heat equation

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2006
Marco Picasso
Abstract The abstract framework of Becker et al. is considered to solve an optimal control problem governed by a parabolic equation. Existence and uniqueness of a solution are proved using the inf-sup framework and space-time functional spaces. A Crank-Nicolson time discretization is proposed, together with continuous, piecewise linear finite elements in space. Existence and uniqueness of a solution to the discretized problem is also proved using the inf-sup framework. An a posteriori error estimate is proposed, the goal being to control the error between the true and computed cost functional. The error estimate remains valid on strongly anisotropic meshes and an anisotropic error indicator is proposed when the time step is small. Finally, the quality of this error indicator is studied numerically on isotropic and anisotropic meshes. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 [source]

### Image analysis using p -Laplacian and geometrical PDEs

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2007
A. KuijperArticle first published online: 29 FEB 200
Minimizing the integral ,,1/p |,L |pd , for an image L under suitable boundary conditions gives PDEs that are well-known for p = 1, 2, namely Total Variation evolution and Laplacian diffusion (also known as Gaussian scale space and heat equation), respectively. Without fixing p, one obtains a framework related to the p -Laplace equation. The partial differential equation describing the evolution can be simplified using gauge coordinates (directional derivatives), yielding an expression in the two second order gauge derivatives and the norm of the gradient. Ignoring the latter, one obtains a series of PDEs that form a weighted average of the second order derivatives, with Mean Curvature Motion as a specific case. Both methods have the Gaussian scale space in common. Using singularity theory, one can use properties of the heat equation (namely. the role of scale) in the full L (x, t) space and obtain a framework for topological image segmentation. In order to be able to extend image analysis aspects of Gaussian scale space in future work, relations between these methods are investigated, and general numerical schemes are developed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Heat flow on Finsler manifolds

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2009
Shin-ichi Ohta
This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, ·) : TxM , ,+ on each tangent space. Mostly, we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold: as gradient flow on L2(M, m) for the energy as gradient flow on the reverse L2 -Wasserstein space ,,2(M) of probability measures on M for the relative entropy Both approaches depend on the choice of a measure m on M and then lead to the same nonlinear evolution semigroup. We prove ,,1, , regularity for solutions to the (nonlinear) heat equation on the Finsler space (M, F, m). Typically solutions to the heat equation will not be ,,2. Moreover, we derive pointwise comparison results à la Cheeger-Yau and integrated upper Gaussian estimates à la Davies. © 2008 Wiley Periodicals, Inc. [source]

### Behavior of the solution of a random semilinear heat equation

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2008
We consider a semilinear heat equation in one space dimension, with a random source at the origin. We study the solution, which describes the equilibrium of this system, and prove that, as the space variable tends to infinity, the solution becomes a.s. asymptotic to a steady state. We also study the fluctuations of the solution around the steady state. [source]

### On Nonexistence of type II blowup for a supercritical nonlinear heat equation

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2004
Hiroshi Matano
In this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut = ,u + |u|p,1u either on ,N or on a finite ball under the Dirichlet boundary conditions. We assume that the exponent p is supercritical in the Sobolev sense, that is, We prove that if ps < p < p*, then blowup is always of type I, where p* is a certain (explicitly given) positive number. More precisely, the rate of blowup in the L, norm is always the same as that for the corresponding ODE dv/dt = |v|p,1v. Because it is known that "type II" blowup (or, equivalently, "fast blowup") can occur if p > p*, the above range of exponent p is optimal. We will also derive various fundamental estimates for blowup that hold for any p > ps and regardless of type of blowup. Among other things we classify local profiles of type I and type II blowups in the rescaled coordinates. We then establish useful estimates for the so-called incomplete blowup, which reveal that incomplete blowup solutions belong to nice function spaces even after the blowup time. © 2004 Wiley Periodicals, Inc. [source]

### Blow-up analysis for a system of heat equations coupled via nonlinear boundary conditions

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2007
Xianfa Song
Abstract In this paper, we study a system of heat equations coupled via nonlinear boundary conditions (1) Here p, q>0. We prove that the solutions always blow up in finite time for non-trivial and non-negative initial values. We also prove that the blow-up occurs only on SR = ,BR for , = BR = {x , ,n:|x|[source]

### A high-order finite difference method for 1D nonhomogeneous heat equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2009
Yuan Lin
Abstract In this article a sixth-order approximation method (in both temporal and spatial variables) for solving nonhomogeneous heat equations is proposed. We first develop a sixth-order finite difference approximation scheme for a two-point boundary value problem, and then heat equation is approximated by a system of ODEs defined on spatial grid points. The ODE system is discretized to a Sylvester matrix equation via boundary value method. The obtained algebraic system is solved by a modified Bartels-Stewart method. The proposed approach is unconditionally stable. Numerical results are provided to illustrate the accuracy and efficiency of our approximation method along with comparisons with those generated by the standard second-order Crank-Nicolson scheme as well as Sun-Zhang's recent fourth-order method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source]

### About smoothness of solutions of the heat equations in closed, smooth space-time domains

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2005
Hongjie Dong
We consider the probabilistic solutions of the heat equation u = u + f in D, where D is a bounded domain in ,2 = {(x1, x2)} of class C2k. We give sufficient conditions for u to have kth -order continuous derivatives with respect to (x1, x2) in D, for integers k , 2. The equation is supplemented with C2k boundary data, and we assume that f , C2(k,1). We also prove that our conditions are sharp by examples in the border cases. © 2005 Wiley Periodicals, Inc. [source]