Quantum Lévy Laplacian and associated heat equation

As a non-commutative extension of the Lévy Laplacian for entire functions on a nuclear space, we define the quantum Lévy Laplacian acting on white noise operators. We solve a heat type equation associated with the quantum Lévy Laplacian and study its relation to the classical Lévy heat equation. The solution to the quantum Lévy heat equation is obtained also from a normal-ordered white noise differential equation involving the quadratic quantum white noise.     

Heat equation with memory in anisotropic and non-homogeneous media

A modified Fourier’s law in an anisotropic and non-homogeneous media results in a heat equation with memory, for which the memory kernel is matrix-valued and spatially dependent. Different conditions on the memory kernel lead to the equation being either a parabolic type or a hyperbolic type. Well-posedness of such a heat equation is established under some general and reasonable conditions. It is shown that the propagation speed for heat pulses could be either infinite or finite, depending on the different types of the memory kernels. Our analysis indicates that, in the framework of linear theory, heat equation with hyperbolic kernel is a more realistic model for the heat conduction, which might be of some interest in physics.     

Similarity variables and reduction of the heat equation on torus

The problem of symmetry classification for the heat equation on torus is studied by means of classical Lie group theory. The Lie point symmetries are constructed and Lie algebra is formed for equation under consideration. Then these algebras are used to classify its subalgebras up to conjugacy classes. In general the heat equation on torus admits one-, two-, three- and four-dimensional algebras. For one-dimensional algebra £₁ and £₂ the heat equation on torus is reduced in independent variables whereas in two-dimensional algebras £₃ and £₄ the considered heat equation is investigated by quadrature. While three- and four-dimensional algebras lead to a trivial solution.     

The heat equation and reflected Brownian motion in time-dependent domains.: II. Singularities of solutions

We study the space-time Brownian motion and the heat equation in non-cylindrical domains. The paper is mostly devoted to singularities of the heat equation near rough points of the boundary. Two types of singularities are identified—heat atoms and heat singularities. A number of explicit geometric conditions are given for the existence of singularities. Other properties of the heat equation solutions are analyzed as well.     

Applications of nonstandard analysis to partial differential equations—I. the diffusion equation

This paper begins by developing a basis for using ¹ finite difference equations to model physical phenomena. Under appropriate conditions the solution F of a ¹ finite difference equation has S-continuous “finite difference derivatives” up to order r. In these circumstances we can show that the standard function °F is a C^r-function and satisfies the differential equation corresponding to the original finite difference equation. The second part of the paper illustrates these techniques by applying them to the heat equation. In particular, we obtain a very nice model of the heat equation with initial conditions corresponding to all the heat concentrated at a single point.     

On Analytical Results for the Optimal Control of the quasi-stationary Radiative Heat Transfer System

We state an optimal control problem of the coupled quasi-stationary radiative heat equations consisting of the radiative transfer equation and the instationary heat transfer equation that model radiative-conductive heat transfer. We give an existence and uniqueness result for the state equations and the adjoint equations of the quasi-stationary radiative heat transfer system. For the optimal control problem the existence of a minimizer is proven. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Probabilistic representations of solutions to the heat equation

In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if ϕ is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition ϕ, is given by the convolution of ϕ with the heat kernel (Gaussian density). Our results also extend the probabilistic representation of solutions of the heat equation to initial conditions that are arbitrary tempered distributions.     

Crystallization waves in thin amorphous layers on heat conducting substrates

A crystallization process in thin films is considered, where, driven by the release of the latent heat of fusion, the transformation from an amorphous state to the crystalline state takes place in a progressing wave of invariant shape. The crystallization rate is determined by a rate equation. The influence of the heat loss due to heat conduction into the substrate is taken into account. The resulting system of an ordinary differential equation and an integro-differential equation is solved numerically using a collocation method. The propagation speed of the wave in dependence on a non-dimensional heat loss parameter is determined. It turns out that the existence of a self-sustaining crystallization wave requires the heat loss parameter to be smaller than a certain critical value. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Solution of the Heat Equation in Two Space Variables using Chebyshev Series

A numerical solution of the heat conduction equation within a closed curve, is obtained in the form of a combinedFourier/Chebyshev series. The method is an extension of themethods for the one-dimensional heat equation given by Knibb& Scraton (1971) and Dew & Scraton (1972).     

A convergent three‐level finite difference scheme for solving a dual‐phase‐lagging heat transport equation in spherical coordinates

Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat diffusion equation since a second‐order derivative of temperature with respect to time and a third‐order mixed derivative of temperature with respect to space and time are introduced. In this study, we consider the heat transport equation in spherical coordinates and develop a three‐level finite difference scheme for solving the heat transport equation in a microsphere. It is shown that the scheme is convergent, which implies that the scheme is unconditionally stable. Results show that the numerical solution converges to the exact solution. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 60–71, 2004.     

The Heat Semigroup for the Jacobi–Dunkl Operator and the Related Markov Processes

This paper is devoted to the heat equation associated with the Jacobi–Dunkl operator on the real line. In particular we show that the heat semigroup has a strictly positive kernel and a finite Green operator. As a direct application, we solve the Poisson equation and we introduce a new family of one-dimensional Markov processes.     

Uncertain partial differential equation with application to heat conduction

This paper first presents a tool of uncertain partial differential equation, which is a type of partial differential equations driven by Liu processes. As an application of uncertain partial differential equation, uncertain heat equation whose noise of heat source is described by Liu process is investigated. Moreover, the analytic solution of uncertain heat equation is derived and the inverse uncertainty distribution of solution is explored. This paper also presents a paradox of stochastic heat equation.     

Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions

A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in n>1 dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of similarity variables given by group foliations of this heat equation, using its admitted group of scaling symmetries. This technique yields explicit similarity solutions as well as other explicit solutions of a more general (non-similarity) form having interesting analytical behavior connected with blow up and dispersion. In contrast, standard similarity reduction of this heat equation gives a semilinear ODE that cannot be explicitly solved by familiar integration techniques such as point symmetry reduction or integrating factors.     

Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations

Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this article; one is He's variational iteration method (VIM) and the other is the homotopy-perturbation method (HPM). The VIM is to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory, and the initial approximations can be freely chosen with unknown constants. The HPM deforms a difficult problem into a simple problem which can be easily solved. Nonlinear convective–radiative cooling equation, nonlinear heat equation (porous media equation) and nonlinear heat equation with cubic nonlinearity are used as examples to illustrate the simple solution procedures. Comparison of the applied methods with exact solutions reveals that both methods are tremendously effective.     

Analytic solution for heat transfer of a third grade viscoelastic fluid in non-Darcy porous media with thermophysical effects

An analytic approximate solution is presented for the natural convective dissipative heat transfer of an incompressible, third grade, non-Newtonian fluid flowing past an infinite porous plate embedded in a Darcy–Forchheimer porous medium. The mathematical model is developed in an $(x\text{,}y)$ coordinate system. Using a set of transformations, the momentum equation is rendered one-dimensional and a partly linearized heat conservation equation is derived. The viscoelastic formulation presented by Akyildiz [Akyildiz FT. A note on the flow of a third grade between heated parallel plates. Int J Non-Linear Mech 2001;36:349–52] is adopted, which generates lateral mass and viscoelastic terms in the heat conservation equation, as well as in the momentum equation. A number of special cases of the general transformed model are discussed. A homotopy analysis method (HAM) is implemented to solve, with appropriate boundary conditions, the coupled third-order, second degree ordinary differential equation for momentum and the second-order, fourth degree heat conservation equation.     

A compact finite difference scheme for solving a three‐dimensional heat transport equation in a thin film

Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation differs from the traditional heat diffusion equation in having a second‐order derivative of temperature with respect to time and a third‐order mixed derivative of temperature with respect to space and time. In this study, we develop a high‐order compact finite difference scheme for the heat transport equation at the microscale. It is shown by the discrete Fourier analysis method that the scheme is unconditionally stable. Numerical results show that the solution is accurate. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 441–458, 2000     

The relation between the Emden–Fowler equation and the nonlinear heat conduction problem with variable transfer coefficient

The main objective of this article is to analyze the RF-pair approach for the relation between the Emden–Fowler equation and the nonlinear heat conduction problem with variable transfer coefficient. The nonlinear heat conduction equation, by means of appropriate series of operators and transformations is transformed into the classical Emden–Fowler equation.     

Sharp Gradient Estimate and Yau's Liouville Theorem for the Heat Equation on Noncompact Manifolds

We derive a sharp, localized version of elliptic type gradientestimates for positive solutions (bounded or not) to the heatequation. These estimates are related to the Cheng–Yauestimate for the Laplace equation and Hamilton's estimate forbounded solutions to the heat equation on compact manifolds.As applications, we generalize Yau's celebrated Liouville theoremfor positive harmonic functions to positive ancient (includingeternal) solutions of the heat equation, under certain growthconditions. Surprisingly this Liouville theorem for the heatequation does not hold even in Rⁿ without such a condition.We also prove a sharpened long-time gradient estimate for thelog of the heat kernel on noncompact manifolds. 2000 MathematicsSubject Classification 35K05, 58J35.     

Representation of green’s function for the heat equation on a compact lie group

We obtain an explicit formula which presents the solution of the heat equation on a compact Lie group as the limit of finite-to-one convolutions of Green’s function for the heat equation in Euclidean space. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 397–413, March, 2000.     

A stochastic heat equation with the distributions of Lévy processes as its invariant measures

We consider a linear heat equation on a half line with an additive noise chosen properly in such a manner that its invariant measures are a class of distributions of Lévy processes. Our assumption on the corresponding Lévy measure is, in general, mild except that we need its integrability to show that the distributions of Lévy processes are the only invariant measures of the stochastic heat equation.