Solution Of Heat Equation Research Articles

Analytic Solution for Two-Dimensional Heat Equation for an Ellipse Region

Analytic Solution for Two-Dimensional Heat Equation for an Ellipse Region

Exact solution of Guyer–Krumhansl type heat equation by operational method

We construct particular solutions for some heat transport differential equations, in particular, for extended forms of hyperbolic heat equation and of Guyer–Krumhansl (GK) equation. The operational approach, integral transforms, generalized orthogonal polynomials and special functions are used. Examples of heat propagation in non-Fourier models are studied and compared with each other. Analytical solutions for some three-dimensional heat transport equations are obtained. The exact analytical solutions for GK type heat equation with linear term are derived. The description of an instant heat surge propagation and of power-exponential pulse is given in heat transport models of Fourier, Cattaneo and Guyer–Krumhansl. Space–time propagation of a periodic function, obeying telegraph and GK equations with linear terms is studied by the operational technique. The exact bounded analytical solutions are obtained. The role of various terms in the equations is illustrated and their influence on the solutions is elucidated. The application for ballistic heat flow study with account for Knudsen number is provided.

A new approximate method for an inverse time-dependent heat source problem using fundamental solutions and RBFs

This paper presents a meshless numerical scheme to solve the inverse heat source time dependent problem. Fundamental solutions of heat equations and radial basis functions (RBFs) are used to obtain a numerical solution. Since the coefficient matrix may be ill-conditioned, the Tikhonov regularization (TR) method is employed to solve the resulting system of linear equations. Therefore, the generalized cross-validation (GCV) criterion is applied to choose a regularization parameter. The accuracy and efficiency of the proposed method is illustrated by some numerical examples.

Numerical Solutions of Heat and Mass Transfer with the First Kind Boundary and Initial Conditions in Capillary Porous Cylinder Using Programmable Graphics Hardware

Recently, heat and mass transfer simulation is more and more important in various engineering fields. In order to analyze how heat and mass transfer in a thermal environment, heat and mass transfer simulation is needed. However, it is too much time-consuming to obtain numerical solutions to heat and mass transfer equations. Therefore, in this paper, one of acceleration techniques developed in the graphics community that exploits a graphics processing unit (GPU) is applied to the numerical solutions of heat and mass transfer equations. The nVidia Compute Unified Device Architecture (CUDA) programming model provides a straightforward means of describing inherently parallel computations. This paper improves the performance of solving heat and mass transfer equations over capillary porous cylinder with the first boundary and initial conditions numerically running on GPU. Heat and mass transfer simulation using the novel CUDA platform on nVidia Quadro FX 4800 is implemented. Our experimental results clearly show that GPU can accurately perform heat and mass transfer simulation. GPU can significantly accelerate the performance with the maximum observed speedups 10 times. Therefore, the GPU is a good approach to accelerate the heat and mass transfer simulation

An analytical model for top-hat long transient mode-mismatched thermal lens spectroscopy

It has been shown that a top-hat excitation beam gives rise to a more sensitive signal for the thermal lens spectroscopy (TLS). Recently, a numerical model has been presented for a top- hat excitation beam in a dual-beam mod-mismatched TLS [Opt. Lett. 33(13), 1464-1466 (2008)]. In this work, we present a full analytical version of this model. Our model was based on a new solution of time-dependent heat equation for a finite radius cylindrical sample exposed to a top-hat excitation laser beam. The Fresnel diffraction integration method was then used to calculate on-axis probe-beam intensity variations due to thermal lensing by taking the aberrant nature of the thermal lens into account. The model was confirmed with experimental data of LSCAS-2 with an excellent agreement.

Stability of Fourier Solutions of Nonlinear Stochastic Heat Equations in 1D

The main focus of this article is studying the stability of solutions of nonlinear stochastic heat equation and give conclusions in two cases: stability in probability and almost sure exponential stability. The main tool is the study of related Lyapunov-type functionals. The analysis is carried out by a natural N-dimensional truncation in isometric Hilbert spaces and uniform estimation of moments with respect to N. Nonlinear stochastic heat equation, additive space-time noise, Lyapunov functional, Fourier solution, finitedimensional approximations, moments, stability.

Similarity solutions of fractional order heat equations with variable coefficients

Similarity solutions of fractional order heat equations with variable coefficients

Hitting probabilities and the Hausdorff dimension of the inverse images of a class of anisotropic random fields

Let X = {X(t): t ∈ RN} be an anisotropic random field with values in Rd. Under certain conditions on X, we establish upper and lower bounds on the hitting probabilities of X in terms of respectively Hausdorff measure and Bessel–Riesz capacity. We also obtain the Hausdorff dimension of its inverse image, and the Hausdorff and packing dimensions of its level sets. These results are applicable to non-linear solutions of stochastic heat equations driven by a white in time and spatially homogeneous Gaussian noise and anisotropic Guassian random fields.

The solutions of time and space conformable fractional heat equations with conformable Fourier transform

Abstract In this paper our aim is to find the solutions of time and space fractional heat differential equations by using new definition of fractional derivative called conformable fractional derivative. Also based on conformable fractional derivative definition conformable Fourier Transform is defined. Fourier sine and Fourier cosine transform definitions are given and space fractional heat equation is solved by conformable Fourier transform.

Exponential growth with Lp-norm of solutions for nonlinear heat equations with viscoelastic term

In this work, we study an initial boundary value problem related to the nonlinear viscoelastic equation(1+a|u|q−2)ut−Δu+∫0tg(t−s)Δu(s)ds=b|u|p−2u.We show the exponential growth of solutions with Lp-norm for negative or positive initial energy.

Averaged controllability for random evolution Partial Differential Equations

We analyze the averaged controllability properties of random evolution Partial Differential Equations. We mainly consider heat and Schrödinger equations with random parameters, although the problem is also formulated in an abstract frame. We show that the averages of parabolic equations lead to parabolic-like dynamics that enjoy the null-controllability properties of solutions of heat equations in an arbitrarily short time and from arbitrary measurable sets of positive measure. In the case of Schrödinger equations we show that, depending on the probability density governing the random parameter, the average may behave either as a conservative or a parabolic-like evolution, leading to controllability properties, in average, of very different kind.

Stability of stationary solutions for semilinear heat equations with concave nonlinearity

This paper is concerned with the stability analysis of stationary solutions of the Cauchy–Dirichlet problem for some semilinear heat equation with concave nonlinearity. The instability of sign-changing solutions is verified under some variational assumption. Moreover, the exponential stability of the positive stationary solution at an optimal rate is proved by exploiting a super–subsolution method as well as the parabolic regularity theory. The base of our analysis relies on the linearization of the equation at each stationary solution and spectral analysis of the corresponding linearized operator. The main difficulties reside in the singularity of the linearized operator due to the concave nonlinearity.

Existence and multiplicity of self-similar solutions for heat equations with nonlinear boundary conditions

This paper is concerned with self-similar solutions in the half-space for linear and semilinear heat equations with nonlinear boundary conditions. Existence, multiplicity and positivity of these solutions are analyzed. Self-similar profiles are obtained as solutions of a nonlinear elliptic PDE with drift term and a nonlinear Neummann boundary condition. For that, we employ a variational approach and derive some compact weighted embeddings for the trace operator.

Numerical method for discrete double barrier option pricing with time-dependent parameters

In this paper, the main purpose is pricing of discrete double barrier option under Black–Scholes framework with time dependent parameters. By some conventional transforms, in each monitoring interval, the problem is reduced to well-known Black–Scholes partial differential equations with convenient constant coefficient such that the solution can be expressed recursively upon the heat equation solution. Finally a numerical method is proposed to compute the obtained recursive formula efficiently. Also, the Greeks of contract are calculated.

Global and blowup solutions of semilinear heat equation involving the square root of the Laplacian

In this paper, we consider nonlinear parabolic equations involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary condition. We use the method on harmonic extension to study the existence and asymptotic estimates of global solutions, as well as the blowup of the parabolic equation.

An efficient numerical technique for the solution of nonlinear heat equation via spectral method

Nonlinear wave equations are more difficult to study mathematically, and no general analytical method exists for their solution. It is found that the Exponential Time Differencing (ETD) scheme requires the steps to achieve a given accuracy, offers a speedy method in calculation time, and has exceptional stability properties in solving a nonlinear equation. This article solves the diagonal example of nonlinear heat equation via the exponential time difference Runge-Kutta 4 methods (ETDRK4). Implementation of the method is proposed by short Matlab programs.

Smoothness of local times and self-intersection local times of Gaussian random fields

This paper is concerned with the smoothness (in the sense of Meyer-Watanabe) of the local times of Gaussian random fields. Sufficient and necessary conditions for the existence and smoothness of the local times, collision local times, and self-intersection local times are established for a large class of Gaussian random fields, including fractional Brownian motions, fractional Brownian sheets and solutions of stochastic heat equations driven by space-time Gaussian noise.

Alternating Direction Implicit Orthogonal Spline Collocation on Non-Rectangular Regions

The alternating direction implicit (ADI) method is a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems on rectangles. When the ADI technique is coupled with orthogonal spline collocation (OSC) for discretization in space we not only obtain the global solution efficiently but the discretization error with respect to space variables can be of an arbitrarily high order. In [2], we used a Crank Nicolson ADI OSC method for solving general nonlinear parabolic problems with Robin’s boundary conditions on rectangular polygons and demonstrated numerically the accuracy in various norms. A natural question that arises is: Does this method have an extension to non-rectangular regions? In this paper, we present a simple idea of how the ADI OSC technique can be extended to some such regions. Our approach depends on the transfer of Dirichlet boundary conditions in the solution of a two-point boundary value problem (TPBVP). We illustrate our idea for the solution of the heat equation on the unit disc using piecewise Hermite cubics.

English

In this article analytical solution of one-dimensional heat equation in relaxation mode of heat generation and conduction using Laplace transforms method is presented. The model adopted takes into account finite velocity of heat propagation, and relaxation of heat source capacity. The properties of heat source terms in four different cases are incorporated in the model and investigated. Temperature distributions and variations with conduction mode and relaxation time are analyzed. High relaxation time is observed to lowers the temperature profile, whereas enhanced temperature distribution changes at particular values of α, andfor source capacity proportional to temperature. How the steady state solution is achieved for some selected values of coefficients is also discussed.   Key words: Relaxation time, conduction mode, pulsed heat source, Laplace transforms.

Pathwise uniqueness for the stochastic heat equation with Hölder continuous drift and noise coefficients

We study the solutions of the stochastic heat equation with multiplicative space–time white noise. We prove a comparison theorem between the solutions of stochastic heat equations with the same noise coefficient which is Hölder continuous of index γ>3/4, and drift coefficients that are Lipschitz continuous. Later we use the comparison theorem to get sufficient conditions for the pathwise uniqueness for solutions of the stochastic heat equation, when both the white noise and the drift coefficients are Hölder continuous.