Nonhomogeneous Heat Equation Research Articles

A quasi-reversibility method for solving nonhomogeneous sideways heat equation

Abstract We consider solving a severely ill-posed problem for determining the surface temperature and heat flux distribution of the nonhomogeneous sideways heat equation by a quasi-reversibility regularization method. An analytical solution is deduced based on the Fourier transform, and then, the logarithmic-type error estimate for the regularized solution is achieved. Finally, three numerical examples, including smooth and non-smooth functions, validate the feasibility and effectiveness of the proposed approach.

On a nonhomogeneous heat equation on the complex plane

In this paper, we investigate the existence, uniqueness, and asymptotic behaviors of mild solutions of parabolic evolution equations on complex plane, in which the diffusion operator has the form [Formula: see text], where [Formula: see text], the function [Formula: see text] is smooth and subharmonic on [Formula: see text], and [Formula: see text] is the formal adjoint of [Formula: see text]. Our method combines certain estimates of heat kernel associating with the homogeneous linear equation of Raich [Heat equations in [Formula: see text], J. Funct. Anal. 240(1) (2006) 1–35] and a fixed point argument.

Identification of Time-Wise Thermal Diffusivity, Advection Velocity on the Free-Boundary Inverse Coefficient Problem

This paper is concerned with finding solutions to free-boundary inverse coefficient problems. Mathematically, we handle a one-dimensional non-homogeneous heat equation subject to initial and boundary conditions as well as non-localized integral observations of zeroth and first-order heat momentum. The direct problem is solved for the temperature distribution and the non-localized integral measurements using the Crank–Nicolson finite difference method. The inverse problem is solved by simultaneously finding the temperature distribution, the time-dependent free-boundary function indicating the location of the moving interface, and the time-wise thermal diffusivity or advection velocities. We reformulate the inverse problem as a non-linear optimization problem and use the lsqnonlin non-linear least-square solver from the MATLAB optimization toolbox. Through examples and discussions, we determine the optimal values of the regulation parameters to ensure accurate, convergent, and stable reconstructions. The direct problem is well-posed, and the Crank–Nicolson method provides accurate solutions with relative errors below 0.006% when the discretization elements are M=N=80. The accuracy of the forward solutions helps to obtain sensible solutions for the inverse problem. Although the inverse problem is ill-posed, we determine the optimal regularization parameter values to obtain satisfactory solutions. We also investigate the existence of inverse solutions to the considered problems and verify their uniqueness based on established definitions and theorems.

BVP with a Load in the Form of a Fractional Integral

A boundary value problem for a nonhomogeneous heat equation with a load in the form of a fractional Riemann–Liouville integral of an order β∈0,1 is considered. By inverting the differential part, the problem is reduced to an integral equation with a kernel with a special function. The special function is presented as a generalized hypergeometric function. The limiting cases of the order β of the fractional derivative are studied: it is shown that the interval for changing the order of the fractional derivative can be expanded to integer values β∈0,1. The results of the study remain unchanged. The kernel of the integral equation is estimated. Conditions for the solvability of the integral equation are obtained.

Null controllability for degenerate parabolic equations with a nonlocal space term

<p style='text-indent:20px;'>We consider two degenerate heat equations with a nonlocal space term, studying, in particular, their null controllability property. To this aim, we first consider the associated nonhomogeneous degenerate heat equations: we study their well posedness, the Carleman estimates for the associated adjoint problems and, finally, the null controllability. Then, as a consequence, using the Kakutani's fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems.</p>

Discrete heat equation with irregular thermal conductivity and tempered distributional data

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb {Z}^n$ . We establish the well-posedness of such Cauchy problems in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces $\ell ^{2}_{s}(\hbar \mathbb {Z}^n)$ , $s \in \mathbb {R}$ , which is enough to prove the well-posedness in the space of tempered distributions $\mathcal {S}'(\hbar \mathbb {Z}^n)$ . Notably, when $s=0$ , we show that for $\hbar \rightarrow 0$ , the classical (resp. very weak) solution of the heat equation in the Euclidean setting $\mathbb {R}^n$ is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting $\hbar \mathbb {Z}^n$ .

Diffusion asymptotics of a coupled model for radiative transfer in a unit disk

We consider the diffusion asymptotics of a coupled model arising in radiative transfer in a unit disk in R2 with one-speed velocity. The model consists of a kinetic equation satisfied by the specific intensity of radiation coupled a nonhomogeneous heat equation satisfied by the material temperature. For the well-prepared data, we prove the existence and the nonequilibrium diffusion limit of solutions to the initial boundary problem for the coupled model.

New regularity results for the heat equation and application to non-homogeneous Burgers equation

Abstract This article deals with the regularity of a solution for a non-homogeneous heat equation in Sobolev spaces. The problem is subject to Cauchy–Dirichlet boundary conditions considered in a rectangle or domain that can be transformed into a rectangle.

Regularization for a Sideways Problem of the Non-Homogeneous Fractional Diffusion Equation

In this article, we investigate a sideways problem of the non-homogeneous time-fractional diffusion equation, which is highly ill-posed. Such a model is obtained from the classical non-homogeneous sideways heat equation by replacing the first-order time derivative by the Caputo fractional derivative. We achieve the result of conditional stability under an a priori assumption. Two regularization strategies, based on the truncation of high frequency components, are constructed for solving the inverse problem in the presence of noisy data, and the corresponding error estimates are proved.

On The Analytical Solutions of Nonhomogeneous Heat equations

In this paper, we consider the analytical solution of the Nonhomogeneous mixed problem of the Heat equation. Namely, we use the separation of variables and Duhamel’s principle to derive the formula of the general solution to this problem. Moreover, we find the solution for a special case of this problem as an application to our result

Application of fuzzy finite difference scheme for the non-homogeneous fuzzy heat equation

A numerical framework based on fuzzy finite difference is presented for approximating fuzzy triangular solutions of fuzzy partial differential equations by considering the type of $$[gH-p]-$$ differentiability. The fuzzy triangle functions are expanded using full fuzzy Taylor expansion to develop a new fuzzy finite difference method. By considering the type of gH-differentiability, we approximate the fuzzy derivatives with a new fuzzy finite difference. In particular, we propose using this method to solve non-homogeneous fuzzy heat equation with triangular initial-boundary conditions. We examine the truncation error and the convergence conditions of the proposed method. Several numerical examples are presented to demonstrate the performance of the methods. The final results demonstrate the efficiency and the ability of the new fuzzy finite difference method to produce triangular fuzzy numerical results which are more consistent with existing reality.

Analytical Solution of Heat Conduction from Turbulence with an Isotropic Example

Aerodynamic heating due to turbulence significantly affects the operation of high-speed vehicles and the entrainment of fluid by turbulent plumes. In this paper, the heat generated and convected by fluid turbulence is examined by rearranging the Navier-Stokes equations into a single equation for the fluctuating dependent variables external to unsteady chaotic motions. This equation is similar to the nonhomogeneous heat equation where sources are terms resulting from this rearrangement. Mean and fluctuating quantities are introduced, and under certain circumstances, a heat equation for the fluctuating density results with corresponding mean and fluctuating source terms. The resultant equation is similar to Lighthill’s acoustic analogy and is a “heat analogy.” A solution is obtained with the use of Green’s function as long as the observer is located outside the region of chaotic motion. Predictions for the power spectrum are shown for high Reynolds number isotropic turbulence. The power spectrum decays as the inverse of the wavenumber of the turbulent velocity fluctuations. The developed theory can easily be applied to other turbulent flows if the statistics of unsteady motion can be estimated.

Null controllability of degenerate parabolic equation with memory

In this paper, we analyze the null controllability property for a degenerate parabolic equation involving memory terms with a locally distributed control. We first derive a null controllability result for a nonhomogeneous degenerate heat equation via new Carleman estimates with weighted time functions that do not blow up at . Then, this result is successfully used with a classical fixed point to obtain null controllability for the initial memory system.

Free Convective Flow of Water/Ethylene Glycol Based Micropolar Nanofluid Over a Shrinking Sheet

In this article, the free convective micropolar nanofluid was investigated over a shrinking sheet in the presence of a heat source by tacking into the water/ethylene glycol-based nanofluid account. The physical problem is first modeled. Under the assumptions of Boussinesq's approximation, the governing equations are reduced into non-linear PDEs. The combined non-linear PDEs representing momentum and non-homogeneous heat equations were reduced to a series of regular non-linear differential equations with appropriate similarity transformations. By applying the Runge-Kutta procedure followed by the Shooting technique, the transformed equations are then solved. Via the diagrams, the impact of related parameters characterizing the flow was presented and then addressed. It is observed that volumetric fraction has a substantial influence on the velocity profile and also induces a decrease in the boundary layer because water-based nanofluid has high thermal conductivity relative to Ferro nanofluid based on ethylene glycol.

Null controllability for a singular heat equation with a memory term

In this paper we focus on the null controllability problem for the heat equation with the so-called inverse square potential and a memory term. To this aim, we first establish the null controllability for a nonhomogeneous singular heat equation by a new Carleman inequality with weights which do not blow up at t = 0. Then the null controllability property is proved for the singular heat equation with memory under a condition on the kernel, by means of Kakutani’s fixed-point theorem.

Development of thermal model for the determination of SLM process parameters

In this study, a mathematical model is developed for determining the process parameters used in the manufacturing process of powder materials with selective laser melting method. Although the studies carried out to date cover detailed modeling studies including three-dimensional and time-dependent situations, obtaining quite different approaches from experimental results has led to the idea that a change should be made in the construction of the mathematical problem. Therefore, different from other studies and for the first time, the Eigen-function expansion method was used in the analytical solution of the selective laser melting thermal model. The developed mathematical model includes the steady-state solution with the appropriate boundary conditions of the 2-D non-homogeneous heat equation. The mathematical model was first solved analytically with the Eigen-function expansion method, and the function obtained as a result of the solution was introduced into the MATLAB software. The parametric study was performed numerically over laser power, laser spot size and powder bed thickness. With the developed model, the solution was converged in 30 s and 11.5% more accurate with respect to experimental results were obtained in width, 29% in depth and 3% in temperature as compared to other analytical models exist in the literature.

Solving Fundamental Solution of Non-Homogeneous Heat Equation with Dirichlet Boundary Conditions

In this study, we developed a solution of nonhomogeneous heat equation with Dirichlet boundary conditions. moreover, the non-homogeneous heat equation with constant coefficient. since heat equation has a simple form, we would like to start from the heat equation to find the exact solution of the partial differential equation with constant coefficient. to emphasize our main results, we also consider some important way of solving of partial differential equation specially solving heat equation with Dirichlet boundary conditions. the main results of our paper are quite general in nature and yield some interesting solution of non-homogeneous heat equation with Dirichlet boundary conditions and it is used for problems of mathematical modeling and mathematical physics.

Reducing the near boundary errors of nonhomogeneous heat equations by boundary consistent methods

Abstract In the paper, we point out a drawback of the Fourier sine series method to represent a given odd function, where the boundary Gibbs phenomena would occur when the boundary values of the function are non-zero. We modify the Fourier sine series method by considering the consistent conditions on the boundaries, which can improve the accuracy near the boundaries. The modifications are extended to the Fourier cosine series and the Fourier series. Then, novel boundary consistent methods are developed to solve the 1D and 2D heat equations. Numerical examples confirm the accuracy of the boundary consistent methods, accounting for the non-zeros of the source terms and considering the consistency of heat equations on the boundaries, which can not only overcome the near boundary errors but also improve the accuracy of solution about four orders in the entire domain, upon comparing to the conventional Fourier sine series method and Duhamel’s principle.

Unicidad de Solución de la Ecuación del Calor en Espacios de Sobolev Periódico

In this article, we prove the uniqueness solution of the homogeneous and non-homogeneous heat equation in periodic Sobolev spaces. We do it in a different way from what we did in [3], in this case we perform differential calculus in Hs-per and we take advantage of the immersion and properties of periodic Sobolev spaces. With this proof we gain to visualize the dissipative property of the homogeneous problem and with this we deduce the continuous dependence with respect to the initial data and the uniqueness solution for both cases: homogeneous and non-homogeneous.

A spring-damping regularization of the Fourier sine series solution to the inverse Cauchy problem for a 3D sideways heat equation

For solving the inverse Cauchy problem of a three-dimensional (3D) non-homogeneous sideways heat equation in a cuboid, we first transform it into a partial homogeneous boundary value problem, using a time-dependent two-dimensional (2D) homogenization technique. After a time step-by-step approach of the inverse Cauchy problem, it becomes an inverse Cauchy problem of the 3D modified Helmholtz equation. A sequence of 2D sine functions and a set of second-order ordinary differential equations are derived to determine the expansion coefficients, which can be solved in closed-form. In order to stably solve the inverse Cauchy problem, we simply introduce a spring-damping regularization technique to modify the coefficients, which results in a regularized 2D Fourier sine series solution of the unknown Dirichlet data on the top of the cuboid. The novel method can find the solution in the whole domain, which is quite accurate with small errors, even a large noise up to is imposed on the given data. The present method together with the analyses of regularization parameter and the stability of solution is novel and shows promise in the 3D sideways problems of the heat equation.