Parabolic Heat Equation Research Articles

Numerical Solution for Two-Sided Stefan Problem

In this paper, we consider a two-phase Stefan problem in one-dimensional space for parabolic heat equation with non-homogenous Dirichlet boundary condition. This problem contains a free boundary depending on time. Therefore, the shape of the problem is changing with time. To overcome this issue, we use a simple transformation to convert the free-boundary problem to a fixed-boundary problem. However, this transformation yields a complex and nonlinear parabolic equation. The resulting equation is solved by the finite difference method with Crank-Nicolson scheme which is unconditionally stable and second-order of accuracy in space and time. The numerical results show an excellent accuracy and stable solutions for two test examples.

Regularity “in Large” for the 3DSalmon’s Planetary Geostrophic Model of Ocean Dynamics

AbstractIt is well known, by now, that the three-dimensional non-viscous planetary geostrophic model, with vertical hydrostatic balance and horizontal Rayleigh friction/damping, coupled to the heat diffusion and transport, is mathematically ill-posed. This is because the no-normal flow physical boundary condition implicitly produces an additional boundary condition for the temperature at the lateral boundary. This additional boundary condition is different, because of the Coriolis forcing term, than the no-heat-flux physical boundary condition. Consequently, the second order parabolic heat equation is over-determined with two different boundary conditions. In a previous work we proposed one remedy to this problem by introducing a fourth-order artificial hyper-diffusion to the heat transport equation and proved global regularity for the proposed model. A shortcoming of this higher-oder diffusion is the loss of the maximum/minimum principle for the heat equation. Another remedy for this problem was suggested by R. Salmon by introducing an additional Rayleigh-like friction/damping term for the vertical component of the velocity in the hydrostatic balance equation. In this paper we prove the global, for all time and all initial data, well-posedness of strong solutions to the three-dimensional Salmon’s planetary geostrophic model of ocean dynamics. That is, we show global existence, uniqueness and continuous dependence of the strong solutions on initial data for this model. Unlike the 3D viscous PG model, we are still unable to show the uniqueness of the weak solution. Notably, we also demonstrate in what sense the additional damping term, suggested by Salmon, annihilate the ill-posedness in the original system; consequently, it can be viewed as “regularizing” term that can possibly be used to regularize other related systems.

Приближенные и точные решения вырождающегося нелинейного уравнения теплопроводности с произвольной нелинейностью

The paper deals with the problem of the motion of a heat wave with a specified front for a general nonlinear parabolic heat equation. An unknown function depends on two variables. Along the heat wave front, the coefficient of thermal conductivity and the source function vanish, which leads to a degeneration of the parabolic type of the equation. This circumstance is the mathematical reason for the appearance of the considered solutions, which describe perturbations propagating along the zero background with a finite velocity. Such effects are generally atypical for parabolic equations. Previously, we proved the existence and uniqueness theorem for the problem considered in this paper. Still, it is local and does not allow us to study the properties of the solution beyond the small neighborhood of the heat wave front. To overcome this problem, the article proposes an iterative method for constructing an approximate solution for a given time interval, based on the boundary element approach. Since it is usually not possible to prove strict convergence theorems of approximate methods for nonlinear equations of mathematical physics with a singularity, verification of the calculation results is relevant. One of the traditional ways is to compare them with exact solutions. In this article, we obtain and study an exact solution of the required type, the construction of which is reduced to integrating the Cauchy problem for an ODE. We obtained some qualitative properties, including an interval estimation of the wave amplitude in one particular case. The performed calculations show the effectiveness of the developed computational algorithm, as well as the compliance of the results of calculations with qualitative analysis.

Simultaneous determination of thermal diffusivity and thermal conductivity of a thin layer using double modulated thermal excitations

A theoretical model is developed to determine simultaneously and in different ways thermal diffusivity and thermal conductivity of thin layers. This is done by using the accurate expression of the temperature distribution derived from the parabolic heat equation when the front surface of the thin layer is excited by a periodic heat flux, while the rear surface is maintained at one of three different types of boundary conditions: modulated periodic heat flux, modulated temperature, or constant temperature. Our approach exploits the modulation frequencies at which the normalized front surface temperature reaches its first maximum and first minimum. It is shown that (i) these characteristic frequencies can be used to obtain the thermal diffusivity of the finite layer under three different types of boundary conditions. (ii) The ratio between the values of the maxima and minima of the temperature can be utilized to determine the thermal conductivity of the finite layer. These two thermal properties are sensitive to the nature of the boundary conditions as well as the modulation frequency of the heat excitation. This paper provides a theoretical basis for the determination of the thermal diffusivity and thermal conductivity of the finite layer using laser-based heating photothermal techniques.

Analytical and Numerical Construction of Heat Wave Type Solutions to the Nonlinear Heat Equation with a Source

For a nonlinear parabolic heat equation we construct a heat wave type solution composed of the zero and nonnegative solutions joined continuously along the wave front. We prove the existence and uniqueness of an analytic solution to the problem with a given wave front in the cases of plane, circular, and spherical symmetry. The solution is constructed in the form of a characteristic series with recurrently defined coefficients. In the case of a power source, we show that the original problem can be reduced to the Cauchy problem for a second order ordinary differential equation and the solution is invariant. We present numerical results verified by using the constructed analytic solutions.

Identification of a multi-dimensional space-dependent heat source from boundary data

• Multi-dimensional space-dependent sources are reconstructed from boundary data. • Iterative regularization methods are developed. • Convergence and stability of numerical results are thoroughly investigated. We investigate the linear but ill-posed inverse problem of determining a multi-dimensional space-dependent heat source in the parabolic heat equation from Cauchy boundary data. This model is important in practical applications where the distribution of internal sources is to be monitored and controlled with care and accuracy from non-invasive and non-intrusive boundary measurements only. The mathematical formulation ensures that a solution of the inverse problem is unique but the existence and stability are still issues to be dealt with. Even if a solution exists it is not stable with respect to small noise in the measured boundary data hence the inverse problem is still ill-posed. The Landweber method is developed in order to restore stability through iterative regularization. Furthermore, the conjugate gradient method is also developed in order to speed up the convergence. An alternating direction explicit finite-difference method is employed for discretizing the well-posed problems resulting from these iterative procedures. Numerical results in two-dimensions are illustrated and discussed.

Reconstruction of time-dependent coefficients from heat moments

This paper investigates the inverse problems of simultaneous reconstruction of time-dependent thermal conductivity, convection or absorption coefficients in the parabolic heat equation governing transient heat and bio-heat thermal processes. Using initial and boundary conditions, as well as heat moments as over-determination conditions ensure that these inverse problems have a unique solution. However, the problems are still ill-posed since small errors in the input data cause large errors in the output solution. To overcome this instability we employ the Tikhonov regularization. A discussion of the choice of multiple regularization parameters is provided. The finite-difference method with the Crank–Nicolson scheme is employed as a direct solver. The resulting inverse problems are recast as nonlinear minimization problems and are solved using the lsqnonlin routine from the MATLAB toolbox. Numerical results are presented and discussed.

A Time-Dependent Optimal Harvesting Problem with Measure-Valued Solutions

The paper is concerned with the optimal harvesting of a marine park, which is described by a parabolic heat equation with Neumann boundary conditions and a nonlinear source term. We consider a cost functional, which is linear with respect to the control; hence the optimal solution can belong to the class of measure-valued control strategies. For each control function, we prove existence and stability estimates for solutions of the parabolic equation. Moreover, we prove the existence of an optimal solution. Finally, some numerical simulations conclude the paper.

Properties of a method of fundamental solutions for the parabolic heat equation

We show that a set of fundamental solutions to the parabolic heat equation, with each element in the set corresponding to a point source located on a given surface with the number of source points being dense on this surface, constitute a linearly independent and dense set with respect to the standard inner product of square integrable functions, both on lateral- and time-boundaries. This result leads naturally to a method of numerically approximating solutions to the parabolic heat equation denoted a method of fundamental solutions (MFS). A discussion around convergence of such an approximation is included.

Simultaneous determination of time-dependent coefficients and heat source

ABSTRACTThis article presents a numerical solution to the inverse problems of simultaneous determination of the time-dependent coefficients and the source term in the parabolic heat equation subject to overspecified conditions of integral type. The ill-posed problems are numerically discretized using the finite-difference method. The resulting system of nonlinear equations is solved numerically using the MATLAB toolbox routine lsqnonlin applied to minimizing the nonlinear Tikhonov regularization functional subject to simple physical bounds on the variables. Numerical examples are presented to illustrate the accuracy and stability of the solution.

Intermittence and Space-Time Fractional Stochastic Partial Differential Equations

We consider time fractional stochastic heat type equation $$\partial^{\beta}_{t}u_{t}(x)=-\nu(-{\Delta})^{\alpha/2} u_{t}(x)+I^{1-\beta}_{t}[\sigma(u)\overset{\cdot}{W}(t,x)] $$ in (d + 1) dimensions, where ν > 0, β ∈ (0, 1), α ∈ (0, 2], \(d<\min \{2,\beta ^{-1}\}\alpha \), \(\partial ^{\beta }_{t}\) is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process, \(\overset {\cdot }{W}(t,x)\) is space-time white noise, and \(\sigma :\mathbb {R}\to \mathbb {R}\) is Lipschitz continuous. The time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. We prove: (i) absolute moments of the solutions of this equation grows exponentially; and (ii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. These results extend the results of Foondun and Khoshnevisan (Electron. J. Probab. 14(21), 548–568, 2009) and Conus and Khoshnevisan (Probab. Theory Relat. Fields 152(3–4), 681–701, 2012) on the parabolic stochastic heat equations.

Simultaneous determination of time and space-dependent coefficients in a parabolic equation

This paper investigates a couple of inverse problems of simultaneously determining time and space dependent coefficients in the parabolic heat equation using initial and boundary conditions of the direct problem and overdetermination conditions. The measurement data represented by these overdetermination conditions ensure that these inverse problems have unique solutions. However, the problems are still ill-posed since small errors in the input data cause large errors in the output solution. To overcome this instability we employ the Tikhonov regularization method. The finite-difference method (FDM) is employed as a direct solver which is fed iteratively in a nonlinear minimization routine. Both exact and noisy data are inverted. Numerical results for a few benchmark test examples are presented, discussed and assessed with respect to the FDM mesh size discretization, the level of noise with which the input data is contaminated, and the chosen regularization parameters.

Identification of a time-dependent source term in nonlinear hyperbolic or parabolic heat equation

A problem of identification of the heat transfer sources in nonlinear media is considered with account for the time of heat flux relaxation. In order to solve the identification problem we use the approach based on the theory of inverse infinite-dimensional open dynamical systems (DS) or, in other words, input-state-output systems. For solving the identification problems we used finite-difference methods. The results of numerical simulation of the problems of internal heat sources identification in nonlinear hyperbolic or parabolic equation are presented.

An Accurate Numerical Integrator for the Solution of Black Scholes Financial Model Equation

In this paper the Black Scholes differential equation is transformed into a parabolic heat equation by appropriate change in variables. The transformed equation is semi-discretized by the Method of Lines (MOL). The evolving system of ordinary differential equations (ODEs) is integrated numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10–10.

Fourier, hyperbolic and relativistic heat transfer equations: a comparative analytical study

Parabolic heat equation based on Fourier's theory (FHE), and hyperbolic heat equation (HHE), has been used to mathematically model the temperature distributions of biological tissue during thermal ablation. However, both equations have certain theoretical limitations. The FHE assumes an infinite thermal energy propagation speed, whereas the HHE might possibly be in breach of the second law of thermodynamics. The relativistic heat equation (RHE) is a hyperbolic-like equation, whose theoretical model is based on the theory of relativity and which was designed to overcome these theoretical impediments. In this study, the three heat equations for modelling of thermal ablation of biological tissues (FHE, HHE and RHE) were solved analytically and the temperature distributions compared. We found that RHE temperature values were always lower than those of the FHE, while the HHE values were higher than the FHE, except for the early stages of heating and at points away from the electrode. Although both HHE and RHE are mathematically hyperbolic, peaks were only found in the HHE temperature profiles. The three solutions converged for infinite time or infinite distance from the electrode. The percentage differences between the FHE and the other equations were larger for higher values of thermal relaxation time in HHE.

Modeling of ultrashort pulsed laser irradiation in the cornea based on parabolic and hyperbolic heat equations using electrical analogy

Hyperbolic and parabolic heat equations are formulated to study a nonperfused homogeneous transparent cornea irradiated by high power and ultrashort pulsed laser in the Laser Thermo Keratoplasty (LTK) surgery. Energy absorption inside the cornea is modeled using the Beer–Lambert law that is incorporated as an exponentially decaying heat source. The hyperbolic and parabolic bioheat models of the tissue were solved by exploiting the mathematical analogy between thermal and electrical systems, by using robust circuit simulation program called Hspice to get the solutions of simultaneous RLC and RC transmission line networks. This method can be used to rapidly calculate the temperature in laser-irradiated tissue at time and space domain. It is found that internal energy gained from the irradiated field results in a rapid rise of temperature in the cornea surface during the early heating period, while the hyperbolic wave model predicts a higher temperature rise than the classical heat diffusion model. In addition, this paper investigates and examines the effect of some critical parameters such as relaxation time, convection coefficient, radiation, tear evaporation and variable thermal conductivity of cornea. Accordingly, it is found that a better accordance between hyperbolic and parabolic models will be achieved by time.

Application of the homotopy perturbation method to an inverse heat problem

Purpose – The purpose of this paper is to present a general framework of Homotopy perturbation method (HPM) for analytic inverse heat source problems. Design/methodology/approach – The proposed numerical technique is based on HPM to determine a heat source in the parabolic heat equation using the usual conditions. Then this shows the pertinent features of the technique in inverse problems. Findings – Using this HPM, a rapid convergent sequence which tends to the exact solution of the problem can be obtained. And the HPM does not require the discretization of the inverse problems. So HPM is a powerful and efficient technique in finding exact and approximate solutions without dispersing the inverse problems. Originality/value – The essential idea of this method is to introduce a homotopy parameter p which takes values from 0 to 1. When p=0, the system of equations usually reduces to a sufficiently simplified form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.

Time-fractional heat equations and negative absolute temperatures

The classical parabolic heat equation based on Fourier’s law implies infinite heat propagation speed. To remedy this physical flaw, the hyperbolic heat equation is used, but it may instead predict temperatures less than absolute zero. In recent years, fractional heat equations have been proposed as generalizations of heat equations of integer order. By simulating a 1D model problem of size on the order of a thermal energy carrier’s mean free path length, we have done a study of four fractional generalized Cattaneo equations from Compte and Metzler (1997) called GCE, GCE I, GCE II, and GCE III and also a fractional version of the parabolic heat equation. We have observed that when the fractional order is large enough, these equations give temperatures less than absolute zero. But if the fractional order is small enough, GCE I does not have this problem when the domain length is comparable to the mean free path length. With larger size, GCE I and GCE III also give non-oscillating solutions for both small and large values of the fractional order.

Numerical study of partial differential equations to estimate thermoregulation in human dermal regions for temperature dependent thermal conductivity

The paper deals with the temperature distribution in multi-layered human skin and subcutaneous tissues (SST). The model suggests the solution of parabolic heat equation together with the boundary conditions for the temperature distribution in SST by assuming the thermal conductivity as a function of temperature. The model formulation is based on singular non-linear boundary value problem and has been solved using finite difference method. The numerical results were found similar to clinical and computational results.

A projected iterative method based on integral equations for inverse heat conduction in domains with a cut

The Cauchy problem for the parabolic heat equation, consisting of the reconstruction of the solution from knowledge of the temperature and heat flux on a part of the boundary of the solution domain, is investigated in a planar region containing a cut. This linear inverse ill-posed problem is numerically solved using an iterative regularization procedure, where at each iteration step mixed Dirichlet–Neumann problems for the parabolic heat equation are used. Using the method of Rothe these mixed problems are reduced to a sequence of boundary integral equations. The integral equations have a square root singularity in the densities and logarithmic and hypersingularities in the kernels. Moreover, the mixed parabolic problems have singularities near the endpoints of the cut. Special techniques are employed to handle each of these (four) types of singularities, and analysis is performed in weighted spaces of square integrable functions. Numerical examples are included showing that the proposed regularizing procedure gives stable and accurate approximations.