Classical Heat Equation Research Articles

Contractivity properties of a quantum diffusion semigroup

We consider a quantum generalization of the classical heat equation and study contractivity properties of its associated semigroup. We prove a Nash inequality and a logarithmic Sobolev inequality. The former leads to an ultracontractivity result. This in turn implies that the largest eigenvalue and the purity of a state with positive Wigner function, evolving under the action of the semigroup, decrease at least inverse polynomially in time, while its entropy increases at least logarithmically in time.

On weighted mixed-norm Sobolev estimates for some basic parabolic equations

Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation \begin{document} $ \partial_tu=\Delta u+f, $ \end{document} the harmonic oscillator evolution equation \begin{document}$\partial_tu=\Delta u-|x|^2u+f, $ \end{document} and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-\Delta)^su=f$ and $(\partial_t-\Delta+|x|^2)^su=f$, for $0

Controllability of a class of heat equations with memory in one dimension

This paper addresses a study of the controllability for a class of heat equations with memory in one spacial dimension. Unlike the classical heat equation, a heat equation with memory in general is not null controllable. There always exists a set of initial values such that the property of the null controllability fails. Also, one does not know whether there are nontrivial initial values, which can be driven to zero with a boundary control. In this paper, we give a characterization of the set of such nontrivial initial values. On the other hand, if a moving control is imposed on this system with memory, we prove the null controllability of it in a suitable state space for any initial value. Copyright © 2016 John Wiley & Sons, Ltd.

Lie Symmetries of (1+2) Nonautonomous Evolution Equations in Financial Mathematics

We analyse two classes of ( 1 + 2 ) evolution equations which are of special interest in Financial Mathematics, namely the Two-dimensional Black-Scholes Equation and the equation for the Two-factor Commodities Problem. Our approach is that of Lie Symmetry Analysis. We study these equations for the case in which they are autonomous and for the case in which the parameters of the equations are unspecified functions of time. For the autonomous Black-Scholes Equation we find that the symmetry is maximal and so the equation is reducible to the ( 1 + 2 ) Classical Heat Equation. This is not the case for the nonautonomous equation for which the number of symmetries is submaximal. In the case of the two-factor equation the number of symmetries is submaximal in both autonomous and nonautonomous cases. When the solution symmetries are used to reduce each equation to a ( 1 + 1 ) equation, the resulting equation is of maximal symmetry and so equivalent to the ( 1 + 1 ) Classical Heat Equation.

Memory Effects Due to Fractional Time Derivative and Integral Space in Diffusion Like Equation Via Haar Wavelets

Memory and hereditary effects due to fractional time derivative are combined with the global behaviours due to space integral term. Haar wavelet operational matrix is adjusted to solve diffusion like equations with time fractional derivative, space derivatives and integral terms. The fractional derivative is understood in the Caputo sense. The memory behaviours is included in all the points of the domain due to the existence of space integral term and the inverse fractional operator treatment and this is ilustrated in error graphs introduced. A general example with four subproblems ranging from the simple classical heat equation to the fractional time diffusion equation with global integral term is proposed and the calculated results are displayed graphically.

Studying One of the Problems of Wave Filtration

Motions of particles through a permeable barrier are studied from the standpoint of Schrodinger wave dynamics. We consider time evolution of a substance concentration spot in the vicinity of one or two potential barriers. To solve the problem we developed a computer technology for integration of the non-stationary Schrodinger equation. The technology is based on symbol mapping and the matrix exponential method which was applied earlier to solve the classical heat equation. The developed theory is relevant to development of nanotechnology, in particular, to studies on permeability of nanoporous membranes.

Controllability of linear and semilinear non-diagonalizable parabolic systems

This paper is concerned with the controllability of some (linear and semilinear) non- diagonalizable parabolic systems of PDEs. We will show that the well known null controllability prop- erties of the classical heat equation are also satisfied by these systems at least when there are as many scalar controls as equations and some (maybe technical) conditions are satisfied. We will also show that, in some particular situations, the number of controls can be reduced. The minimal amount is then determined by a Kalman rank condition.

A mathematical model of multi-dimensional freeze-drying for food products

A simple mathematical model was developed to predict the kinetics of freeze-drying where sublimation progresses multi-dimensionally in a product, and the model was adopted to the freeze-drying of peeled apple cubes. This model consists of classical heat and mass transfer equations, which are solved by assuming a quasi-steady-state energy balance at the sublimation interface. The radiative heat coefficient and mass transfer coefficient between the chamber and the condenser are included in the model, and were experimentally obtained by carrying out the ice sublimation tests with the freeze-dryer employed. The mass transfer property is a specification of a given freeze-dryer. This work suggested that it strongly influenced the drying kinetics and it would be an important parameter to meet scaling-up issue. When the sublimation progresses multi-dimensionally, the surface areas of the product exterior, the product bottom and the sublimation interface varied as a function of the extent of drying. These relationships were obtained from experiments with freeze-drying of peeled apple cubes and employed for estimating the mean thickness of the dried and frozen layers with a hollow spherical model proposed in the study. The model equations were solved using commercial spreadsheet software, and the weight loss during drying was predicted as a function of time. The calculation results were in good agreement with the experimental data. This approach allows to set a heating program and/or localized variances of microstructural parameters by manual input, and easily simulate drying kinetics and temperature history. This makes big advantage for planning a desired drying operation.

The Non-Archimedean Stochastic Heat Equation Driven by Gaussian Noise

We introduce and study a new class of non-Archimedean stochastic pseudodifferential equations. These equations are the non-Archimedean counterparts of the classical stochastic heat equations. We show the existence and uniqueness of mild random field solutions for these equations.

Fundamental Solution via Invariant Approach for a Brain Tumor Model and its Extensions

Abstract We firstly show how one can use the invariant criteria for a scalar linear (1+1) parabolic partial differential equations to perform reduction under equivalence transformations to the first Lie canonical form for a class of brain tumor models. Fundamental solution for the underlying class of models via these transformations is thereby found by making use of the well-known fundamental solution of the classical heat equation. The closed-form solution of the Cauchy initial value problem of the model equations is then obtained as well. We also demonstrate the utility of the invariant method for the extended form of the class of brain tumor models and find in a simple and elegant way the possible forms of the arbitrary functions appearing in the extended class of partial differential equations. We also derive the equivalence transformations which completely classify the underlying extended class of partial differential equations into the Lie canonical forms. Examples are provided as illustration of the results.

Well-posedness for a model derived from an attraction–repulsion chemotaxis system

In this paper, we are interested in a model derived from an attraction–repulsion chemotaxis system in high dimensions:{∂tu−Δu=−β1∇⋅(u∇v)+β2∇⋅(u∇w),x∈Rn,t>0,λ1v−Δv=u,x∈Rn,t>0,λ2w−Δw=u,x∈Rn,t>0,u(x,0)=u0(x),x∈Rn, with the parameters β1≥0, β2≥0, λ1>0, λ2>0 and nonnegative initial data u0(x)∈L1(Rn)∩L∞(Rn). We prove that a global bounded solution exists when the repulsion prevails over the attraction in the sense of β1<β1. Moreover, we give the smoothness of the solution and obtain its decay rates in Ws,p(Rn), which coincide with the ones of the classical heat equation. Conversely, when n=2, β1>β2, we prove that the finite time blow-up may occur.

Time-stepping error bounds for fractional diffusion problems with non-smooth initial data

We apply the piecewise constant, discontinuous Galerkin method to discretize a fractional diffusion equation with respect to time. Using Laplace transform techniques, we show that the method is first order accurate at the nth time level tn, but the error bound includes a factor tn−1 if we assume no smoothness of the initial data. We also show that for smoother initial data the growth in the error bound as tn decreases is milder, and in some cases absent altogether. Our error bounds generalize known results for the classical heat equation and are illustrated for a model problem.

A semi-analytical approach to Green׳s functions for heat equation in regions of irregular shape

Initial-boundary-value problems are considered for the classical two-dimensional heat equation in regions of irregular configuration. A semi-analytical algorithm is proposed to accurately compute profiles of Green׳s function for such problems. The algorithm is based on a modification of the standard boundary integral equation method. To make the modification efficient, analytical representations of Green׳s functions are required for relevant regularly shaped regions. These are obtained in a closed form and employed then as kernels of the corresponding heat potentials, reducing the problem to a regular integral equation on a part of a boundary of the considered region.

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.

Diffusion processes on an interval under linear moment conditions

We discuss a class of diffusion-type partial differential equations on a bounded interval and discuss the possibility of replacing the boundary conditions by certain linear conditions on the moments of order 0 (the total mass) and of another arbitrarily chosen order n. Each choice of n induces the addition of a certain potential in the equation, the case of zero potential arising exactly in the special case of n=1 corresponding to a condition on the barycenter. In the linear case we exploit smoothing properties and perturbation theory of analytic semigroups to obtain well-posedness for the classical heat equation (with said conditions on the moments). Long time behavior is studied for both the linear heat equation with potential and certain nonlinear equations of porous medium or fast diffusion type. In particular, we prove polynomial decay in the porous medium range and exponential decay in the fast diffusion range, respectively.

Solving Schrödinger equation via Tartaglia/Pascal triangle: a possible link between stochastic processing and quantum mechanics

In a recent paper (Farina et al. in Signal Image Video Process 1–16, 2011), it was shown a clean connection between the solution of the classical heat equation and the Tartaglia/Pascal/Yang-Hui triangle. When the time variable in the heat equation is substituted with the imaginary time, the heat equation becomes the Schrodinger equation of the quantum mechanics. So, a conjecture was put forward about a connection between the solution of the Schrodinger equation and a suitable generalization of the Tartaglia triangle. This paper proves that this conjecture is true and shows a new—as far as the authors are aware—result concerning the generalization of the classical Tartaglia triangle by introducing the “complex-valued Tartaglia triangle.” A “complex-valued Tartaglia triangle” is just the square root of an ordinary Tartaglia triangle, with a suitable phase factor calculated via a discretized version of the ordinary continuous case of the Schrodinger equation. So, taking the square of this complex-valued Tartaglia triangle gives back exactly the probability distribution of a discrete random walk. We also discuss about potential connections between the theories of stochastic processes and quantum mechanics: a connection debated since the inception of the theories and still lively hot today.

On conduction heat transfer in metals

The classical heat equation is characterized both by the instantaneous propagation of a physical interaction and by the lack of a particle that carries the heat. Metals have a particular feature: their atoms form a lattice with electrons traveling through the solid. This paper provides a conduction heat model for metals. The model replaces atoms by potential barrier and assigns the heat transport to electrons and, in some cases, photons. To emulate the transitory solution of Schrödinger's equation, the potential barrier is fragmented in others so that, the electron is losing energy packets while interacting with the sub-barriers. With this approach, the energy distribution is obtained for electrons and, in addition, it is derived a velocity expression for the heat spread through a metal. Last but not least, the classical solution of the heat equation is obtained but now the electron energy limits the heat diffusion avoiding the infinite range of the classical solution.

Symmetry properties of the causal heat equations

Symmetry properties of the causal (hyperbolic) heat equations are studied. Symmetry groups of two variants of hyperbolic equations are calculated. The results obtained are analysed and compared with known properties of the classical heat equation. These findings may be considered as possible arguments in favour of one of the causal heat conduction models.

A weighted quasilinear equation related to the mean curvature operator

This paper studies a weighted quasilinear perturbation, through the mean curvature flow operator, of the classical linear heat equation. The mean curvature has the effect of maintaining bounded all classical positive steady states of the model, though their derivatives must be somewhere unbounded. The dynamics of the positive solutions of the model is fully described.

Superconvergence of a Discontinuous Galerkin Method for Fractional Diffusion and Wave Equations

We consider an initial-boundary value problem for $\partial_tu-\partial_t^{-\alpha}\nabla^2u=f(t)$, that is, for a fractional diffusion ($-1<\alpha<0$) or wave ($0<\alpha<1$) equation. A numerical solution is found by applying a piecewise-linear, discontinuous Galerkin (DG) method in time combined with a piecewise-linear, conforming finite element method in space. The time mesh is graded appropriately near $t=0$, but the spatial mesh is quasi-uniform. Previously, we proved that the error, measured in the spatial $L_2$-norm, is of order $k^{2+\alpha_-}+h^2\ell(k)$, uniformly in $t$, where $k$ is the maximum time step, $h$ is the maximum diameter of the spatial finite elements, $\alpha_-=\min(\alpha,0)\le0$, and $\ell(k)=\max(1,|\log k|)$. Here, we prove convergence of order $k^{3+2\alpha_-}\ell(k)+h^2$ at each time level $t_n$ for $-1<\alpha<1$. Thus, if $-1/2<\alpha<1$, then the DG solution is superconvergent, which generalizes a known result for the classical heat equation (i.e., the case $\alpha=0$). A simple postprocessing step employing Lagrange interpolation leads to superconvergence for any $t$. Numerical experiments indicate that our theoretical error bound is pessimistic if $\alpha<0$. Ignoring logarithmic factors, we observe that the error in the DG solution at $t=t_n$, and after postprocessing at all $t$, is of order $k^{3+\alpha_-}+h^2$ for $-1<\alpha<1$.