Discrete Heat Equation Research Articles

Discrete heat equation for a periodic layered system with allowance for the interfacial thermal resistance: General formulation and dispersion analysis.

Discrete heat equations for the multilayered periodic systems with allowance for the thermal resistance between the layers and corresponding dispersion relations in ω-k space have been derived and analyzed. The discrete equations imply a finite velocity of thermal disturbances and guarantee the positiveness of the solutions. Analytical expressions for the attenuation distance, and phase and group velocities have been obtained as functions of frequency and thermal resistance between the discrete layers. These functions demonstrate unusual behavior at high frequency compared to the continuum case. Furthermore, the maximum allowed frequency and wave number for the discrete heat equation are limited, whereas there are no such limits in a continuum. The discrete equation contains an infinite hierarchy of continuous partial differential equations, which starts with the Fourier law, proceeds with the hyperbolic equation, the Guyer-Krumhansl (or Jeffreys type) equation, and then with higher-order equations. The partial differential equations with a finite number of terms are only approximations of the discrete equation, which implies that on the ultrashort space and timescales the discrete approach is preferable. This work provides a relatively simple, easy-to-adopt, conceptual tool, together with analytical expressions allowing one to study ultrafast wavelike heat conduction regimes in periodic multilayered metamaterials.

Artificial patterns in spatially discrete models of cell migration and how to mitigate them

Several discrete models for diffusive motion are known to exhibit checkerboard artifacts, absent in their continuous analogues. We study the origins of the checkerboard artifact in the discrete heat equation and show that this artifact decays exponentially in time when following either of two strategies: considering the present state of each lattice site to determine its own future state (self-contributions), or using non-square lattice geometries. Afterwards, we examine the effects of these strategies on nonlinear models of biological cell migration with two kinds of cell-cell interactions: adhesive and polar velocity alignment. In the case of adhesive interaction, we show that growing modes related to pattern formation overshadow artifacts in the long run; nonetheless, artifacts can still be completely prevented following the same strategies as in the discrete heat equation. On the other hand, for polar velocity alignment we show that artifacts are not only strengthened, but also that new artifacts can arise in this model which were not observed in the previous models. We find that the lattice geometry strategy works well to alleviate artifacts. However, the self-contribution strategy must be applied more carefully: lattice sites should contribute to both their own density and velocity values, and their own velocity contribution should be high enough. With this work, we show that these two strategies are effective for preventing artifacts in spatial models based on the discrete continuity equation.

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 $\hbar \mathbb {Z}^n$ .

Stability Analysis for Discrete Fractional Order Steady-State Heat Equation with Neumann Boundary Conditions

Stability Analysis for Discrete Fractional Order Steady-State Heat Equation with Neumann Boundary Conditions

Inverse Problems for Discrete Heat Equations and Random Walks for a Class of Graphs

We study the inverse problem of determining a finite weighted graph from the source-to-solution map on a vertex subset for heat equations on graphs, where the time variable can be either discrete or continuous. We prove that this problem is equivalent to the discrete version of the inverse interior spectral problem, provided that there does not exist a nonzero eigenfunction of the weighted graph Laplacian vanishing identically on . In particular, we consider inverse problems for discrete-time random walks on finite graphs. We show that under a novel geometric condition (called the Two-Points Condition), the graph structure and the transition matrix of the random walk can be uniquely recovered from the distributions of the first passing times on , or from the observation on of one realization of the random walk.

Construction and analysis of a discrete heat equation using dynamic consistency: The meso-scale limit

We present and analyze a new derivation of the meso-level behavior of a discrete microscopic model of heat transfer. This construction is based on the principle of dynamic consistency. Our work reproduces and corrects, when needed, all the major previous expressions which provide modifications to the standard heat PDE. However, unlike earlier efforts, we do not allow the microscopic level parameters to have zero limiting values. We also give insight into the difficulties of constructing physically valid heat equations within the framework of the general mathematically inequivalent of difference and differential equations.

Non-intersecting Path Constructions for TASEP with Inhomogeneous Rates and the KPZ Fixed Point

We consider a discrete-time TASEP, where each particle jumps according to Bernoulli random variables with particle-dependent and time-inhomogeneous parameters. We use the combinatorics of the Robinson–Schensted–Knuth correspondence and certain intertwining relations to express the transition kernel of this interacting particle system in terms of ensembles of weighted, non-intersecting lattice paths and, consequently, as a marginal of a determinantal point process. We next express the joint distribution of the particle positions as a Fredholm determinant, whose correlation kernel is given in terms of a boundary-value problem for a discrete heat equation. The solution to such a problem finally leads us to a representation of the correlation kernel in terms of random walk hitting probabilities, generalizing the formulation of Matetski et al. (Acta Math. 227(1):115–203, 2021) to the case of both particle- and time-inhomogeneous rates. The solution to the boundary value problem in the fully inhomogeneous case appears with a finer structure than in the homogeneous case.

The q-deformed heat equation and q-deformed diffusion equation with q-translation symmetry

In this paper we consider the discrete heat equation with a certain non-uniform space interval which is related to q-addition appearing in the non-extensive entropy theory. By taking the continuous limit, we obtain the q-deformed heat equation. Similarly, we obtain the solution of the q-deformed diffusion equation

A Parabolic Harnack Principle for Balanced Difference Equations in Random Environments

We consider difference equations in balanced, i.i.d. environments which are not necessary elliptic. In this setting we prove a parabolic Harnack inequality (PHI) for non-negative solutions to the discrete heat equation satisfying a (rather mild) growth condition, and we identify the optimal Harnack constant for the PHI. We show by way of an example that a growth condition is necessary and that our growth condition is sharp. Along the way we also prove a parabolic oscillation inequality and a (weak) quantitative homogenization result, which we believe to be of independent interest.

Optimal H2 Moment Matching-Based Model Reduction for Linear Systems through (Non)convex Optimization

In this paper, we compute a (local) optimal reduced order model that matches a prescribed set of moments of a stable linear time-invariant system of high dimension. We fix the interpolation points and parametrize the models achieving moment-matching in a set of free parameters. Based on the parametrization and using the H2-norm of the approximation error as the objective function, we derive a nonconvex optimization problem, i.e., we search for the optimal free parameters to determine the model yielding the minimal H2-norm of the approximation error. Furthermore, we provide the necessary first-order optimality conditions in terms of the controllability and the observability Gramians of a minimal realization of the error system. We then propose two gradient-type algorithms to compute the (local) optimal models, with mathematical guarantees on the convergence. We also derive convex semidefinite programming relaxations for the nonconvex Problem, under the assumption that the error system admits block-diagonal Gramians, and derive sufficient conditions to guarantee the block diagonalization. The solutions resulting at each step of the proposed algorithms guarantee the achievement of the imposed moment matching conditions. The second gradient-based algorithm exhibits the additional property that, when stopped, yields a stable approximation with a reduced H2-error norm. We illustrate the theory on a CD-player and on a discretized heat equation.

Nonlocal thermal diffusion in one-dimensional periodic lattice

Heat conduction in micro-structured rods can be simulated with unidimensional periodic lattices. In this paper, scale effects in conduction problems, and more generally in diffusion problems, are studied from a one-dimensional thermal lattice. The thermal lattice is governed by a mixed differential-difference equation, accounting for some spatial microstructure. Exact solutions of the linear time-dependent spatial difference equation are derived for a thermal lattice under initial uniform temperature field with some temperature perturbations at the boundary. This discrete heat equation is approximated by a continuous nonlocal heat equation built by continualization of the thermal lattice equations. It is shown that such a nonlocal heat equation may be equivalently obtained from a nonlocal Fourier's law. Exact solutions of the thermal lattice problem (discrete heat equation) are compared with the ones of the local and nonlocal heat equation. An error analysis confirms the accurate calibration of the length scale of the nonlocal diffusion law with respect to the lattice spacing. The nonlocal heat equation may efficiently capture scale effect phenomena in periodic thermal lattices.

Electromagnetic and Thermal Analysis of a ZnO Element of Transmission‐Line Arrester for a Lightning Surge Current

The voltage generated across a zinc‐oxide (ZnO) element, in which a lightning impulse current flows, has been computed with the finite‐difference time‐domain (FDTD) method. Also, the distribution of temperature in the ZnO element has been computed on the basis of FDTD‐computed conduction current densities with a discretized heat equation. The ZnO element having a thickness of 37 mm and a diameter of 34 mm is represented with many 1 mm × 1 mm × 1 mm cubic cells, each of which has a nonlinear resistivity in each of the x, y, and z directions dependent on the electric field in each direction and the temperature in each cell. Waveforms of voltage appearing across the ZnO element and the temperature on the side surface of the ZnO element computed for a short lightning impulse current having a magnitude of 66 kA agree reasonably well with the corresponding measured ones. © 2021 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Large time behaviour for the heat equation on [formula omitted], moments and decay rates

The paper is devoted to understand the large time behaviour and decay of the solution of the discrete heat equation in the one dimensional mesh Z on ℓp spaces, and its analogies with the continuous-space case. We do a deep study of the moments of the discrete gaussian kernel (which is given in terms of Bessel functions), in particular the mass conservation principle; that is reflected on the large time behaviour of solutions. We prove asymptotic pointwise and ℓp decay results for the fundamental solution. We use that estimates to get rates on the ℓp decay and large time behaviour of solutions. For the ℓ2 case, we get optimal decay by use of Fourier techniques.

An algebraic approach to discrete time integrability

We propose the systematic construction of classical and quantum two-dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical and quantum algebras. This is a novel construction that leads to the derivation of fully discrete integrable systems governed by sets of consistent integrable non-linear space-time difference equations. To illustrate the proposed methodology, we derive two versions of the fully discrete non-linear Schrödinger type system. The first one is based on the existence of a rational r-matrix, whereas the second one is the fully discrete Ablowitz–Ladik model and is associated to a trigonometric r-matrix. The Darboux-dressing method is also applied for the first discretization scheme, mostly as a consistency check, and solitonic as well as general solutions, in terms of solutions of the fully discrete heat equation, are also derived. The quantization of the fully discrete systems is then quite natural in this context and the two-dimensional quantum lattice is thus also examined.

HEAT EQUATION OBTAINED BY l -DIFFERENCE OPERATOR CHANGE OF TIME WITH TWO VARIABLES

We investigate the generalized partial difference operator and propose a model of it in discrete heat equation. For example, the temperature in an object changes with time and the position within the object. In this paper Generalized heat equation obtained by two dimensional l -difference operator change of time with two variables

On a population model in discrete periodic habitat. I. Spreading speed and optimal dispersal strategy

Starting from an age-structured diffusive population growth law for single species in a discrete and periodic habitat, we formulate a stage structured population model with spatially periodic dispersal, mortality and recruitment. With a KPP type setting, after establishing the fundamental solution of a discretized heat equation with spatially periodic dispersal, we apply some recently developed dynamical system theories to obtain the existence of the spreading speed and its coincidence with the minimal speed of pulsating waves, as well as the variational characterization of the speed, by which we further analyze how the habitat periodicity influences the speed. In particular, there is a unique optimal dispersal strategy to maximize the speed.

Bounds on Rate of Convergence for the Shuffled Discrete Heat Equation in Zd

Bounds on Rate of Convergence for the Shuffled Discrete Heat Equation in Zd

Electromagnetic and thermal analysis of a multilayer CFRP panel struck by lightning with the FDTD method

The transient distribution of heat generated in a 16‐layer carbon fiber‐reinforced plastic (CFRP) panel, in which a lightning current with a magnitude of 3 kA, a rise time of 6.4 µs, and a time to half‐peak value of 69 µs is injected, has been computed. Each layer of the CFRP panel has a fiber axis direction of 45, 90, −45, or 0° and, therefore, has anisotropic electrical and thermal conductivities. This multilayer CFRP panel has been represented with a conductivity matrix consisting of off‐diagonal elements in the finite‐difference time domain (FDTD) simulation. The heat is computed, on the basis of FDTD‐computed conduction current densities, using a discretized heat equation. © 2019 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

A discrete Hardy’s uncertainty principle and discrete evolutions

In this paper we give a discrete version of Hardy’s uncertainty principle, by using complex variable arguments, as in the classical proof of Hardy’s principle. Moreover, we give an interpretation of this principle in terms of decaying solutions to the discrete Schrodinger and heat equations.

Discrete Heat Equation model of Rod by Partial Fibonacci Difference Operator with shift values

Partial Fibonacci difference equation is introduced and subjected to investigation in discrete heat equation by having recourse to Fibonacci difference operator with shift values in this paper. By having Fourier law of cooling as its basis, the heat transfer in the long rod is investigated and the solutions obtained are validated by MATLAB.