Heat Equation Research Articles

Fujita exponent for the global-in-time solutions to a semilinear heat equation with non-homogeneous weights

Fujita exponent for the global-in-time solutions to a semilinear heat equation with non-homogeneous weights

Analog quantum simulation of partial differential equations

Abstract Quantum simulators were originally proposed for simulating one partial differential equation in particular -- Schrodinger's equation. Can quantum simulators also efficiently simulate other partial differential equations? While most computational methods for partial differential equations -- both classical and quantum -- are digital (they must be discretised first), partial differential equations have continuous degrees of freedom. This suggests that an analog representation can be more natural. While digital quantum degrees of freedom are usually described by qubits, the analog or continuous quantum degrees of freedom can be captured by qumodes. Based on a method called Schrodingerisation, we show how to directly map D-dimensional linear partial differential equations onto a (D+1)-qumode quantum system where analog or continuous-variable Hamiltonian simulation on D+1 qumodes can be used. This very simple methodology does not require one to discretise partial differential equations first, and it is not only applicable to linear partial differential equations but also to some nonlinear partial differential equations and systems of nonlinear ordinary differential equations. We show some examples using this method, including the Liouville equation, heat equation, Fokker-Planck equation, Black-Scholes equations, wave equation and Maxwell's equations. We also devise new protocols for linear partial differential equations with random coefficients, important in uncertainty quantification, where it is clear how the analog or continuous-variable framework is most natural. This also raises the possibility that some partial differential equations may be simulated directly on analog quantum systems by using Hamiltonians natural for those quantum systems.

Temporal properties of the stochastic fractional heat equation with spatially-colored noise

Consider the stochastic partial differential equation ∂ ∂ t u t ( x ) = − ( − Δ ) α 2 u t ( x ) + b ( u t ( x ) ) + σ ( u t ( x ) ) F ˙ ( t , x ) , t ≥ 0 , x ∈ R d , \begin{equation*} \frac {\partial }{\partial t}u_t(\boldsymbol {x})= -(-\Delta )^{\frac {\alpha }{2}}u_t(\boldsymbol {x}) +b\left (u_t(\boldsymbol {x})\right )+\sigma \left (u_t(\boldsymbol {x})\right ) \dot F(t, \boldsymbol {x}), \quad t\ge 0,\: \boldsymbol {x}\in \mathbb R^d, \end{equation*} where − ( − Δ ) α 2 -(-\Delta )^{\frac {\alpha }{2}} denotes the fractional Laplacian with power α 2 ∈ ( 1 2 , 1 ] \frac {\alpha }{2}\in (\frac 12,1] , and the driving noise F ˙ \dot F is a centered Gaussian field which is white in time and has a spatial homogeneous covariance given by the Riesz kernel. We study the detailed behavior of the approximation of the temporal gradient u t + ε ( x ) − u t ( x ) u_{t+{\varepsilon }}(\boldsymbol {x})-u_t(\boldsymbol {x}) at any fixed t > 0 t > 0 and x ∈ R d \boldsymbol {x}\in \mathbb R^d , as ε ↓ 0 {\varepsilon }\downarrow 0 . As applications, we deduce Khintchin’s law of iterated logarithm, Chung’s law of iterated logarithm, and a result on the q q -variations of the temporal process { u t ( x ) } t ≥ 0 \{u_t(\boldsymbol {x})\}_{ t \ge 0} of the solution, where x ∈ R d \boldsymbol {x}\in \mathbb R^d is fixed.

The Burgers-type equation driven by a stochastic measure

We study the one-dimensional equation driven by a stochastic measure μ \mu . For μ \mu we assume only σ \sigma -additivity in probability. Our results imply the global existence and uniqueness of the solution to the heat equation and the local existence and uniqueness of the solution to the Burgers equation. The averaging principle for such equation is studied.

On inverse source term for heat equation with memory term

Abstract In this article, we first study the inverse source problem for parabolic with memory term. We show that our problem is ill-posed in the sense of Hadamard. Then, we construct the convergence result when the parameter tends to zero. We also investigate the regularized solution using the Fourier truncation method. The error estimate between the regularized solution and the exact solution is obtained.

Sparse representations of approximation to identity via time–space fractional heat equations

In this paper, we investigate sparse representations of approximation to identity via time–space fractional heat equations: { ∂ t β u ( t , x ) = − ν ( − Δ ) α / 2 u ( t , x ) , ( t , x ) ∈ R + 1 + n ; u ( 0 , x ) = f ( x ) ,   x ∈ R n . Due to the time-fractional derivative, the semigroup property is invalid for the solutions u ( ⋅ , ⋅ ) to the above problem. This deficiency makes it difficult to verify the boundary vanishing condition of u ( ⋅ , ⋅ ) , which is essential for getting the sparse representations. We develop a new method to avoid using the semigroup property. The analogous results are obtained for stochastic time–space fractional heat equations. As an application, we apply the adaptive Fourier decomposition to establish sparse representations of the solutions to the concerned equations.

What Rayleigh numbers are achievable under Oberbeck–Boussinesq conditions?

The validity of the Oberbeck–Boussinesq (OB) approximation in Rayleigh–Bénard (RB) convection is studied using the Gray & Giorgini (Intl J. Heat Mass Transfer, vol. 19, 1976, pp. 545–551) criterion that requires that the residuals, i.e. the terms that distinguish the full governing equations from their OB approximations, are kept below a certain small threshold $\hat {\sigma }$ . This gives constraints on the temperature and pressure variations of the fluid properties (density, absolute viscosity, specific heat at constant pressure $c_p$ , thermal expansion coefficient and thermal conductivity) and on the magnitudes of the pressure work and viscous dissipation terms in the heat equation, which all can be formulated as bounds regarding the maximum temperature difference in the system, $\varDelta$ , and the container height, $L$ . Thus for any given fluid and $\hat {\sigma }$ , one can calculate the OB-validity region (in terms of $\varDelta$ and $L$ ) and also the maximum achievable Rayleigh number ${{Ra}}_{max,\hat {\sigma }}$ , and we did so for fluids water, air, helium and pressurized SF $_6$ at room temperature, and cryogenic helium, for $\hat {\sigma }=5\,\%$ , $10\,\%$ and $20\,\%$ . For the most popular fluids in high- ${{Ra}}$ RB measurements, which are cryogenic helium and pressurized SF $_6$ , we have identified the most critical residual, which is associated with the temperature dependence of $c_p$ . Our direct numerical simulations (DNS) showed, however, that even when the values of $c_p$ can differ almost twice within the convection cell, this feature alone cannot explain a sudden and strong enhancement in the heat transport in the system, compared with its OB analogue.

Analysis of a positivity-preserving splitting scheme for some semilinear stochastic heat equations

We construct a positivity-preserving Lie--Trotter splitting scheme with finite difference discretization in space for approximating the solutions to a class of semilinear stochastic heat equations with multiplicative space-time white noise. We prove that this explicit numerical scheme converges in the mean-square sense, with rate $1/4$ in time and rate $1/2$ in space, under appropriate CFL conditions. Numerical experiments illustrate the superiority of the proposed numerical scheme compared with standard numerical methods which do not preserve positivity. p, li { white-space: pre-wrap; }

Transient Heat Transfer Analysis in Metal Plates with Variable Thickness

Nonlinear transient heat transfer via conduction–radiation is a dynamic topic of long-standing interest with applications ranging from aeronautical and mechanical engineering to industrial and civil security. To gain a better understanding of the performances of materials having thermal proprieties that change during nonlinear heat transfer, several studies using the finite element method (FEM) have been conducted. Such studies apply nonlinear thermal material characteristics to describe the complete system under different loading conditions in each region by adjusting the temperature values for the other three edges and the thickness parameter with Dirichlet boundary conditions. As a result, while modeling and simulating temperature distributions for such situations, nonlinearities generated by temperature-dependent thermal conductivity must be considered. In this work, we focus on the analysis of coupled transient heat transfer through two metal plates with temperature-dependent thermal characteristics in which the temperature is fixed along the bottom edge and heat is transferred from both the top and bottom faces of the two plates. FEM is employed to solve the nonlinear heat equation and compute the temperature as a function of time for variable thickness. The study examines the effect of modifying the thickness parameter values on the temperature distribution over time for various edge values over 5,000 s.

Local solvability and dilation-critical singularities of supercritical fractional heat equations

We consider the Cauchy problem for fractional semilinear heat equations with supercritical nonlinearities and establish both necessary conditions and sufficient conditions for local-in-time solvability. We introduce the notion of a dilation-critical singularity (DCS) of the initial data and show that such singularities always exist for a large class of supercritical nonlinearities. Moreover, we provide exact formulae for such singularities.

Thermo-mechanical modeling of pancakelike domes on Venus

In this paper, we present a mathematical model aimed at describing both the effusive and relaxing phase of pancakelike lava domes on the Venus surface. Our model moves from the recent paper by Quick et al. [“New approaches to inferences for steep-sided domes on Venus,” J. Volcanol. Geotherm. Res. 319, 93–105 (2016)] but generalizes it under several respects. Indeed, we consider a temperature field, playing a fundamental role in the flow evolution, whose dynamics is governed by the heat equation. In particular, we suggest that the main mechanism that drives cooling is radiation at the dome surface. We obtain a generalized form of the equation describing the dome shape, where the dependence of viscosity on temperature is taken into account. Still following Quick et al. [“New approaches to inferences for steep-sided domes on Venus,” J. Volcanol. Geothermal Res. 319, 93–105 (2016)], we distinguish an isothermal relaxing phase preceded by a non-isothermal (cooling) effusive phase, but the fluid mechanical model, developed in an axisymmetric thin-layer approximation, takes into account both shear thinning and thermal effects. In both cases (relaxing and effusive phase), we show the existence of self-similar solutions. In particular, this allows to obtain a likely scenario of the volumetric flow rate which originated this kind of domes. Indeed, the model predicts a time varying discharge, which is maximum at the beginning of the formation process and decreases until vanishing when the effusive phase is over. The model, in addition to fitting well the dome shape, suggests a possible forming scenario, which may help the largely debated questions about the emplacement and lava composition of these domes.

Numerical simulation and optimization of Pervaporation process based on Heat-Mass-Flow coupling

Pervaporation (PV) has the advantages of low energy consumption and high separation efficiency, but the trade-off effect between flux and separation factor restricts its large-scale application. Based on the conservation equations of mass, momentum and heat, a transient model of the whole PV process coupled with heat-mass-flow was established in this paper. The effects of feed temperature, feed concentration and feed flow rate on heat and mass transfer in PV process were analyzed. TEOS crosslinked PDMS/PTFE-PVDF membrane was prepared by scraping coating method. The reliability of the model was verified by taking n-butanol/water separation system as an example. The concentration polarization and temperature boundary layer under different operating conditions are discussed. The effect of operating conditions on the separation performance was evaluated by response surface method, and the optimal operating conditions are obtained. The results show that the concentration polarization of the membrane surface becomes more obvious with the increase of feed concentration. The temperature boundary layer decreases with the increase of feed flow rate. Surprisedly, there is a significant temperature gradient in the membrane, and the maximum temperature difference between the two sides of the membrane is as high as 1.02 K. Diffusion in the membrane is the rate controlling factor of PV process. The contribution of operating conditions to the separation performance is in the order of feed temperature > feed concentration > feed flow rate. The best operating conditions are: feed temperature 350.2 K, feed concentration 3.8 % n-butanol/water and feed flow rate 23.6L/h. This work provides a theoretical basis for the design of PV membrane materials and the optimization of PV operating conditions.

Duality in non-Hermitian random matrix theory

We consider 9 Gaussian matrix ensembles characterized by single symmetry among the 38-fold symmetry classification classes of non-Hermitian random matrices, and establish exact duality formulae of certain observables between them. Particularly, averaged products of K characteristic polynomials in an N×N matrix ensemble can be expressed in terms of another K×K matrix ensemble. Our method is to combine matrix-valued heat equations and differential identities for determinants and Pfaffians and has more possible applications.

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

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

A lemma on C 0-semigroups on time scales and approximate controllability of the heat dynamic equation

In this paper, we present a lemma that allows us to characterize a broad class of C 0-semigroups on time scales, which can be applied to prove existence and uniqueness of solutions of systems of partial differential equations where the time domain is a time scale. The result obtained is applied to study the controllability of the heat equation on time scales. The lemma presented in this paper can be seen as a unification of the one proved by H. Leiva in [28], for semigroups in [0,∞).

Comparison principle for stochastic heat equations driven by α-stable white noises

Comparison principle for stochastic heat equations driven by α-stable white noises

Internal null controllability for the one-dimensional heat equation with dynamic boundary conditions

Abstract The primary focus of this paper is to establish the internal null controllability for the one-dimensional heat equation featuring dynamic boundary conditions. This achievement is realized by introducing a new Carleman estimate and an observability inequality for the corresponding backward system. In conclusion, the paper includes a set of numerical experiments that serve to confirm the validity of the theoretical findings and underscore the effectiveness of the designed control with a minimal $L^{2}$-norm.

Multi-Objective Topology Optimization of Conjugate Heat Transfer Using Level Sets and Anisotropic Mesh Adaptation

This study proposes a new computational framework for the multi-objective topology optimization of conjugate heat transfer systems using a continuous adjoint approach. It relies on a monolithic solver for the coupled steady-state Navier–Stokes and heat equations, which combines finite elements stabilized by the variational multi-scale method, level set representations of the fluid–solid interfaces and immersed modeling of heterogeneous materials (fluid–solid) to ensure that the proper amount of heat is exchanged to the ambient fluid by solid objects in arbitrary geometry. At each optimization iteration, anisotropic mesh adaptation is applied in near-wall regions automatically captured by the level set. This considerably cuts the computational effort associated with calling the finite element solver, in comparison to traditional topology optimization algorithms operating on isotropic grids with a comparable refinement level. Given that we operate within the constraint of a specified number of nodes in the mesh, this allows not only to improve the accuracy of interface representation and motion but also to retain the high fidelity of the numerical solutions at the grid points just adjacent to the interface. Finally, the remeshing and resolution steps both run within a highly parallel environment, which makes it possible for the proposed algorithm to tackle large-scale problems in three dimensions with several tens of millions of state degrees of freedom. The developed solver is validated first by minimizing dissipation in a flow splitter device, for which the method delivers relevant optimal designs over a wide range of volume constraints and flow rate distributions over the multiple outlet orifices but yields better accuracy compared to reference data from literature obtained using uniform meshes (in the sense that the layouts are more smooth, and the solutions are better resolved). The scheme is then applied to a two-dimensional heat transfer problem, using bi-objective cost functionals combining flow resistance and thermal recoverable power. A comprehensive parametric study reveals a complex arrangement of optimal solutions on the Pareto front, with multiple branches of symmetric and asymmetric designs, some of them previously unreported. Finally, the algorithmic developments are substantiated with several three-dimensional numerical examples tackled under fixed weights for heat transfer and flow resistance, for which we show that the optimal layouts computed at low Reynolds number, that are intrinsically relevant to a broad range of microfluidic application, can also serve as smooth solutions to high-Reynolds-number engineering problems of practical interest.

Characterization of Pseudo-Differential Operators Associated with the Coupled Fractional Fourier Transform

The main aim of this article is to derive certain continuity and boundedness properties of the coupled fractional Fourier transform on Schwartz-like spaces. We extend the domain of the coupled fractional Fourier transform to the space of tempered distributions and then study the mapping properties of pseudo-differential operators associated with the coupled fractional Fourier transform on a Schwartz-like space. We conclude the article by applying some of the results to obtain an analytical solution of a generalized heat equation.

On a family of nonzero solutions to the heat equation ut = △u on ℝn+1 which vanish on an arbitrary n-dimensional hyperplane P ⊂ ℝn+1

Motivated by the paper Rosenbloom–Widder [A temperature function which vanishes initially, Amer. Math. Mon. 65(8) (1958) 607–609], we construct a more general family of nonzero solutions to the [Formula: see text]-dimensional heat equation [Formula: see text] on [Formula: see text] with zero initial data. These solutions are analytically more manageable than the well-known Tychonoff power series solution. Nonzero solutions to the [Formula: see text]-dimensional heat equation [Formula: see text] on [Formula: see text] with zero initial data (i.e. vanishing on the hyperplane [Formula: see text]) can be easily obtained as a consequence of the examples on [Formula: see text] Finally, given an arbitrary hyperplane [Formula: see text], we can construct a family of nonzero solutions which vanish on [Formula: see text]