Inhomogeneous Heat Equation Research Articles

On the space-time analyticity of the inhomogeneous heat equation on the half space with Neumann boundary conditions

We consider the inhomogeneous heat equation on the half-space R + d with Neumann boundary conditions. We prove a space-time Gevrey regularity of the solution, with a radius of analyticity uniform up to the boundary of the half-space. We also address the case of homogeneous Robin boundary conditions. Our results generalize the case of homogeneous Dirichlet boundary conditions established by Kukavica and Vicol [A direct approach to Gevrey regularity on the half-space. In: Partial differential equations in fluid mechanics. Cambridge: Cambridge Univ. Press; 2018. p. 268–288. (London math. soc. lecture note ser.; vol. 452).].

Asymptotic profiles for inhomogeneous heat equations with memory

We study the large-time behavior in all Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} norms of solutions to an inhomogeneous nonlocal heat equation in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^N$$\\end{document} involving a Caputo α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-time derivative and a power β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} of the Laplacian when the dimension is large, N>4β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N> 4\\beta $$\\end{document}. The asymptotic profiles depend strongly on the space-time scale and on the time behavior of the spatial L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} norm of the forcing term.

Spatio-temporal LPV model of 2D workpiece temperature for Direct Laser Deposition

Additive manufacturing of metallic components presents challenges regarding quality assurance and consistency, owing to the workpiece's complex spatially-dependent temperature history and a lack of suitable models for closed-loop temperature control of the fabrication process. The selected case study is the laying of cladding material (workpiece) through Direct Laser Deposition. The paper proposes identifying a linear spatio-temporal parameter-varying model for the workpiece temperature for control-oriented applications. For this, the temperature distribution is reconstructed from data measured by an infrared camera. The model structure is derived from the inhomogeneous heat equation. Finally, the model parameters are identified using a nonlinear least-squares solver. The relationships between theoretical and estimated model parameters are presented.

Numerical analysis of heating by a current pulse of a niobium nitride membrane in its longitudinal section

A numerical calculation of the evolution of the temperature distribution in the longitudinal section of a niobium nitride membrane when it is heated by an electric current pulse is performed. Mathematical modeling was carried out on the basis of a two-dimensional initial-boundary value problem for an inhomogeneous heat equation. In the initial boundary value problem, it was taken into account that current and potential contacts to the membrane serve simultaneously as contacts for heat removal. The case was considered for the third from the left and the first from the right initial-boundary value problem. Analysis of the numerical solution showed that effective heat removal from the membrane can be provided by current-carrying and potential clamping contacts made, for example, of beryllium bronze. This makes it possible to study the current-voltage characteristics of superconducting membranes near the critical temperature of the transition to the superconducting state by currents close to the critical density without significant heating.

Gevrey index theorem for the inhomogeneous n-dimensional heat equation with a power-law nonlinearity and variable coefficients

We are interested in the Gevrey properties of the formal power series solution in time of the inhomogeneous semilinear heat equation with a power-law nonlinearity in $1$-dimensional time variable $t\in\mathbb{C}$ and $n$-dimensional spatial variable $x\in\mathbb{C}^n$ and with analytic initial condition and analytic coefficients at the origin $x=0$. We prove in particular that the inhomogeneity of the equation and the formal solution are together $s$-Gevrey for any $s\geq1$. In the opposite case $s<1$, we show that the solution is $1$-Gevrey at most while the inhomogeneity is $s$-Gevrey, and we give an explicit example in which the solution is $s'$-Gevrey for no $s'<1$.

Decay/growth rates for inhomogeneous heat equations with memory. The case of large dimensions

&lt;abstract&gt;&lt;p&gt;We study the decay/growth rates in all $ L^p $ norms of solutions to an inhomogeneous nonlocal heat equation in $ \mathbb{R}^N $ involving a Caputo $ \alpha $-time derivative and a power $ \beta $ of the Laplacian when the dimension is large, $ N &amp;gt; 4\beta $. Rates depend strongly on the space-time scale and on the time behavior of the spatial $ L^1 $ norm of the forcing term.&lt;/p&gt;&lt;/abstract&gt;

Обратные задачи для двумерного уравнения теплопроводности по отысканию правой части

Based on the solution of the first initial-boundary value problem for an inhomogeneous two-dimensional heat equation, we state and study inverse problems, whose right-hand sides contain unknown factors depending on spatial and time variables. Preliminarily, we explicitly construct a solution to the direct initial-boundary-value problem. We prove the uniqueness of the solution to direct and inverse problems, making use of the completeness property of the system of eigenfunctions of the corresponding homogeneous Dirichlet problem for the Laplace operator. We also establish existence theorems for solutions of inverse problems and construct solutions explicitly.

A modified quasi-boundary value method for a backward problem for the inhomogeneous time conformable fractional heat equation in a cylinder

ABSTRACT The time conformable heat equation is a generalization of classical heat equation involved local and limit-based derivative, which is called conformable fractional derivative. In this paper, we study a backward problem for the time conformable fractional heat equation defined in cylindrical coordinates for the axis-symmetric case which is a severely ill-posed problem. By using a modified quasi-boundary value method, the problem is regularized and some new error estimates are obtained. The numerical experiment shows that the method is feasible and effective.

On the summability of the solutions of the inhomogeneous heat equation with a power-law nonlinearity and variable coefficients

In this article, we investigate the summability of the formal power series solutions in time of the inhomogeneous heat equation with a power-law nonlinearity of degree two, and with variable coefficients. In particular, we give necessary and sufficient conditions for the 1-summability of the solutions in a given direction. These conditions generalize the ones given for the linear heat equation by W. Balser and M. Loday-Richaud in a 2009 article [5].

Singular limit for the magnetohydrodynamics of the damped wave type in the critical Fourier–Sobolev space

We study the Cauchy problem of the incompressible damped wave type magnetohydrodynamic system in RN(N≥2). The purpose of this paper is to show the global well-posedness and a singular limit of the problem in Fourier–Sobolev spaces. For the proof of the results, we use the Lp-Lq type estimates for the fundamental solutions of the damped wave equation and end-point maximal regularity for the inhomogeneous heat equation in that space with a detailed estimate of difference between the symbol of the heat kernel and fundamental solution of the damped wave equation.

The Stabilization Rate of Solutions to Parabolic Equations

We estimate the difference between the solution to the Cauchy problem for a inhomogeneous heat equation and the Riesz means for large time provided that the right-hand side of the heat equation has at most power growth at infinity. Bibliography: 8 titles.

Determination of temperature distribution and thermal flux for two-dimensional inhomogeneous sideways heat equations

We investigate an inverse and ill-posed problem for the two-dimensional inhomogeneous heat equation in the presence of a general source term. The goal here consists of recovering not only the temperature distribution but also the thermal flux from the measure data. With the appearance of the general source term, this model gets far worse than its homogeneous counterpart. Based on an analysis of the instability caused in the solution, we propose a kernel regularization scheme to stabilize the investigated problem. The Holder-type convergence estimates are achieved under some appropriate a priori assumptions. We further numerically demonstrate the theoretical results by proposing a robust algorithm based on the 2-D fast Fourier transform. The numerical outputs exemplify the feasibility and efficiency of the proposed method.

Existence of solutions for an inhomogeneous fractional semilinear heat equation

We obtain necessary conditions and sufficient conditions on the existence of solutions to the Cauchy problem for a fractional semilinear heat equation with an inhomogeneous term. We identify the strongest spatial singularity of the inhomogeneous term for the solvability of the Cauchy problem.

Автоматизація досліджень багатошарових систем

The article considers the issues of calculating and setting parameters of multilayer microbiological systems containing local concentrated discrete sources of temperature fields. As an object of research in the work, the considered multilayer, unlimited microbiological material subjected to local point thermal influence. The novelty of the author’s research is to take into account the specific features of the process of thermal action and the multilayer structure of microbiological material. The purpose of the research is to build physically justified computational mathematical models that describe the state of the research object, as well as to outline ways for the optimization process of a whole class of multilayer microbiological systems. The author has constructed a calculated mathematical model of the process of point thermal action on a multilayer unlimited microbiological material. It should be noted that a feature of the object of study is that its state is described by a boundary-value problem for a system of non-stationary, inhomogeneous differential heat equations (a system with distributed parameters). This specificity significantly complicates the process of constructing the appropriate mathematical models for optimizing multilayer microbiological systems, as well as their use to improve the quality of the biotechnological process of dividing (cutting, segmentation) microbiological material. In this regard, the author proposes to implement the procedure for solving a series of boundary value problems and ensuring an iterative process of searching for rational parameters of heat exposure on a specialized grid analog or hybrid model. This will allow at each iteration to significantly reduce the time it takes to solve a boundary value problem, which will make it possible to increase the accuracy of solving the entire problem of searching for parameters of the systems under consideration. Key words : mathematical models, optimization, boundary value problem, analog model, hybrid model

Critical behavior for a semilinear parabolic equation with forcing term depending on time and space

We investigate the large-time behavior of the sign-changing solution of the inhomogeneous semilinear heat equation ∂tu=Δu+|u|p+tσw(x) in (0,T)×RN, where N≥2, p>1, σ>−1, σ≠0 and w≢0. The novelty of this paper lies in considering a forcing term (tσw(x)) which depends on both of time and space. We show that there is an exponent p⁎(σ) which is critical in the following sense: the solution of the above problem blows up in finite time when 1<p<p⁎(σ) and ∫RNw(x)dx>0, while global solution exists for suitably small initial data and w belonging to certain Lebesgue spaces when p≥p⁎(σ). Our obtained results show that the forcing term induces an interesting phenomenon of discontinuity of the critical exponent p⁎(σ). Namely, we found that limσ→0−⁡p⁎(σ)≠limσ→0+⁡p⁎(σ). Furthermore, limσ→0−⁡p⁎(σ) coincides with the critical exponent of the above problem with σ=0.

Local well‐posedness and blow‐up for an inhomogeneous nonlinear heat equation

In this paper, we consider the nonlinear heat equation with inhomogeneous nonlinearity where having either a polynomial growth or exponential growth, and is a function satisfying some assumptions to be stated later. We first prove the local well‐posedness in suitable Lebesgue spaces when belongs to some Lebesgue space and has polynomial growth. We also obtain some blow‐up results.

Задача Кошi для рiвняння теплопровiдностi на площинi з випадковою правою частиною з простору Орлiча

The heat equation with random conditions is a classical problem of mathematical physics. Recently, a number of works appeared, which in many ways investigated this equation according to the type of random initial conditions. We consider a Cauchy problem for the heat equations with a random right part. We study the inhomogeneous heat equation on the plane with a random right part. We consider the right part as a random function of the Orlicz space. The conditions of existence with probability one classical solution of the problem are investigated. For such a problem has been got the estimation for the distribution of the supremum solution.

Властивостi узагальненого розв’язку задачi Кошi для рiвняння теплопровiдностi з випадковою правою частиною

The subject of this work is at the intersection of two branches of mathematics: mathematical physics and stochastic processes. The physical formulation of problems of mathematical physics with random factors was studied by Kampe de Feriet. In the works by E. Beisenbaev, Yu.V. Kozachenko and V.V. Buldygin a new approach studying the solutions of partial differential equations with random initial conditions was proposed. The authors investigate the convergence in probability of the sequence of function spaces of partial sums approximating the solution of a problem. The mentioned approach was used in the worksby E. Barrasa de La Krus, Endzhyrgly, Ya.A. Kovalchuk. In the paper by V.V. Buldygin and Yu.V. Kozachenko the application of the Fourier method for the homogeneous hyperbolic equation with Gaussian initial conditions is justified and existence conditions in terms of correlation functions are studied. Homogeneous hyperbolic equation with random initial conditions from the space Sub ϕ (Ω) are considered in works by B. V. Dovgay, G.I.Slyvka-Tylyshchak. The model of a solution of a hyperbolic type equation with random initial conditions was investigated in the papers by G.I. Slyvka-Tylyshchak. There is studied a boundary value problem of mathematical physics for the inhomogeneous hyperbolic equation with ϕ-subgaussian in right part in works by B. V. Dovgay. The parabolic typeequations of Mathematical Physics with random factors of Orlicz spaces have been studied in the papers by Yu.V. Kozachenko and K.J. Veresh. Properties of the classical solution of the heat equation on a line with a random right part are considered in works by Yu.V. Kozachenko and G.I. Slyvka-Tylyshchak.We consider a Cauchy problem for the heat equations with a random right part. We study the inhomogeneous heat equation on a line with a random right part. We consider the right part as a random function of the space L p (Ω). The conditions of existence with probability one generalized solution of the problem are investigated.

Моделирование аппроксимирующего 16-точечного отсека поверхности отклика применительно к решению неоднородного уравнения теплопроводности

The paper proposes a computational method for solving differential equations of mathematical physics by approximating the desired solution using geometric objects of multidimensional space passing through predetermined points. The essence of the method is to simulate an approximating geometric object of a multidimensional affine space constructed on a regular multidimensional network of points. In this case, the response function values satisfying the solution of the original differential equation are calculated at the nodal points of the network. Modeling of approximating geometric object is carried out by means the arcs of algebraic curves passing through predetermined points. It should be noted that taking into account the boundary conditions does not require changes in the geometric algorithm or point equations. It is sufficient to use the necessary coordinates of the nodal boundary points corresponding to the boundary conditions of the solution of the differential equation. To achieve the required accuracy of the solution of differential equations, it is sufficient to compact the reference network of points. Under such conditions, it is possible to use as a single geometric object to approximate the solution of the differential equation, and composite, based on the simulation of multidimensional contours on a regular network of points of multidimensional space. A geometric classification of differential equations depending on the number of parameters determining the approximating geometric object in multidimensional space is proposed. An example of solving the inhomogeneous heat equation by means of an approximating response surface passing through 16 predetermined points is given. In this case, the required approximating compartment of the response surface passes through 3 straight lines that correspond to the boundary conditions and satisfies the solution of the original differential equation at the nodal points of the 16-point network. A comparison of the results of solving the inhomogeneous heat equation approximated by a 16-point compartment of the response surface with the reference compartment of the surface obtained by the method of separating variables is also presented.

Asymptotics for a class of heat equations with inhomogeneous nonlinearity

Abstract The initial value problem for an inhomogeneous nonlinear heat equation with a pure power nonlinearity is investigated. In the radial energy space, global and non-global well-posedness are discussed.