The Fourier Transform, Heat Conduction, and the Wave Equation

Approximation of exact controls for semilinear wave and heat equations through space-time methods

The solutions to certain stochastic partial differential equations with linear Gaussian noise constitute interesting examples of self-similar processes. In this chapter we analyze these classes of self-similar processes. We focus on the solution to the linear heat and wave equation driven by a Gaussian noise which behaves as a Brownian motion or fractional Brownian motion with respect to the time variable and is white or colored with respect to the space variable. We consider various aspects of these self-similar processes. In particular we present the conditions for the existence of the solution, the sharp regularity of their trajectories, we study the law of the solution to the linear heat equation and its connection with the bifractional Brownian motion.

We consider three one-dimensional continuous-time Markov processes on a lattice, each of which models the conduction of heat: the family of Brownian Energy Processes with parameter m (BEP(m)), a Generalized Brownian Energy Process, and the Kipnis-Marchioro-Presutti (KMP) process. The hydrodynamic limit of each of these three processes is a parabolic equation, the linear heat equation in the case of the BEP(m) and the KMP, and a nonlinear heat equation for the Generalized Brownian Energy Process with parameter a (GBEP(a)). We prove the hydrodynamic limit rigorously for the BEP(m), and give a formal derivation for the GBEP(a). We then formally derive the pathwise large-deviation rate functional for the empirical measure of the three processes. These rate functionals imply gradient-flow structures for the limiting linear and nonlinear heat equations. We contrast these gradient-flow structures with those for processes describing the diffusion of mass, most importantly the class of Wasserstein gradient-flow systems. The linear and nonlinear heat-equation gradient-flow structures are each driven by entropy terms of the form −log ρ; they involve dissipation or mobility terms of order ρ2 for the linear heat equation, and a nonlinear function of ρ for the nonlinear heat equation.

We consider the linear heat equation with potential in a n‐dimensional thin cilinder Ωε=Ω×(0,ε) where Ω is a bounded open smooth set of $\mathbb{R}^{n-1}$ with n≥2 and ε is a small parameter. We study the null controllability problem when the control acts in a cylindrical region ωε=ω×(0,ε), where ω⊂Ω is an open and non‐empty subset of Ω. We prove that, under appropriate boundary conditions, for a suitable class of potentials the heat equation is uniformly null controllable as ε→0. We also prove the convergence of the controls to a null control for the n−1‐dimensional heat equation in Ω. Similar results are proved for the semilinear heat equation with globally Lipschitz nonlinearities.

In this paper we address the problem of null-controllability of heat equations in two different cases: (a) The semilinear heat equation in bounded domains and (b) The linear heat equation in the half line. Concerning the first problem (a) we show that a number of systems in which blow-up arises may be controlled by means of external forces which are localized in an arbitrarily small open set. In the frame of problem (b) we prove that compactly supported initial data may not be driven to zero if the control is supported in a bounded set. This shows that although the velocity of propagation in the heat equation is infinite, this is not sufficient to guarantee null-controllability properties.We also include a list of open problems.KeywordsHeat EquationMoment ProblemApproximate ControllabilityCarleman EstimateOpen Nonempty SubsetThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The use of the local truncation error to improve arbitrary-order finite elements for the linear wave and heat equations

Stability analysis of a linear system coupling wave and heat equations with different time scales

A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation developed in Part 1 is applied to the simulation of 1-D and 2-D test problems on regular and irregular domains. Trivial conforming and non-conforming Cartesian meshes with 3-point stencils in the 1-D case and 9-point stencils in the 2-D case are used in calculations. The numerical solutions of the 1-D wave equation as well as the 2-D wave and heat equations for a simple rectangular plate show that the accuracy of the new approach on non-conforming meshes is practically the same as that on conforming meshes for both the Dirichlet and Neumann boundary conditions. Moreover, very small distances ( $$0.1 h - 10^{-9}h$$ where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not decrease the accuracy of the new technique. The application of the new approach to the 2-D problems on an irregular domain shows that the order of accuracy is close to four for the wave and heat equations and is close to five for the Poisson equation. This is in agreement with the theoretical results of Part 1 of the paper. The comparison of the numerical results obtained by the new approach and by FEM shows that at similar 9-point stencils, the accuracy of the new approach on irregular domains is two orders higher for the wave and heat equations and three orders higher for the Poisson equation than that for the linear finite elements. Moreover, the new approach yields even much more accurate results than the quadratic and cubic finite elements with much wider stencils. An example of a problem with a complex irregular domain that requires a prohibitively large computation time with the finite elements but can be easily solved with the new approach is presented.

In this paper, we study the null controllability of linear heat and wave equations with spatial nonlocal integral terms. Under some analyticity assumptions on the corresponding kernel, we show that the equations are controllable. We employ compactness-uniqueness arguments in a suitable functional setting, an argument that is harder to apply for heat equations because of its very strong time irreversibility. Some possible extensions and open problems concerning other nonlocal systems are presented.

In the present paper, preliminary results towards the generalization to the infinite-dimensional setting of some known robust finite-dimensional control algorithms are illustrated. The main focus of the present paper is on the rejection of non-vanishing external disturbances. The generalization to the infinite-dimensional setting of some known finite-dimensional controllers, namely the “Power-fractional” controller [5], and two “Second-order sliding-mode” control algorithms (the “Twisting” and “Super-Twisting” algorithms [10]) is performed to address control problems involving the uncertain wave and heat equations. Constructive proofs of stability are developed via Lyapunov functional technique, which leads to simple tuning rules for the controller parameters. Simulation results are discussed to verify the effectiveness of the proposed schemes.

We study various aspects of stochastic partial differential equations driven by Levy white noise. This driving noise, which is a generalization of Gaussian white noise, can be viewed either as a generalized random process or as an independently scattered random measure. After unifying these approaches and establishing appropriate stochastic integral representations, we show that a necessary and sufficient condition for a Levy white noise to have values in the space of tempered Schwartz distributions, is that the underlying Levy measure have a positive absolute moment. In the case of a linear stochastic partial differential equation with a general differential operator and driven by a symmetric pure jump Levy white noise, we show that when the mild solution is locally Lebesgue integrable, then it is equal to the generalized solution, and that a random field representation exists for the generalized solution if and only if the fundamental solution of the operator has certain integrability properties. In that case, we show that the random field representation is equal to the mild solution. For this purpose, a new stochastic Fubini theorem is proved. These results are applied to the linear stochastic heat and wave equations driven by a symmetric alpha-stable noise. We then study the non-linear stochastic heat equation driven by a general type of Levy white noise, possibly with heavy tails and non-summable small jumps. Our framework includes in particular the alpha-stable noise. In the case of the equation on the whole space, we show that the law of the solution that we construct does not depend on the space variable. Then we show in various domains D that the solution u to the stochastic heat equation is such that t -> u(t,·) has a cadlag version in a fractional Sobolev space of order r < -d /2. Finally, we show that the partial functions have a continuous version under some optimal moment conditions. In the alpha-stable case, we show that for the choices of alpha for which this moment condition is not satisfied, the sample paths of the partial functions are unbounded on any non-empty open subset.

It is known that, if the time variable in the heat equation is complex and belongs to a sector in ℂ, then the theory of analytic semigroups becomes a powerful tool of study. The same is true for the Laplace equation on an infinite strip in the plane, regarded as an initial value boundary value problem. Also, it is known that if both variables, time and spatial, are complex, then, e.g. the Cauchy problem for the heat equation admits as solution, only a formal power series which, in general, converges nowhere. In a recent paper (C.G. Gal, S.G. Gal, and J.A. Goldstein, Evolution equations with real time variable and complex spatial variables, Complex Var. Elliptic Eqns. 53 (2008), pp. 753–774), a complementary approach was made: the study of the complex versions of the classical heat and Laplace equations, obtained by ‘complexifying’ the spatial variable only (and keeping the time variable real). The goal of this article is to extend that study to the higher-order heat and Laplace-type equations. This ‘complexification’ is based on integral representations of the solutions in the case of a real spatial variable, by complexifying the spatial variable in the corresponding semigroups of operators. It is of interest to note that these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness.

In this article, a modified cubic B-spline differential quadrature method (MCB-DQM) is proposed to solve some of the basic differential equations. Here we have considered an ordinary differential equation of order two along with heat equation and one- and two-dimensional wave equations. A nonlinear ordinary differential equation of order two is also considered. The ordinary differential equation is reduced to a system of nonhomogeneous linear equations which is then solved by using the Gauss elimination method, whereas the heat equation and the one-dimensional and two-dimensional heat and wave equations are reduced to a system of ordinary differential equations. The system is then solved by the optimal four-stage three-order strong stability preserving time stepping Runge–Kutta (SSP-RK43) scheme. The reliability and efficiency of the method have been tested on six examples.KeywordsOrdinary differential equationHeat equationWave equation cubic B-spline functionsModified cubic B-spline quadrature methodSystem of ordinary differential equationsGauss elimination methodRunge–Kutta fourth-order method

In this article we make a new connection between the linear wave equation and the linear heat equation. In this way we are able to construct new solutions of the linear wave equation, using symmetries and conditional symmetries of the heat equation.

In this paper, we consider Harnack inequalities (the gradient estimates) of positive solutions for two different heat equations via the use of the maximum principle. In the first part, we obtain the gradient estimate for positive solutions to the following nonlinear heat equation problem $$\partial _{t}u={\Delta } u+au\log u+Vu,~~u>0$$ on the compact Riemannian manifold (M, g) of dimension n and with Ric(M) ≥−K. Here a > 0 and K are some constants and V is a given smooth positive function on M. Similar results are showed to be true in case when the manifold (M, g) has compact convex boundary or (M, g) is a complete non-compact Riemannian manifold. In the second part, we study Harnack inequality (gradient estimate) for positive solution to the following linear heat equation on a compact Riemannian manifold with non-negative Ricci curvature: $$ \partial _{t}u={\Delta } u+\sum W_{i}u_{i}+Vu, $$ where Wi and V only depend on the space variable x ∈ M. The novelties of our paper are the refined global gradient estimates for the corresponding evolution equations, which are not previously considered by other authors such as Yau (Math. Res. Lett. 2(4), 387–399, 1995), Ma (J. Funct. Anal. 241(1), 374–382, 2006), Cao et al. (J. Funct. Anal. 265, 2312–2330, 2013), Qian (Nonlinear Anal. 73, 1538–1542, 2010).