Parabolic differential equations with bounded delay

In this paper, we consider a class of optimal control problems with control and terminal inequality constraints, where the system dynamics is governed by a linear second-order parabolic partial differential equation with first boundary condition. A feasible direction algorithm for solving this class of optimal control problems has already been obtained in the literature. The aim of this paper is to improve the convergence result by using a topology arising in the study of relaxed controls.

This study presents an observer‐based output feedback compensator design for a linear parabolic partial differential equation (PDE), where a finite number of actuators and sensors are active over partial areas and at specified points in the spatial domain, respectively. In the proposed design method, a Luenberger‐type PDE observer is first constructed by using the non‐collocated pointwise observation to exponentially track the PDE state. Based on the estimated state, a collocated local piecewise state feedback controller is then proposed. By employing a Lyapunov direct method, integration by parts, Wirtinger's inequality and first mean value theorem for definite integrals, sufficient conditions on the exponential stability of the resulting closed‐loop coupled PDEs are presented in terms of standard linear matrix inequalities. Furthermore, both open‐loop and closed‐loop well‐posedness analysis results are also established by the ‐semigroup method. Numerical simulation results are presented to show the effectiveness of the proposed design method.

This paper presents an observer-based dynamic feedback control design for a linear parabolic partial differential equation (PDE) system, where a finite number of actuators and sensors are active over part thereof and at specified points in the spatial domain, respectively. In the proposed design method, a Luenberger-type PDE observer is first constructed by using the non-collocated pointwise measurements to exponentially tract the state of the PDE system. Based on the estimated state, a collocated local piecewise state feedback controller is then proposed. By employing Lyapunov direct method, integration by parts, Wirtinger's inequality and first mean value theorem for integrals, a sufficient condition on exponential stability of the resulting closed-loop system is presented in term of standard linear matrix inequalities (LMIs). Numerical simulation results are presented to show the effectiveness of the proposed design method.

191 - Parabolic Equations: Monte Carlo Method

This work proposes a hybrid control methodology that integrates feedback and switching for fault-tolerant constrained control of linear parabolic partial differential equations (PDEs) for which the spectrum of the spatial differential operator can be partitioned into a finite “slow” set and an infinite stable “fast” complement. Modal decomposition techniques are initially used to derive a finite-dimensional system (set of ordinary differential equations (ODEs) in time) that captures the dominant dynamics of the PDE. The ODE system is then used as the basis for the integrated synthesis, via Lyapunov techniques, of a stabilizing nonlinear feedback controller together with a switching law that orchestrates the switching between the admissible control actuator con- figurations, based on their constrained regions of stability, in a way that respects actuator constraints and maintains closed-loop stability in the event of actuator failure. Precise conditions that guarantee stability of the constrained closed-loop PDE system under actuator switching are provided. The proposed method is applied to the problem of stabilizing an unstable, spatially unifrom steady-state of a linear parabolic PDE under constraints and actuator failures.

I consider the problem of computing the space of conservation laws for a second-order parabolic partial differential equation for one function of three independent variables. The PDE is formulated as an exterior differential system \({\cal I}\) on a 12-manifold M, and its conservation laws are identified with the vector space of closed 3-forms in the infinite prolongation of \({\cal I}\) modulo the so-called "trivial" conservation laws. I use the tools of exterior differential systems and Cartan's method of equivalence to study the structure of the space of conservation laws. My main result is: Theorem.Any conservation law for a second-order parabolic PDE for one function of three independent variables can be represented by a closed 3-form in the differential ideal ${\cal I}$ on the original 12-manifold M. I show that if a nontrivial conservation law exists, then \({\cal I}\) has a deprolongation to an equivalent system \({\cal J}\) on a 7-manifold N, and any conservation law for \({\cal I}\) can be expressed as a closed 3-form on N that lies in \({\cal J}\). Furthermore, any such system in the real analytic category is locally equivalent to a system generated by a (parabolic) equation of the formA (uxxuyy-u2xy)+Buxx+2Cuxy+Duyy+E = 0 where A, B, C, D, E are functions of x, y, t, u, ux, uy, ut. I compute the space of conservation laws for several examples, and I begin the process of analyzing the general case using Cartan's method of equivalence. I show that the non-linearizable equation \( u_t = {1 \over 2} e^{-u} (u_{xx}+u_{yy}) \) has an infinite-dimensional space of conservation laws. This stands in contrast to the two-variable case, for which Bryant and Griffiths showed that any equation whose space of conservation laws has dimension 4 or more is locally equivalent to a linear equation, i.e., is linearizable.

Convergence analysis of constraint energy minimizing generalized multiscale finite element method for a linear stochastic parabolic partial differential equation driven by additive noises

In this paper, a special form of extrapolation formula is presented and its formation and analysis show its uniqueness as a highly accurate combination of extrapolation schemes of the method of lines for the solution of linear parabolic partial differential equations.

This study presents numerical solution of time-fractional linear parabolic partial differential equations (PDEs) using the Caputo finite difference scheme. The discretization process is based on the second-order implicit finite difference scheme and the Caputo fractional derivative operator. The resulting system of linear approximation equations is solved using the Kaudd Successive Over Relaxation (KSOR) iterative method. A comparison is made with the Gauss-Seidel (GS) iterative method through three numerical examples. The results demonstrate that the KSOR method requires fewer iterations and reduced computational time compared to the GS method.

Within the last decade, attention has been devoted to the introduction of several fast computational methods for solving the linear difference equations which are derived from the finite difference discretisation of many standard partial differential equations of Mathematical Physics. In this paper, the authors develop and extend an exact factorisation technique previously applied to parabolic equations in one space dimension to the implicit difference equations which are derived from the application of alternating direction implicit methods when applied to elliptic and parabolic partial differential equations in 2 space dimensions under a variety of boundary conditions.

Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential equations (PDEs), which are widely used in finance, engineering, and natural sciences. In particular, simulations indicate that algorithms based on deep learning overcome the curse of dimensionality in the numerical approximation of solutions of semilinear PDEs. For certain linear PDEs it has also been proved mathematically that deep neural networks overcome the curse of dimensionality in the numerical approximation of solutions of such linear PDEs. The key contribution of this article is to rigorously prove this for the first time for a class of nonlinear PDEs. More precisely, we prove in the case of semilinear heat equations with gradient-independent nonlinearities that the numbers of parameters of the employed deep neural networks grow at most polynomially in both the PDE dimension and the reciprocal of the prescribed approximation accuracy. Our proof relies on recently introduced full history recursive multilevel Picard approximations for semilinear PDEs.

The H ∞ control problem is considered for linear parabolic partial differential equation (PDE) systems with completely unknown system dynamics. We propose a model-free policy iteration (PI) method for learning the H ∞ control policy by using measured system data without system model information. First, a finite-dimensional system of ordinary differential equation (ODE) is derived, which accurately describes the dominant dynamics of the parabolic PDE system. Based on the finite-dimensional ODE model, the H ∞ control problem is reformulated, which is theoretically equivalent to solving an algebraic Riccati equation (ARE). To solve the ARE without system model information, we propose a least-square based model-free PI approach by using real system data. Finally, the simulation results demonstrate the effectiveness of the developed model-free PI method.

Partial differential equation is an equation containing a partial derivative of one or more dependent variables on more than one independent variable. In the differential equation there are coefficients in the form of constants and functions. The solution of a partial differential equation whose coefficients are constants is easily determined. However, the solution of the differential equations whose coefficients are functions is quite difficult to determine. One method that can be used to determine the solution is by using Taylor polynomial. This method can be used in second-order linear partial differential equation with coefficient of function with two independent variables. The purpose of this research is to determine the Taylor polynomial solution on second-order linear partial differential equation. In this research we get solution from second-order linear partial differential equation by assuming solution in the form of polynomial of Taylor having degree 𝑁𝑁 𝑢𝑢 (𝑥𝑥, 𝑦𝑦) = 􀷍􀷍αα𝑟𝑟, 𝑠𝑠 (𝑥𝑥-𝑐𝑐0) 𝑟𝑟 (𝑦𝑦-𝑐𝑐1) 𝑠𝑠, 𝑁𝑁𝑠𝑠 = 0𝑁𝑁𝑟𝑟 = 0 with αα𝑟𝑟, 𝑠𝑠 = 1𝑟𝑟! 𝑠𝑠! 𝑢𝑢 (𝑟𝑟, 𝑠𝑠) (𝑐𝑐0, 𝑐𝑐1) is the Taylor polynomial coefficient, or can be expressed in terms of the matrix equation 𝑢𝑢 (𝑥𝑥, 𝑦𝑦) = 𝑋𝑋𝑋𝑋𝑋𝑋

The heat conduction equation is a parabolic differential equation and a type of second-order linear partial differential equation. By applying the finite difference scheme in the Crank-Nicolson method, the numerical solution of the heat conduction equation can be calculated. Obtaining numerical solutions with a high level of accuracy, Richardson extrapolation is required. The Crank-Nicolson approach scheme has a high level of accuracy, because the gap between numerical and analytical solutions is very small. Richardson extrapolation greatly influences the accuracy of numerical solutions, because the gap between analytical solution and numerical solutions with Richardson extrapolation is smaller than disparity in numerical solutions without Richardson extrapolation.

<div class="htmlview paragraph">Many methods have been suggested for the solution of parabolic partial differential equations. Prior efforts have heavily focused on explicit difference equations, implicit difference equations, and a combination of these two methods. Some work has been done in exponential methods and other obscure methods. This paper introduces an advanced exponential method for obtaining solutions of parabolic partial differential equations.</div> <div class="htmlview paragraph">The explicit, implicit, Crank-Nicolson, DuFort-Frankel, modified DuFort-Frankel, original exponential method, and the advanced exponential method are briefly described and compared. The advanced exponential method is unconditionally stable with whatever time step is taken. Accuracy and speed of solution for this method is compared to other solution methods for various example test cases. For the example test cases, the method gave excellent results when compared to the analytical solution and to other methods. The method has been introduced into the Systems Improved Numerical Differencing Analyzer and Fluid Integrator (SINDA/FLUINT) code as a solver. The advanced exponential method gives excellent speed, accuracy, and stability for solving parabolic partial differential equations.</div>