Time-stepping algorithms for semidiscretized linear parabolic PDEs based on rational approximants with distinct real poles

Time dependent problems in Partial Differential Equations (PDEs) are often solved by the Method Of Lines (MOL). For linear parabolic PDEs, the exact solution of the resulting system of first order Ordinary Differential Equations (ODEs) satisfies a recurrence relation involving the matrix exponential function. In this paper, we consider the development of a fourth order rational approximant to the matrix exponential function possessing real and distinct poles which, consequently, readily admits a partial fraction expansion, thereby allowing the distribution of the work in solving the corresponding linear algebraic systems in essentially Backward Euler-like solves on concurrent processors. The resulting parallel algorithm possesses appropriate stability properties, and is implemented on various parabolic PDEs from the literature including the forced heat equation and the advection-diffusion equation.Dedicated to Professor J. Crank on the occasion of his 80th birthday     

An exponential matrix method for solving systems of linear differential equations

This paper presents an exponential matrix method for the solutions of systems of high‐order linear differential equations with variable coefficients. The problem is considered with the mixed conditions. On the basis of the method, the matrix forms of exponential functions and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown coefficients are determined and thus the approximate solutions are obtained. Also, an error estimation based on the residual functions is presented for the method. The approximate solutions are improved by using this error estimation. To demonstrate the efficiency of the method, some numerical examples are given and the comparisons are made with the results of other methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Two-derivative Runge–Kutta methods for PDEs using a novel discretization approach

We develop a novel and general approach to the discretization of partial differential equations. This approach overcomes the rigid restriction of the traditional method of lines (MOL) and provides flexibility in the treatment of spatial discretization. This method is essential for developing efficient numerical schemes for PDEs based on two-derivative Runge–Kutta (TDRK) methods, where the first and second derivatives must be discretized in an efficient way. This is unlikely to be achieved by using MOL. We then apply the explicit TDRK methods to the advection equations and analyze the numerical stability in the linear advection equation case. We conduct numerical experiments on the Burgers’ equation using the TDRK methods developed. We also apply a two-stage semi-implicit TDRK method of order-4 and stage-order-4 to the heat equation. A very significant improvement in the efficiency of this TDRK method is observed when compared to the popular Crank-Nicolson method. This paper is partially based on the work in Tsai’s PhD thesis (2011) [10].     

A high‐order finite difference method for 1D nonhomogeneous heat equations

In this article a sixth‐order approximation method (in both temporal and spatial variables) for solving nonhomogeneous heat equations is proposed. We first develop a sixth‐order finite difference approximation scheme for a two‐point boundary value problem, and then heat equation is approximated by a system of ODEs defined on spatial grid points. The ODE system is discretized to a Sylvester matrix equation via boundary value method. The obtained algebraic system is solved by a modified Bartels‐Stewart method. The proposed approach is unconditionally stable. Numerical results are provided to illustrate the accuracy and efficiency of our approximation method along with comparisons with those generated by the standard second‐order Crank‐Nicolson scheme as well as Sun‐Zhang's recent fourth‐order method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre multiwavelets

An effective method based upon Legendre multiwavelets is proposed for the solution of Fredholm weakly singular integro-differential equations. The properties of Legendre multiwavelets are first given and their operational matrices of integral are constructed. These wavelets are utilized to reduce the solution of the given integro-differential equation to the solution of a sparse linear system of algebraic equations. In order to save memory requirement and computational time, a threshold procedure is applied to obtain the solution to this system of algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation.     

The construction of operational matrix of fractional derivatives using B-spline functions

Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. Here we construct the operational matrix of fractional derivative of order α in the Caputo sense using the linear B-spline functions. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus we can solve directly the problem. The method is applied to solve two types of fractional differential equations, linear and nonlinear. Illustrative examples are included to demonstrate the validity and applicability of the new technique presented in the current paper.     

Integración temporal explícita con grandes pasos de tiempo de la ecuación de transmisión del calor

We present a novel fully explicit time integration method that remains stable for large time steps, requires neither matrix inversions nor solving a system of equations and therefore allows for nearly effort-less parallelization. In this first paper the proposed approach is applied to solve conduction heat transfer problems, showing that it is stable for any time step as is the case with implicit methods but with a much lower computation time.     

A compact local one‐dimensional scheme for solving a 3D N‐carrier system with Neumann boundary conditions

We consider a mathematical model for thermal analysis in a 3D N‐carrier system with Neumann boundary conditions, which extends the concept of the well‐known parabolic two‐step model for micro heat transfer. To solve numerically the complex system, we first reduce 3D equations in the model to a succession of 1D equations by using the local one‐dimensional (LOD) method. The obtained 1D equations are then solved using a fourth‐order compact finite difference scheme for the interior points and a second‐order combined compact finite difference scheme for the points next to the boundary, so that the Neumann boundary condition can be applied directly without discretizing. By using matrix analysis, the compact LOD scheme is shown to be unconditionally stable. The accuracy of the solution is tested using two numerical examples. Results show that the solutions obtained by the compact LOD finite difference scheme are more accurate than those obtained by a Crank‐Nicholson LOD scheme, and the convergence rate with respect to spatial variables is about 2.6. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Design of LMI‐based sliding mode controller with an exponential policy for a class of underactuated systems

In this article, a novel sliding mode control (SMC) approach is proposed for the control of a class of underactuated systems which are featured as in cascaded form with external disturbances. The asymptotic stability conditions on the error dynamical system are expressed in the form of linear matrix inequalities. The control objective is to construct a controller such that would force the state trajectories to approach the sliding surface with an exponential policy. The proposed SMC has a simple structure because it is derived from the associated first‐order differential equation and is capable of handling system disturbances and nonlinearities. The effectiveness of the proposed control method is validated using intensive simulations. © 2014 Wiley Periodicals, Inc. Complexity 21: 117–124, 2016     

关于热传导方程半离散差分格式的一个注记

本文研究了热传导方程初边值问题的半离散化差分格式直接解算法.分别从Dirichlet和Neumann边界条件出发,直接由空间差分格式导出与时间相关的一阶常微分方程组,随后通过正/余弦变换获得了原方程的半解析解,并给出了相关收敛性分析.并对中心差分格式和紧差分格式的精度差异,通过矩阵特征值理论给出了相关原因分析.另外,对于二维热传导方程初边值问题,应用矩阵张量积运算,该直接解算法可直接演变成二重正(余)弦变换.该方法由于不涉及时间上的离散,从而具有较好的计算效率.     

A new scheme for solving nonlinear Stratonovich Volterra integral equations via Bernoulli’s approximation

In this paper, an efficient method for solving nonlinear Stratonovich Volterra integral equations is proposed. By using Bernoulli polynomials and their stochastic operational matrix of integration, these equations can be reduced to the system of nonlinear algebraic equations with unknown Bernoulli coefficient which can be solved by numerical methods such as Newton’s method. Also, an error analysis is valid under fairly restrictive conditions. Furthermore, in order to show the accuracy and reliability of the proposed method, the new approach is compared with the block pulse functions method by some examples. The obtained results reveal that the proposed method is more accurate and efficient than the block pulse functions method.     

Efficient stochastic model for the point kinetics equations

A new stochastic model for the point kinetics equations with I-delayed neutron precursor groups is presented. In this stochastic model, the point kinetics equations are separated into three terms: prompt neutrons, delayed neutrons and external neutrons source. The matrix form of the efficient stochastic model is solved by a semi-analytical method. The semi-analytical method is based on the exponential function of the coefficient matrix. The eigenvalues of the coefficient matrix and Gaussian elimination are used to calculate this exponential function. The mean and standard deviation of neutron and precursor populations of the efficient stochastic model with step, ramp, and sinusoidal reactivities are computed. The results of the efficient stochastic model are compared with the results of Allen's stochastic model for the point kinetics equations. This comparison confirms that the efficient stochastic model is an accurate model compared with the deterministic point kinetics equations. This stochastic model is efficient to study the natural behavior of neutron and precursor populations in the nuclear reactor dynamics.     

Computational Krylov‐based methods for large‐scale differential Sylvester matrix problems

In the present paper, we propose Krylov‐based methods for solving large‐scale differential Sylvester matrix equations having a low‐rank constant term. We present two new approaches for solving such differential matrix equations. The first approach is based on the integral expression of the exact solution and a Krylov method for the computation of the exponential of a matrix times a block of vectors. In the second approach, we first project the initial problem onto a block (or extended block) Krylov subspace and get a low‐dimensional differential Sylvester matrix equation. The latter problem is then solved by some integration numerical methods such as the backward differentiation formula or Rosenbrock method, and the obtained solution is used to build the low‐rank approximate solution of the original problem. We give some new theoretical results such as a simple expression of the residual norm and upper bounds for the norm of the error. Some numerical experiments are given in order to compare the two approaches.     

A new Jacobi operational matrix: An application for solving fractional differential equations

In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.     

Exponential estimates for stochastic delay hybrid systems with Markovian switching

This paper deals with the problem of norm bounds for the solutions of stochastic hybrid systems with Markovian switching and time delay.Based on Lyapunov-Krasovskii theory for functional differential equations and the linear matrix inequality(LMI)approach,mean square exponential estimates for the solutions of this class of linear stochastic hybrid systems are derived.Finally,An example is illustrated to show the applicability and effectiveness of our method.     

Asymptotic stability in thermoelectromagnetism with memory

A model describing a linear homogeneous dielectric with memory which obeys the Cattaneo–Maxwell law for the heat conduction is presented. The restrictions on the constitutive functionals are found as a direct consequence of the Second Law of Thermodynamics and some free energy potentials exhibited. Such potentials allow to determine a domain of dependence theorem for the first‐order integro‐differential system of equations governing the evolution of the thermoelectromagnetic radiation. The dissipativity due to the memory and to the heat conduction allows to establish some estimates on the asymptotic behaviour and prove the exponential decay of the solution of the system in absence of external sources. Copyright © 1999 John Wiley & Sons, Ltd.     

Stability Analysis of a Reduced Heat Exchanger Model

We consider the mathematical model of a heat exchanger in a heat pumping system using carbon dioxide as refridgerant. From the Euler equations of compressible fluid flow we derive a reduced model, called the zero Mach-number limit, by eliminating time scales associated with soundwaves. The reduced model is a highly nonlinear partial differential-algebraic equation (PDAE). We analyse the stability of this model and show that it is possible to use simple spatial discretisations within the method of lines (MOL), a standard practice in many system simulation software, to simulate the system. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

光滑区域上二维无黏性无热传导Boussinesq方程组与三维轴对称不可压Euler方程组的指数增长全局光滑解

研究二维无黏性无热传导Boussinesq方程组和三维轴对称不可压Euler方程组光滑解的增长情况,找各种区域使其上的方程组有快增长的解。对Boussinesq方程组,通过选取初始温度和速度的一个分量,可以把方程去耦为两部分。从关于涡量的部分求出涡量、速度场和使结论成立的区域,从关于温度的部分,可见温度的高阶导的增长仅依赖于速度场的一个分量。通过适当选取该分量,得到温度高阶导有指数增长的全局光滑解。对轴对称Euler方程组做类似的处理,适当选取速度场的径向分量,可把方程组去耦,最终得到一类光滑区域,在其上方程组有指数增长全局光滑解。该研究把Chae、Constantin、Wu对一个二维锥形区域上无黏性无热传导Boussinesq方程的结果,推广到一类光滑区域上, 并把他们的方法应用到三维轴对称不可压Euler方程组, 得到了类似的结果。     

Quasi-stability and upper semicontinuity for coupled parabolic equations with memory

This current study deals with the long-time dynamics of a nonlinear system of coupled parabolic equations with memory. The system describes the thermodiffusion phenomenon where the fluxes of mass diffusion and heat depend on the past history of the chemical potential and the temperature gradients, respectively, according to Gurtin-Pipkin's law. Inspired by the works of Chueshov and Lasiecka on the property of quasi-stability of dynamic systems, we prove this property for the problem considered in this study. This property allows us to analyze certain properties of global and exponential attractors in a more efficient and practical way. This approach is applied for the first time for coupled parabolic equations. We analyze the continuity of global attractors with respect to a pair of parameters in a residual dense set and their upper semicontinuity in a complete metric space. Finally, we analyze the upper semicontinuity of global attractors with respect to small perturbations of the damping terms.     

Sensitivity to Small Delays of Pathwise Stability for Stochastic Retarded Evolution Equations

In this paper, we shall study the almost sure pathwise exponential stability property for a class of stochastic functional differential equations with delays, possibly, in the highest-order derivative terms driven by multiplicative noise. Instead of establishing a moment exponential stability as the first step and then proceeding to investigate the pathwise stability of the system under consideration, we shall develop a direct approach for this problem. As a consequence, we can show that some systems, which are not exponential momently stable, have the exponential stability not sensitive to small delays in the almost sure sense.