A REDUCED-ORDER MFE FORMULATION BASED ON POD METHOD FOR PARABOLIC EQUATIONS

In this paper, we extend the applications of proper orthogonal decomposition （POD） method, i.e., apply POD method to a mixed finite element （MFE） formulation naturally satisfied Brezz-Babu^ka for parabolic equations, establish a reduced-order MFE formulation with lower dimensions and sufficiently high accuracy, and provide the error estimates between the reduced-order POD MFE solutions and the classical MFE solutions and the implementation of algorithm for solving reduced-order MFE formulation. Some numerical examples illustrate the fact that the results of numerical computation are consis- tent with theoretical conclusions. Moreover, it is shown that the new reduced-order MFE formulation based on POD method is feasible and efficient for solving MFE formulation for parabolic equations.

Monotone Iterative Methodfor Nonlinear Discontinuous ParabolicDifferential Equations

In this paper, the existence of solutions for discontinuous nonlinear parabolic differential IBVP is proved by using a more generalized monotone iterative method. Moreover, the convergence of this method is discussed.

SOLVING COUPLED PSEUDO-PARABOLIC EQUATION USING A MODIFIED DOUBLE LAPLACE DECOMPOSITION METHOD

In this paper, the modification of double Laplace decomposition method is pro- posed for the analytical approximation solution of a coupled system of pseudo-parabolic equation with initial conditions. Some examples are given to support our presented method. In addition, we prove the convergence of double Laplace transform decomposition method applied to our problems.

Decomposition method for solving parabolic equations in finite domains

This paper presents a comparison among Adomian decomposition method (ADM), Wavelet-Galerkin method (WGM), the fully explicit (1,7) finite difference technique (FTCS), the fully implicit (7,1) finite difference method (BTCS), (7,7) Crank-Nicholson type finite difference formula (C-N), the fully explicit method (1,13) and 9-point finite difference method, for solving parabolic differential equations with arbitrary boundary conditions and based on weak form functionals in finite domains. The problem is solved rapidly, easily and elegantly by ADM. The numerical results on a 2D transient heat conducting problem and 3D diffusion problem are used to validate the proposed ADM as an effective numerical method for solving finite domain parabolic equations. The numerical results showed that our present method is less time consuming and is easier to use than other methods. In addition, we prove the convergence of this method when it is applied to the nonlinear parabolic equation.

Alternating Direction Finite Volume Element Methods for Three-Dimensional Parabolic Equations

This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The L2 norm and H1 semi-norm error estimates are obtained for the first scheme and second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.

AN A.D.I.GALERKIN METHOD FOR NONLINEAR PARABOLIC INTEGRO-DIFFERENTIAL EQUATION USING PATCH APPROXIMATION

An A. D. I. Galerkin scheme for three-dimensional nonlinear parabolic integro-differen-tial equation is studied. By using alternating-direction, the three-dimensional problem is reduced to a family of single space variable problems, the calculation is simplified; by using a local approxima-tion of the coefficients based on patches of finite elements, the coefficient matrix is updated at each time step; by using Ritz-Volterra projection, integration by part and other techniques, the influence coming from the integral term is treated; by using inductive hypothesis reasoning, the difficulty coming from the nonlinearity is treated. For both Galerkin and A. D. I. Galerkin schemes the con-vergence properties are rigorously demonstrated, the optimal H^1-norm and L^2-norm estimates are obtained.

On L ∞ Stability and Convergence of Fictitious Domain Method for the Numerical Solution to Parabolic Differential Equation with Derivative Boundary Conditions

This paper investigates some known difference schemes for the numerical solution to parabolic differential equation with derivative boundary conditions by the fictitious domain method.The stability and convergence in L ∞ are proven.

Solving multi-object radar cross section based on wide-angle parabolic equation method

Based on a Pade approximation, a wide-angle parabolic equation method is introduced for computing the multiobject radar cross section （RCS） for the first time. The method is a paraxial version of the scalar wave equation, which solves the field by marching them along the paraxial direction. Numerical results show that a single wide-angle parabofic equation run can compute multi-object RCS efficiently for angles up to 45 ° . The method provides anew and efficient numerical method for computation electromagnetics.

Spectral Method for Semilinear Parabolic Integrodifferential Equations

Based on the discussion of the semidiscretization of a parabolic equation with asemilinear memory term,an error estimate is derived for the fully discrete scheme with spectral method in space and the backward Euler method in time The trapezoidal rule is adopted.for the quadrature of the memory term and the quadrature error isestimated.

An O(k²+kh²+h²) Accurate Two-level Implicit Cubic Spline Method for One Space Dimensional Quasi-linear Parabolic Equations

In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k2 + kh2 + h4) for the solution of quasi-linear parabolic equation , 0 0 subject to appropriate initial and Dirichlet boundary conditions, where h > 0, k > 0 are grid sizes in space and time-directions, respectively. The cubic spline approximation produces at each time level a spline function which may be used to obtain the solution at any point in the range of the space variable. The proposed cubic spline method is applicable to parabolic equations having singularity. The stability analysis for diffusion- convection equation shows the unconditionally stable character of the cubic spline method. The numerical tests are performed and comparative results are provided to illustrate the usefulness of the proposed method.

Developments of parabolic equation method in the period of 2000–2016

Parabolic equation （PE） method is an efficient tool for modelling underwater sound propagation, particularly for problems involving range dependence. Since the PE method was first introduced into the field of underwater acoustics, it has been about 40 years, during which contributions to extending its capability has been continuously made. The most recent review paper surveyed the contributions made before 1999. In the period of 2000-2016, the development of PE method basically focuses on seismo-acoustic problems, three-dimensional problems, and realistic applications. In this paper, a review covering the contribution from 2000 to 2016 is given, and what should be done in future work is also discussed.

GLOBAL ASYMPTOTICS TOWARD THE REREFACTION WAVES FOR A PARABOLIC-ELLIPTIC SYSTEM RELATED TO THE CAMASSA-HOLM SHALLOW WATER EQUATION

This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation {ut＋（u^2/2）x＋px=εuxx, t〉0,x∈R, -αPxx＋P=f（u）＋α/2ux^2-1/2u^2, t〉0,x∈R, （E） with the initial data u（0,x）=u0（x）→u±, as x→±∞ （I） Here, u_ 〈 u＋ are two constants and f（u） is a sufficiently smooth function satisfying f＂ （u） 〉 0 for all u under consideration. Main aim of this article is to study the relation between solutions to the above Cauchy problem and those to the Riemann problem of the following nonlinear conservation law It is well known that if u_ 〈 u＋, the above Riemann problem admits a unique global entropy solution u^R（x/t） u^R（x/t）={u_,（f′）^-1（x/t）,u＋, x≤f′（u_）t, f′（u_）t≤x≤f′（u＋）t, x≥f′（u＋）t. Let U（t, x） be the smooth approximation of the rarefaction wave profile constructed similar to that of [21, 22, 23], we show that if u0（x） - U（0,x） ∈ H^1（R） and u_ 〈 u＋, the above Cauchy problem （E） and （I） admits a unique global classical solution u（t, x） which tends to the rarefaction wave u^R（x/t） as → ＋∞ in the maximum norm. The proof is given by an elementary energy method.

Partition of Unity for a Class of Nonlinear Parabolic Equation on Overlapping Non-Matching Grids

A class of nonlinear parabolic equation on a polygonal domain Ω R2 is inves- tigated in this paper. We introduce a finite element method on overlapping non-matching grids for the nonlinear parabolic equation based on the partition of unity method. We give the construction and convergence analysis for the semi-discrete and the fully discrete finite element methods. Moreover, we prove that the error of the discrete variational problem has good approximation properties. Our results are valid for any spatial dimensions. A numerical example to illustrate the theoretical results is also given.

THE HOMOGENEOUS DIRICHLET PROBLEM FOR QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC-HYPERBOLIC EQUATION WITH L^p INITIAL VALUE

The aim of this paper is to prove the well-posedness（existence and uniqueness） of the L p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L p initial value.We use the device of doubling variables and some technical analysis to prove the uniqueness result.Moreover we can prove that the L p entropy solution can be obtained as the limit of solutions of the corresponding regularized equations of nondegenerate parabolic type.

NEW ALTERNATING DIRECTION FINITE ELEMENT SCHEME FOR NONLINEAR PARABOLIC EQUATION

A new alternating direction (AD) finite element (FE) scheme for 3-dimensional nonlinear parabolic equation and parabolic integro-differential equation is studied. By using AD,the 3-dimensional problem is reduced to a family of single space variable problems, calculation work is simplified; by using FE, high accuracy is kept; by using various techniques for priori estimate for differential equations such as inductive hypothesis reasoning, the difficulty arising from the nonlinearity is treated. For both FE and ADFE schemes, the convergence properties are rigorously demonstrated, the optimal H1- and L2-norm space estimates and the O((△t)2) estimate for time variable are obtained.

STABILIZED NUMERICAL APPROXIMATIONS OF THE BACKWARD PROBLEM OF A PARABOLIC EQUATION

In this paper, the backward problem of a parabolic equation is considered. Three new stability estimates are given. Based on the new stability estimates, a regularization method is proposed for which error estimates are available. The regularization method can be used for the numerical approximations of the original problem which will be shown by the numerical examples.

Second-order two-scale analysis and numerical algorithms for the hyperbolic–parabolic equations with rapidly oscillating coefficients

We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, we theoretically explain the importance of the second-order two-scale solution by the error analysis in the pointwise sense. The associated explicit convergence rates are also obtained. Then a second-order two-scale numerical method based on the Newmark scheme is presented to solve the equations. Finally, some numerical examples are used to verify the effectiveness and efficiency of the multiscale numerical algorithm we proposed.

THE NUMERICAL SOLUTION OF A SINGULARLY PERTURBED PROBLEM FOR SEMILINEAR PARABOLIC DIFFERENTIAL EQUATION

The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.

A Note on Homogenization of Parabolic Equation in Perforated Domains

We are concerned with a class of parabolic equations in periodically perforated domains with a homogeneous Neumann condition on the boundary of holes.By using the periodic unfolding method in perforated domains, we obtain the homogenization results under the conditions slightly weaker than those in the corresponding case considered by Nandakumaran and Rajesh（Nandakumaran A K, Rajesh M. Homogenization of a parabolic equation in perforated domain with Neumann boundary condition. Proc. Indian Acad. Sci.（Math. Sci.）, 2002, 112（1）： 195–207）. Moreover,these results generalize those obtained by Donato and Nabil（Donato P, Nabil A. Homogenization and correctors for the heat equation in perforated domains. Ricerche di Matematica L. 2001, 50： 115–144）.

Existence and Nonexistence of Global Solutions of a Fully Nonlinear Parabolic Equation

In the paper, we study the global existence of weak solution of the fully nonlinear parabolic problem (1.1)-(1.3) with nonlinear boundary conditions for the situation without strong absorption terms. Also, we consider the blow up of global solution of the problem (1.1)-(1.3) by using the convexity method.