• Title/Summary/Keyword: Parabolic problems

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SERIES SOLUTIONS TO INITIAL-NEUMANN BOUNDARY VALUE PROBLEMS FOR PARABOLIC AND HYPERBOLIC EQUATIONS

  • Bougoffa, Lazhar;Al-Mazmumy, M.
    • Journal of applied mathematics & informatics
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    • v.31 no.1_2
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    • pp.87-97
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    • 2013
  • The purpose of this paper is to employ a new useful technique to solve the initial-Neumann boundary value problems for parabolic, hyperbolic and parabolic-hyperbolic equations and obtain a solution in form of infinite series. The results obtained indicate that this approach is indeed practical and efficient.

OSCILLATIONS OF CERTAIN NONLINEAR DELAY PARABOLIC BOUNDARY VALUE PROBLEMS

  • Zhang, Liqin;Fu, Xilin
    • Journal of applied mathematics & informatics
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    • v.8 no.1
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    • pp.137-149
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    • 2001
  • In this paper we consider some nonlinear parabolic partial differential equations with distributed deviating arguments and establish sufficient conditions for the oscillation of some boundary value problems.

DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR PARABOLIC PROBLEMS WITH MIXED BOUNDARY CONDITION

  • Ohm, Mi Ray;Lee, Hyun Yong;Shin, Jun Yong
    • Journal of applied mathematics & informatics
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    • v.32 no.5_6
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    • pp.585-598
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    • 2014
  • In this paper we consider the nonlinear parabolic problems with mixed boundary condition. Under comparatively mild conditions of the coefficients related to the problem, we construct the discontinuous Galerkin approximation of the solution to the nonlinear parabolic problem. We discretize spatial variables and construct the finite element spaces consisting of discontinuous piecewise polynomials of which the semidiscrete approximations are composed. We present the proof of the convergence of the semidiscrete approximations in $L^{\infty}(H^1)$ and $L^{\infty}(L^2)$ normed spaces.

LEAST-SQUARES SPECTRAL COLLOCATION PARALLEL METHODS FOR PARABOLIC PROBLEMS

  • SEO, JEONG-KWEON;SHIN, BYEONG-CHUN
    • Honam Mathematical Journal
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    • v.37 no.3
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    • pp.299-315
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    • 2015
  • In this paper, we study the first-order system least-squares (FOSLS) spectral method for parabolic partial differential equations. There were lots of least-squares approaches to solve elliptic partial differential equations using finite element approximation. Also, some approaches using spectral methods have been studied in recent. In order to solve the parabolic partial differential equations in parallel, we consider a parallel numerical method based on a hybrid method of the frequency-domain method and first-order system least-squares method. First, we transform the parabolic problem in the space-time domain to the elliptic problems in the space-frequency domain. Second, we solve each elliptic problem in parallel for some frequencies using the first-order system least-squares method. And then we take the discrete inverse Fourier transforms in order to obtain the approximate solution in the space-time domain. We will introduce such a hybrid method and then present a numerical experiment.

Lp error estimates and superconvergence for finite element approximations for nonlinear parabolic problems

  • LI, QIAN;DU, HONGWEI
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.4 no.1
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    • pp.67-77
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    • 2000
  • In this paper we consider finite element mathods for nonlinear parabolic problems defined in ${\Omega}{\subset}R^d$ ($d{\leq}4$). A new initial approximation is taken. Optimal order error estimates in $L_p$ for $2{\leq}p{\leq}{\infty}$ are established for arbitrary order finite element. One order superconvergence in $W^{1,p}$ for $2{\leq}q{\leq}{\infty}$ are demonstrated as well.

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FINITE VOLUME ELEMENT METHODS FOR NONLINEAR PARABOLIC PROBLEMS

  • LI, QIAN;LIU, ZHONGYAN
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.6 no.2
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    • pp.85-97
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    • 2002
  • In this paper, finite volume element methods for nonlinear parabolic problems are proposed and analyzed. Optimal order error estimates in $W^{1,p}$ and $L_p$ are derived for $2{\leq}p{\leq}{\infty}$. In addition, superconvergence for the error between the approximation solution and the generalized elliptic projection of the exact solution (or and the finite element solution) is also obtained.

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FINITE VOLUME ELEMENT METHODS FOR NONLINEAR PARABOLIC INTEGRODIFFERENTIAL PROBLEMS

  • Li, Huanrong;Li, Qian
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.7 no.2
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    • pp.35-49
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    • 2003
  • In this paper, finite volume element methods for nonlinear parabolic integrodifferential problems are proposed and analyzed. The optimal error estimates in $L^p\;and\;W^{1,p}\;(2\;{\leq}\;p\;{\leq}\;{\infty})$ as well as some superconvergence estimates in $W^{1,p}\;(2\;{\leq}\;p\;{\leq}\;{\infty})$ are obtained. The main results in this paper perfect the theory of FVE methods.

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DUALITY IN THE OPTIMAL CONTROL PROBLEMS OF NONLINEAR PARABOLIC SYSTEMS

  • Lee, Mi-Jin
    • East Asian mathematical journal
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    • v.16 no.2
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    • pp.267-275
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    • 2000
  • In this paper, we study the duality theory of nonlinear parabolic systems. The main objective is to prove the duality theorem under general conditions within an infinite-dimensional framework. As an application, an example is given.

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ERROR ESTIMATIES FOR A FREQUENCY-DOMAIN FINITE ELEMENT METHOD FOR PARABOLIC PROBLEMS WITH A NEUMANN BOUNDARY CONDITION

  • Lee, Jong-Woo
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.2
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    • pp.345-362
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    • 1998
  • We introduce and anlyze a naturally parallelizable frequency-domain method for parabolic problems with a Neumann boundary condition. After taking the Fourier transformation of given equations in the space-time domain into the space-frequency domain, we solve an indefinite, complex elliptic problem for each frequency. Fourier inversion will then recover the solution of the original problem in the space-time domain. Existence and uniqueness of a solution of the transformed problem corresponding to each frequency is established. Fourier invertibility of the solution in the frequency-domain is also examined. Error estimates for a finite element approximation to solutions fo transformed problems and full error estimates for solving the given problem using a discrete Fourier inverse transform are given.

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SYMMETRY AND MONOTONICITY OF SOLUTIONS TO FRACTIONAL ELLIPTIC AND PARABOLIC EQUATIONS

  • Zeng, Fanqi
    • Journal of the Korean Mathematical Society
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    • v.58 no.4
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    • pp.1001-1017
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    • 2021
  • In this paper, we first apply parabolic inequalities and a maximum principle to give a new proof for symmetry and monotonicity of solutions to fractional elliptic equations with gradient term by the method of moving planes. Under the condition of suitable initial value, by maximum principles for the fractional parabolic equations, we obtain symmetry and monotonicity of positive solutions for each finite time to nonlinear fractional parabolic equations in a bounded domain and the whole space. More generally, if bounded domain is a ball, then we show that the solution is radially symmetric and monotone decreasing about the origin for each finite time. We firmly believe that parabolic inequalities and a maximum principle introduced here can be conveniently applied to study a variety of nonlocal elliptic and parabolic problems with more general operators and more general nonlinearities.