• Title/Summary/Keyword: fourth-order equations

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NUMERICAL SOLUTION OF A CONSTRICTED STEPPED CHANNEL PROBLEM USING A FOURTH ORDER METHOD

  • Mancera, Paulo F. de A.;Hunt, Roland
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.3 no.2
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    • pp.51-67
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    • 1999
  • The numerical solution of the Navier-Stokes equations in a constricted stepped channel problem has been obtained using a fourth order numerical method. Transformations are made to have a fine grid near the sharp corner and a long channel downstream. The derivatives in the Navier-Stokes equations are replaced by fourth order central differences which result a 29-point computational stencil. A procedure is used to avoid extra numerical boundary conditions near the solid walls. Results have been obtained for Reynolds numbers up to 1000.

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ON A CLASS OF NONCOOPERATIVE FOURTH-ORDER ELLIPTIC SYSTEMS WITH NONLOCAL TERMS AND CRITICAL GROWTH

  • Chung, Nguyen Thanh
    • Journal of the Korean Mathematical Society
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    • v.56 no.5
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    • pp.1419-1439
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    • 2019
  • In this paper, we consider a class of noncooperative fourth-order elliptic systems involving nonlocal terms and critical growth in a bounded domain. With the help of Limit Index Theory due to Li [32] combined with the concentration compactness principle, we establish the existence of infinitely many solutions for the problem under the suitable conditions on the nonlinearity. Our results significantly complement and improve some recent results on the existence of solutions for fourth-order elliptic equations and Kirchhoff type problems with critical growth.

A FOURTH-ORDER FAMILY OF TRIPARAMETRIC EXTENSIONS OF JARRATT'S METHOD

  • Kim, Young Ik
    • Journal of the Chungcheong Mathematical Society
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    • v.25 no.3
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    • pp.579-587
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    • 2012
  • A fourth-order family of triparametric extensions of Jarratt's method are proposed in this paper to find a simple root of nonlinear algebraic equations. Convergence analysis including numerical experiments for various test functions apparently verifies the fourth-order convergence and asymptotic error constants.

Trends in Researches for Fourth Order Elliptic Equations with Dirichlet Boundary Condition

  • Park, Q-Heung;Yinghua Jin
    • Journal for History of Mathematics
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    • v.16 no.4
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    • pp.107-115
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    • 2003
  • The nonlinear fourth order elliptic equations with jumping nonlinearity was modeled by McKenna. We investigate the trends for the researches of the existence of solutions of a fourth order semilinear elliptic boundary value problem with Dirichlet boundary Condition, ${\Delta}^2u{+}c{\Delta}u=b_1[(u+1)^{-}1]{+}b_2u^+$ in ${\Omega}$, where ${\Omega}$ is a bounded open set in $R^N$ with smooth boundary ${\partial}{\Omega}$.

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MULTIPLICITY RESULTS FOR SOME FOURTH ORDER ELLIPTIC EQUATIONS

  • Jin, Yinghua;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.18 no.4
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    • pp.489-496
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    • 2010
  • In this paper we consider the Dirichlet problem for an fourth order elliptic equation on a open set in $R^N$. By using variational methods we obtain the multiplicity of nontrivial weak solutions for the fourth order elliptic equation.

ANALYTICAL SOLUTION OF SINGULAR FOURTH ORDER PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS OF VARIABLE COEFFICIENTS BY USING HOMOTOPY PERTURBATION TRANSFORM METHOD

  • Gupta, V.G.;Gupta, Sumit
    • Journal of applied mathematics & informatics
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    • v.31 no.1_2
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    • pp.165-177
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    • 2013
  • In this paper, we apply Homotopy perturbation transform method (HPTM) for solving singular fourth order parabolic partial differential equations with variable coefficients. This method is the combination of the Laplace transform method and Homotopy perturbation method. The nonlinear terms can be easily handled by the use of He's polynomials. The aim of using the Laplace transform is to overcome the deficiency that is mainly caused by unsatisfied conditions in other semi-analytical methods such as Homotopy perturbation method (HPM), Variational iteration method (VIM) and Adomain Decomposition method (ADM). The proposed scheme finds the solutions without any discretization or restrictive assumptions and avoids the round-off errors. The comparison shows a precise agreement between the results and introduces this method as an applicable one which it needs fewer computations and is much easier and more convenient than others, so it can be widely used in engineering too.

INFINITELY MANY SOLUTIONS FOR A CLASS OF MODIFIED NONLINEAR FOURTH-ORDER ELLIPTIC EQUATIONS ON ℝN

  • Che, Guofeng;Chen, Haibo
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.3
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    • pp.895-909
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    • 2017
  • This paper is concerned with the following fourth-order elliptic equations $${\Delta}^2u-{\Delta}u+V(x)u-{\frac{k}{2}}{\Delta}(u^2)u=f(x,u),\text{ in }{\mathbb{R}}^N$$, where $N{\leq}6$, ${\kappa}{\geq}0$. Under some appropriate assumptions on V(x) and f(x, u), we prove the existence of infinitely many negative-energy solutions for the above system via the genus properties in critical point theory. Some recent results from the literature are extended.

HYPERBOLIC EQUATION FOR FOURTH ORDER WITH MULTIPLE CHARACTERISTICS

  • Bougoffa, Lazhar
    • Journal of applied mathematics & informatics
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    • v.28 no.3_4
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    • pp.837-844
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    • 2010
  • In this paper, a class of initial value problem of hyperbolic equations for fourth order with multiple characteristics is considered and can be solved analytically by variable transforms. Also, similar to Goursat's problem we present a direct integration technique for finding a new solutions of an inhomogeneous hyperbolic equation of fourth order such that the attached conditions are given on its multiple characteristics.

INFINITELY MANY HOMOCLINIC SOLUTIONS FOR DIFFERENT CLASSES OF FOURTH-ORDER DIFFERENTIAL EQUATIONS

  • Timoumi, Mohsen
    • Communications of the Korean Mathematical Society
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    • v.37 no.1
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    • pp.137-161
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    • 2022
  • In this article, we study the existence and multiplicity of homoclinic solutions for the following fourth-order differential equation (1) u(4)(x) + ωu''(x) + a(x)u(x) = f(x, u(x)), ∀x ∈ ℝ where a(x) is not required to be either positive or coercive, and F(x, u) = ∫u0 f(x, v)dv is of subquadratic or superquadratic growth as |u| → ∞, or satisfies only local conditions near the origin (i.e., it can be subquadratic, superquadratic or asymptotically quadratic as |u| → ∞). To the best of our knowledge, there is no result published concerning the existence and multiplicity of homoclinic solutions for (1) with our conditions. The proof is based on variational methods and critical point theory.