A Quadratic $C^0$ Interior Penalty Method for Linear Fourth Order Boundary Value Problems with Boundary Conditions of the Cahn--Hilliard Type

2012 ◽  
Vol 50 (4) ◽  
pp. 2088-2110 ◽  
Author(s):  
Susanne C. Brenner ◽  
Shiyuan Gu ◽  
Thirupathi Gudi ◽  
Li-yeng Sung
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Penalty Method ◽  
Boundary Value ◽  
Interior Penalty ◽  
Interior Penalty Method

scholarly journals Solvability of a fourth order boundary value problem with periodic boundary conditions

1988 ◽  
Vol 11 (2) ◽  
pp. 275-284
Author(s):  
Chaitan P. Gupta
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Elastic Beam ◽  
Periodic Boundary Conditions ◽  
Uniqueness Of Solutions

Fourth order boundary value problems arise in the study of the equilibrium of an elastaic beam under an external load. The author earlier investigated the existence and uniqueness of the solutions of the nonlinear analogues of fourth order boundary value problems that arise in the equilibrium of an elastic beam depending on how the ends of the beam are supported. This paper concerns the existence and uniqueness of solutions of the fourth order boundary value problems with periodic boundary conditions.

scholarly journals Existence and Global Behavior of Positive Solutions for Some Fourth-Order Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Ramzi S. Alsaedi
Keyword(s):  
Continuous Function ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Global Behavior ◽  
Nonnegative Continuous Function

We establish the existence and uniqueness of a positive solution to the following fourth-order value problem:u(4)(x)=a(x)uσ(x),x∈(0,1)with the boundary conditionsu(0)=u(1)=u'(0)=u'(1)=0, whereσ∈(-1,1)andais a nonnegative continuous function on (0, 1) that may be singular atx=0orx=1. We also give the global behavior of such a solution.

scholarly journals Numerical Solutions of General Fourth Order Two Point Boundary Value Problems by Galerkin Method with Legendre Polynomials

2015 ◽  
Vol 62 (2) ◽  
pp. 103-108 ◽  
Author(s):  
Md Bellal Hossain ◽  
Md Shafiqul Islam
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Legendre Polynomials ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Equivalent Form ◽  
Linear And Nonlinear

In this paper, Galerkin weighted residual method is presented to find the numerical solutions of the general fourth order linear and nonlinear differential equations with essential boundary conditions. For this, the given differential equations and the boundary conditions over arbitrary finite domain [a, b] are converted into its equivalent form over the interval [0, 1]. Here the Legendre polynomials, over the interval [0, 1], are chosen as trial functions satisfying the corresponding homogeneous form of the Dirichlet boundary conditions. Details matrix formulations are derived for solving linear and nonlinear boundary value problems (BVPs). Numerical examples for both linear and nonlinear BVPs are considered to verify the proposed formulation and the results obtained are compared. DOI: http://dx.doi.org/10.3329/dujs.v62i2.21973 Dhaka Univ. J. Sci. 62(2): 103-108, 2014 (July)

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