Galerkin's finite element formulation of the system of fourth-order boundary-value problems

Abstract

In this article, a Galerkin's finite element approach based on weighted-residual is presented to find approximate solutions of a system of fourth-order boundary-value problems associated with obstacle, unilateral and contact problems. The approach utilizes a piece-wise cubic approximations utilizing cubic Hermite interpolation polynomials. Numerical studies have shown the superior accuracy and lesser computational cost of the scheme in comparison to cubic spline, non-polynomial spline and cubic non-polynomial spline methods. Numerical examples are presented to illustrate the applicability of the method.

A study of obstacle problems using homotopy perturbation method

“…ℋ; 1 = ; 1 + ℎ = 0 (8) ;  =        (9)  = lim → ;  =    +    +    … (10) (11) ... (3) with the boundary conditions  =  , , , ,  ≤  ≤  , , , ,  ≤  ≤  , , , ,  ≤  ≤ …”

“…... (11) with boundary conditions u(0) = u(π) = 0. The analytical solution for the above mentioned example is given by  = lim → ;  =    +    +    … (10)…”

“…The solution of the fourth order system of boundary value problem was presented in 2007 using the non polynomial spline technique (Siddiqi & Akram, 2007a) and cubic non polynomial spline technique (Siddiqi & Akram, 2007b). Also a solution of fourth order system was suggested using Galerkin's finite element method in 2011 (Iqbal, 2011).…”

OHAM and FEM solutions of concentric n-layer flows of incompressible third-grade fluids in a horizontal cylindrical pipe

“…In this section, we apply Galerkin's FEM [32][33][34][35][36] to solve governing differential Eq. 17, with respect to the boundary conditions (18) and (19) for finding fluid velocity in comparison with OHAM.…”

“…The significant plan is that the source functions can be characterized piecewise over subregions of the domain called finite elements and that over any sub-domain, the source functions can be selected to be very straightforward functions such a polynomials of low degree. In this way, the Galerkin's definition in the finite element model requires, see [33][34][35][36], that we choose a suitable preliminary or source approximation ũ(r) to the true solution u(r) that is connected locally over an ordinary finite element in the complete r domain. Now, in order to solve the governing differential Eq.…”