Numerical solution of singularly perturbed fifth order two point boundary value problem

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Abstract

We will consider Adomain decomposition method and the homotopy method to solve a fifth order singularly perturbed BVP arising in viscoelastic flows. The success and pitfalls of the methods will be investigated. Numerical testing will be provided to show the efficiency of the methods proposed. Comparison with the work of others will also be done.

Introduction

The problem under consideration has the formd4udx4+ϵdudxd5udx5=f,where u = u(x) and f = f(ϵ,x)  are defined on the interval -12x12, and u(x) satisfies the boundary conditionsu±12=dudx±12=0,d2udx2-12=c.

Here ϵ and c are positive constants that represents the elasticity parameter and a boundary stress, respectively. The problem given by (1), (2), (3) captures, in one dimension, the essential character of the primary hyperbolic-elliptic operator of the form(1+ϵv.)Δ2encountered in 2-dimensional creeping viscoelastic flows, where v is the velocity and Δ2 is the biharmonic operator, see Davies, Karageorghis and Phillips [5].

Numerical consideration of viscoelastic flows has been the subject of interest of many researchers. Such flows are normally modeled by coupled quasi-linear partial differential systems of mixed hyperbolic-elliptic type. There is always difficulty in the numerical treatment of such problems by conventional methods due to the presence of high gradients in velocity, pressure and stress. In their work, Marchal and Crochet [8], and through a mixed finite element formulation with higher order (Hermitian) trial functions, avoided the appearance of spurious limit points when the elasticity parameter is large. The presence of such limit points often leads to a significant loss of accuracy. Davies, Karageorghis and Phillips [5] used Galerkin method with very high order trial functions; that is, they extended the work of Marchal and Crochet [8]. In a different paper, Karageorghis, Phillips and Davies [6], used expansions in terms of beam functions and Chebysheve polynomials to solve the two point boundary value problem using spectral collocation. The resulting linear-nonlinear system was then solved using Newton’s method. Due to the presence of a small parameter ϵ the problem can be considered a singularly perturbed problem. Attili [4] regularized the two point boundary value problem first and then employed initial value solvers through the shooting to solve the resulting regular problem. The analytical aspects of such perturbed systems through asymptotic analysis are well established, see Kevorkin and Cole [7], Nayfeh [9] and O’Mally [10], [11]. Studies conclude that the solution has two states; namely, one varies slowly and the other rapidly with respect to x. Also, the presence of the small parameter ϵ leads to what is called boundary layer phenomenon where the solution does not converge uniformly at end points.

We will consider the application of the Adomain decomposition method and the homotopy method for solving the problem given by (1), (2), (3). No regularization or special treatment of the original problem is needed. Different right hand sides that depends on both x and ϵ will be tested. The two methods will be tested and we will report their successes or pitfalls.

The outline of the paper will be as follows. In Section 2, we will describe the Adomain decomposition method in its general format and its suitability for our problem. The homotopy method will be explained in Section 3. Numerical details and experimentation will be done in the last section together with comparison with the work of others.

Section snippets

Adomain decomposition method

The Adomain decomposition method usually defines the given equation in an operator form by considering the highest-order derivative in the problem. Consider the nonlinear problemH(u)=f,we separate the operator H into three parts asH=L+R+N,where N is the nonlinear part of the operator and L + R form the linear term. Here L is usually chosen to be easily invertible and R is the remainder of the linear term. The method is based on applying the operator L−1 formally to the equationL(u)=f-R(u)-N(u).

Homotopy method

Consider our problem againd4udx4+ϵdudxd5udx5=f(ϵ,x).

Rewrite it asL(u)+N(u)=f1(x)+f2(ϵ,x),where f(ϵ,x) =  f1(x) + f2(ϵ,x), L(u)=d4udx4 is the linear part of the operator and N(u)=ϵdudxd5udx5 is the nonlinear part. Construct a homotopy in the formH(u,λ)=(1-λ)(L(u)-f1(x))+λ(L(u)+N(u)-f(ϵ,x))=L(u)-f1(x)+λ(N(u)-f2(ϵ,x)).

If λ = 0, then H(u,0) = 0 is the linear ordinary differential equationd4udx4=f1(x)which can be solved directly and the solution is given byu0=f1(x)dxdxdxdx.

If λ = 1, then H(u,1) = 0 is our

Numerical results

To study the performance of the initial value solvers described in Sections 2 Adomain decomposition method, 3 Homotopy method, three test problems were considered. They are the same test problems considered by [5], [6], and [4]. Comparison with their results will be presented. The general form of the problems is given byd4udx4+ϵdudxd5udx5=f(ϵ,x),-12<x<12,u±12=dudx±12=0,d2udx2-12=c,where the difference from one problem to another is in the right hand side f(ϵ,x).

Example 4.1

Consider the initial value

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