On a numerical method for a homogeneous, nonlinear, nonlocal, elliptic boundary value problem

https://doi.org/10.1016/j.na.2010.10.042Get rights and content

Abstract

In this work we develop a numerical method for the equation: α(01u(t)dt)u(x)+[u(x)]2n+1=0,x(0,1),u(0)=a,u(1)=b. We begin by establishing a priori estimates and the existence and uniqueness of the solution to the nonlinear auxiliary problem via the Schauder fixed point theorem. From this analysis, we then prove the existence and uniqueness to the problem above by defining a continuous compact mapping, utilizing the a priori estimates and the Brouwer fixed point theorem. Next, we analyze a discretization of the above problem and show that a solution to the nonlinear difference problem exists and is unique and that the numerical procedure converges with error (h). We conclude with some examples of the numerical process.

Introduction

In [1] the authors analyzed the analytic and numerical solution to the nonlocal elliptic B.V.P.: α(01u(t)dt)u(x)=f(x),x(0,1),u(0)=a,u(1)=b, which was a one-dimensional problem similar to those discussed in [2], [3], [4], [5]. The crux of the research in [1] essentially amounted to analyzing conditions on the coefficient and data, which lead to the existence and uniqueness of the solution, the existence and uniqueness of a numerical approximation and the convergence of the numerical approximation to the analytic solution.

This analysis also motivated the consideration of the following equation: α(01u(t)dt)u(x)+[u(x)]2n+1=0,x(0,1),u(0)=a,u(1)=b, which is a nonlinear, nonlocal boundary value problem whose u coefficient is dependent upon the integral of the solution over the domain of the solution, analogous to the nonlocal problem in [1]. We ultimately show that the solution of the analytic problem (1.1) exists and is unique, and also demonstrate that a suitable numerical approximation yields similar existence and uniqueness results as well. In addition, we establish that an approximation to (1.1) converges to the analytic solution of (1.1) with a satisfactory degree of accuracy, as was discovered for the problem defined in [1], and also supply examples of the numerical process.

It is also important to mention that the numerical methods developed in this paper function as a paradigm for further numerical methods to be developed for various other analytic problems with physical applications. In fact, much research has been done on nonlocal modeling problems with structures very similar to (1.1). Below we discuss some of these problems and their applications. In each example, we also display the modeling equation to emphasize the similarity with (1.1).

A nonlocal problem modeling Ohmic heating with variable thermal conductivity was studied in [6], including an analysis of the asymptotic behavior and the blow-up of solutions. This model has the following form: ut=(u3ux)x+λf(u)(11f(u)dx)2,1<x<1,t>0,u(1,t)=u(1,t)=0,t>0u(x,0)=u0(x),1<x<1, where λ is a positive parameter. This work was motivated by the problem studied in [7], [8] that models Ohmic heating, which contains a standard linear diffusion term of the form uxx and is also a nonlocal problem.

In addition, Stańczy [9] studied nonlocal elliptic equations that arise in physical models including systems of particles in thermodynamical equilibrium interacting via gravitational (Coulomb) potential and a similar problem was studied in [10]. The equations that Stańczy studied also arose in fully turbulent behavior of a real flow, thermal runway in Ohmic heating, shear bands in metals deformed under high strain rates and one dimensional fluid flows with rate of strain proportional to a power of stress multiplied by a function of temperature. The model in [9] is defined as follows: Δφ=Mf(φ)α(Ωf(φ))βin ΩRn, where φ=0 on Ω.

Lastly, a nonlocal problem that arises as a local model for the temperature in a thin region, which occurs during linear friction welding, was studied in [11]. A similar model was addressed in [12], which models thermo–viscoelastic flows. The model in [11] has the structure seen below: ut=uxx+f(u)(0f(u)dy)(1+a)for 0<x<, with ux=0 on x=0 and ux1 as x.

Section snippets

A priori estimates for the solution of the nonlinear auxiliary equation

We first assume αα(q) is a continuous positive function defined over <q< and bounded below by a positive real constant α0. We define the nonlinear auxiliary problem as α(q)u(x)+[u(x)]2n+1=0,x(0,1),u(0)=a,u(1)=b, where a and b are positive real constants. We then have the following theorem.

Theorem 2.1

For uu(x) a solution of (2.1) we have 0<umax(a,b).

Proof

First assume u>max(a,b). Then, there must exist a number, say x0(0,1), so that u(x0) is a positive maximum. Therefore, α(q)u(x0)+[u(x0)]2n+1>0,

Existence and uniqueness of the solution to the nonlinear auxiliary problem

We consider the problem of (2.1) and ultimately show the existence of its solution via the Schauder fixed point theorem and conclude with a unicity argument. To begin, let f2(01(f(x))2dx)1/2 be the norm induced by the inner product f,g01f(x)g(x)dx. We examine the linear problem α(q)u(x)+[ϕ(x)]2nu(x)=0,x(0,1),u(0)=a,u(1)=b, which defines the mapping (Fϕ)(x)u(x;ϕ) via the existence and uniqueness for the linear elliptic two-point boundary value problem defined in (3.1). By the a

Existence and uniqueness of the solution to the nonlinear, nonlocal elliptic equation

We first define the mapping T(q)01u(x;q)dx, where u is a solution to α(q)u(x)+[u(x)]2n+1=0,u(0)=a,u(1)=b,0<x<1 for arbitrary positive constants a and b, where α(q) is continuously differentiable on 0qmax(a,b) and α(q)<0 with α(q)α0>0. From Theorem 2.1, we see that 0<T(q)<max(a,b) and from the existence and uniqueness results established in Section 3, the mapping T(q) is well defined for 0qmax(a,b).

Lemma 4.1

If ψ is defined by ψ(x)M|α(q1)α(q2)|x(1x)±z, for zu1(x)u2(x) with uithe solution of 

Discretization of the nonlinear, nonlocal elliptic B.V.P.

In this section we discretize the problem defined by (1.1). We first rewrite the problem as follows: u(x)+1α(01u(t)dt)[u(x)]2n+1=0,x(0,1)u(0)=a,u(1)=b. Next, we note that as u has two continuous derivatives with respect to x and α(01u(t)dt) is a constant, it follows from Eq. (5.1) that u is infinitely differentiable and its derivatives are bounded.

Now let xi=i/N for i=0,,N and uiu(xi), then (5.1) can be written as ui+1α(01u(t)dt)ui2n+1=0. We discretize 01u(t)dt as Q(u)i=0N1ui+1+ui

The existence and uniqueness of the solution to the nonlinear difference equation

We begin by showing the existence and uniqueness of the solution w(w0,w1,,wN) of the auxiliary problem: α(q)Δh2wi+wi2n+1=0,w0=a,wN=b,i=1,,N1 where Δh2=(wi12wi+wi+1)/h2, for h=1/N and N a positive integer. We then have the following.

Theorem 6.1

For wi a solution to (6.1), we have 0<wimax(a,b), for a,b>0.

Proof

First, suppose wi>max(a,b) for some interior i, say i0, then w must have a positive maximum, therefore α(q)Δh2wi0+wi02n+1>0, which is a contradiction of Eq. (6.1). Thus, it follows wimax(a,b) for

Convergence of the solution of the difference equation to the solution of the nonlinear, nonlocal B.V.P.

Next, we analyze the error ziuiwi noting that z0=zN=0. By subtracting (5.4) from (5.3) we immediately see that Δh2zi+1α(Q(u))ui2n+11α(Q(w))wi2n+1=O(h2), which implies that Δh2zi+1α(Q(u))[ui2n+1wi2n+1]=wi2n+1[1α(Q(w))1α(Q(u))]+O(h2). It then follows that Δh2zi+1α(Q(u))[k=02nui2nkwik]zi=wi2n+1α(ξ)Q(z)α(Q(u))α(Q(w))+O(h2), where ξ(min(Q(u),Q(w)),max(Q(u),Q(w))).

We now partition the analysis into three parts, which are based on the tricotomy of Q(z). For ease of notation,

Numerical experiments

In this section we present the results of two numerical examples that were based on an interval-halving scheme. These results are displayed in Table 1, Table 2. The numerical scheme searched for a change of sign in qk=Q(w(qk)) proceeded by a interval-halving procedure until |Q(w(qk))q| diminished below a pre-set precision. In each table below, N represents the number of iterations and ALPHA corresponds to the numerical computation of the value α(01udx). The J-VALUE symbolizes the number of

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