On a numerical method for a homogeneous, nonlinear, nonlocal, elliptic boundary value problem
Introduction
In [1] the authors analyzed the analytic and numerical solution to the nonlocal elliptic B.V.P.: 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: which is a nonlinear, nonlocal boundary value problem whose 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: where 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 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: where on .
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: with on and as .
Section snippets
A priori estimates for the solution of the nonlinear auxiliary equation
We first assume is a continuous positive function defined over and bounded below by a positive real constant . We define the nonlinear auxiliary problem as where and are positive real constants. We then have the following theorem.
Theorem 2.1 For a solution of (2.1) we have .
Proof First assume . Then, there must exist a number, say , so that is a positive maximum. Therefore,
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 be the norm induced by the inner product We examine the linear problem which defines the mapping 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 where is a solution to for arbitrary positive constants and , where is continuously differentiable on and with . From Theorem 2.1, we see that and from the existence and uniqueness results established in Section 3, the mapping is well defined for .
Lemma 4.1 If is defined by , for with the 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: Next, we note that as has two continuous derivatives with respect to and is a constant, it follows from Eq. (5.1) that is infinitely differentiable and its derivatives are bounded.
Now let for and , then (5.1) can be written as We discretize as
The existence and uniqueness of the solution to the nonlinear difference equation
We begin by showing the existence and uniqueness of the solution of the auxiliary problem: where , for and a positive integer. We then have the following.
Theorem 6.1 For a solution to (6.1), we have , for .
Proof First, suppose for some interior , say , then must have a positive maximum, therefore , which is a contradiction of Eq. (6.1). Thus, it follows for
Convergence of the solution of the difference equation to the solution of the nonlinear, nonlocal B.V.P.
Next, we analyze the error noting that . By subtracting (5.4) from (5.3) we immediately see that which implies that It then follows that where .
We now partition the analysis into three parts, which are based on the tricotomy of . 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 proceeded by a interval-halving procedure until diminished below a pre-set precision. In each table below, represents the number of iterations and ALPHA corresponds to the numerical computation of the value . The -VALUE symbolizes the number of
References (13)
On positive solutions of nonlocal and nonvariational elliptic problems
Nonlinear Anal.
(2004)- et al.
On a class of problems involving a nonlocal operator
Appl. Math. Comput.
(2004) - et al.
On a class of nonlocal elliptic problems via Galerkin method
J. Math. Anal. Appl.
(2005) - et al.
A nonlocal problem modeling ohmic heating with variable thermal conductivity
Nonlinear Anal. Real World Appl.
(2001) Nonlocal elliptic equations
Nonlinear Anal.
(2001)A survey of numerical methods for parabolic differential equations
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