_{1}

^{*}

We consider the nonlinear boundary value problems for elliptic partial differential equations and using a maximum principle for this problem we show uniqueness and continuous dependence on data. We use the strong version of the maximum principle to prove that all solutions of two-point BVP are positives and we also show a numerical example by applying finite difference method for a two-point BVP in one dimension based on discrete version of the maximum principle.

In this paper, we have considered a simple two-point boundary value problem (BVP) for a second order linear ordinary differential equation. Using a maximum principle for this problem, we show uniqueness and continuous dependence on data.

We write the BVP in variational form and use this together (with elements from functional analysis) we prove existence, uniqueness and continuous dependence on data. The finite difference method is a method for early development of numerical analysis to differential equations. In such a method, an approximate solution is sought at the points of a finite grid of points reducing the problem to a finite linear system of algebraic equations [

In this paper, we illustrate this for a two-point BVP in one dimension in which the analysis is based on discrete versions of maximum principle. Much attention has been paid to the development, analysis and implementation of accurate methods for the numerical solution of this problem in the literature. Many problems are modeled by smooth initial conditions and Dirichlet boundary conditions, see [

We treat the two-point boundary value problem in Hilbert space

in

Let

By using integration by parts with

Here (4) is the variational formulation of the problem (1). If we introduce the bilinear form

with the functional

for all

Next we show two theorems that demonstrate the existence of a solution of the variational equation (6).

Theorem 1. Let

Theorem 2. Let

In theorem 1, we obtain the weak solution u of (6) and this solution is more regular than stated there, therefore

This expression together with

We can see that the weak solution of (1) is a strong solution, but we can also see that with

The next nonlinear boundary problem shows that all solutions are positive by using the strong version of maximum principle.

We consider the nonlinear boundary value problem

in

for all

Therefore, we have

u and

The strong version of the principle refers to the case when there is a minimum interior point inside_{0} then

After this assumption, the expression

In this case, we are going to solve

with the following boundary conditions:

Let

Next, integrate over

By using integration by parts, we can write the left hand side as

The last term comes down to 0,

therefore, the Equation (8) becomes

which is the variational form of the (7).

Lax-Milgram lemma may allow us to prove existence of a solution. First we consider the LHS and the RHS as a bilinear form and a linear functional respectively, in fact

*

then

Coercivity of

*

The Equation (9) can be written as

since

Similar to (7), this equation can be written using the auxiliary function

Therefore, we can see that this function has the same variational formulation, i.e.

Then, we can use the bilinear form and linear functional. Lax-Milgram lemma shows can be subsequently applied to prove the existence of solution in a similar way as before.

Let the auxiliary function

We can see that this quantity represents still a bilinear form

therefore,

then,

hence Lax-Milgram lemma can be applied to

where

In this section, we give the variational formulation for the beam equation and we prove the existence and uniqueness of solution. We consider the beam equation

with the boundary conditions

Now, using an auxiliary function

therefore

Thus, the variational form of the beam equation is

Again, let the bilinear form and linear functional

We can see that

Here, the Lax-Milgram theorem can be applied to this system and show existence of a solution for u.

Note: In mechanical representations, the boundary conditions

deflection and the slope of the deflection at the boundaries is 0 which means that the ends of the beam are fixed.

In this example, we consider the two-point boundary value problem [

with

with

^{−}^{5} vs. 8.829 × 10^{−}^{5}. The Logarithmic plot in

We would like to thank the referee for his valuable suggestions that improved the presentation of this paper and group GEDNOL of the Universidad Tecnológica de Pereira-Colombia.