A Special Case of Variational Formulation for Two-Point Boundary Value Problem in ( ) Ω L 2

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.


Introduction
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 [1].
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 [2] and [3].We can say that three classes of solution techniques have emerged for solution of BVP for differential equations: the finite difference techniques, the finite element methods and the spectral techniques (see [4] and [5]).The last one has the advantage of high accuracy attained by the resulting discretization for a given number of nodes [6] [7] and [8].

Variational Formulation
We treat the two-point boundary value problem in Hilbert space ( )

L
. We consider the problem ( ) Here the coefficients , α β and γ are smooth and Let φ an auxiliary function on ( ) , so multiplying Equation ( 1) by φ and integrating over Ω we have ( ) ( ) By using integration by parts with ( ) ( ) Here ( 4) is the variational formulation of the problem (1).If we introduce the bilinear form , we can write (3) as H φ ∈ .We can say that function u is a weak solution of the problem (1) provided that In theorem 1, we obtain the weak solution u of (6) and this solution is more regular than stated there, therefore u′′ exists as a weak function (derivative) and ( ) This expression together with We can see that the weak solution of ( 1) is a strong solution, but we can also see that with α less smooth we still obtain a weak solution in 1 0 H .The next nonlinear boundary problem shows that all solutions are positive by using the strong version of maximum principle.

Numerical Examples
We consider the nonlinear boundary value problem In the maximum principle, we need to consider a differential operator  , i.e. in this case we have β = .Now, the maximum principle states that for a differential operator  and a function ( )  respect the conditions of the principle, the boundary conditions of the problem are ( ) ( ) Thus, all solutions of u are The strong version of the principle refers to the case when there is a minimum interior point inside Ω , which results in a constant value of the function u inside the domain Ω .We know already that 0 u ≥  , and if we assume that u has a minimum point inside Ω say x 0 then ( ) ( ) ( ) After this assumption, the expression 0 by the initial hypothesis.Here, we have a contradiction, because the assumption that there is a minimum interior point is false.In the other words, the minimum 0 is attained only at the boundaries and all solutions inside the interval ( )

Variation Formulation and Existence of ′′ u g − =
In this case, we are going to solve ( ) with the following boundary conditions: 2.2.1.Solution with BCs: ( ) ( ) an auxiliary function with the homogeneous boundary conditions as u, ( ) ( ) . Multiplying the initial differential operator u′′ − and the function f by φ , then expression (7) becomes Next, integrate over Ω we have ( ) ( ) ( ) ( ) 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 * ( ) , a v w is symmetric.To show coercivity of ( ) , a ⋅ ⋅ , we can apply Cauchy-Schwarz inequality ( ) ( ) ( ) (RHS).We can see that ( ) The Equation ( 9) can be written as , a ⋅ ⋅ is a bounded bilinear form, coercive in the Hilbert space Ω and ( ) L ⋅ is a bounded linear form in the same space, so the Lax-Milgram theorem states that there exists a unique vector Ω u ∈ i.e. a solution of u exists.

Solution with BCs
Similar to (7), this equation can be written using the auxiliary function φ and integration over the domain Ω and ( ) ( ) 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.

Solution with BCs:
( ) ( ) ( ) Let the auxiliary function φ with the same boundary conditions ( ) In this case, the LHS of (7) can be written as ) We can see that this quantity represents still a bilinear form ( ) , a v w is also symmetric.In this case the coercivity on the Hilbert space ω also applies, in fact hence Lax-Milgram lemma can be applied to where ( ) , a ⋅ ⋅ is a bilinear, symmetric coercive form and L a linear functional.A solution of u needs to exist in the domain Ω .

Case of the Beam Equation
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 ( )

C φ ∈
with the same BCs as u, and integrating on Ω we get ( ) ( ) Thus, the variational form of the beam equation is ( ) Again, let the bilinear form and linear functional ( ) for all u ∈  .We can see that u  usually takes the general form

Figure 1 .
Figure 1.Solutions using the finite difference method (11) for (a) h = 1/10 and (b) h = 1/10.(c) Error when comparing these 2 grid choices with the exact solution.(d) Logarithmic plot of the error vs. the choice of h.

4 .we show in Figure 1
Maximum of the Error at the Mesh-Points for 2-Point BVPIn this example, we consider the two-point boundary value problem[9] the exact solution of the error at the mesh-points.Figure1(a) and Figure1(b) present similar plots.However, a h twice as small decreases the maximal error a 4 fold, shown in Figure1(c).At the same point x = 0.6, the error is 2.209 × 10 −5 vs. 8.829 × 10 −5 .The Logarithmic plot in Figure1(d)shows a linear relationship between the error and h with slope 8 units.