APPROXIMATE SOLUTION OF BOUNDARY VALUE PROBLEM WITH BERNSTEIN POLYNOMIAL LAPLACE DECOMPOSITION METHOD

Abstract: In this paper, a new approach enabling Laplace Adomian decomposition method in combination with Bernstein polynomials for numerical solution of boundary value problem is introduced. The accuracy of the method is dependent on the size of the set of Bernstein polynomials. Numerical results with comparisons are given to establish the reliability of the proposed method for solving singular and nonsingular boundary value problem. Convergence analysis of the presented technique is also discussed.


Introduction
Nonlinear boundary value problems (BVPs) eminently emerge in the modelling of real life problems.A huge research has been conducted to find the numerical solutions of these problems by considering their applications in many branches of science and engineering for example, fluid mechanics, quantum mechanics, aero-dynamics, reaction-diffusion process, chemical reactor, geophysics [10], [21], astrophysics, boundary layer theory, the study of stellar interiors, op-timal control theory and flow networks in biology [1].In recent times such kinds of problems have been solved analytically and numerically by some methods like Galerkin and collocation method, Sinc-Galerkin method, homotopy perturbation method, homotopy analysis method, homotopy asymptotic method, He-Laplace method, Adomian decomposition method, Laplace differential transform method and shooting method, etc.
The aim of the present research is motivated to achieve the numerical solutions by using a new reliable modification in Laplace Adomian decomposition method.Due to the effectiveness of the technique, Laplace decomposition method is adopted by many researchers to solve various functional equations [15], [17], [2], [19], [7].Mainly the method is based on representing the series solution of the problem and decomposing the nonlinear term by Adomian polynomials.Various authors investigated and modified this technique [4], [8] to find the solutions of different nonlinear functional equations.In [5] Khuri extended the method by combining Laplace transform with Adomian decomposition method for solving nonlinear differential equations.Diversified modifications and improvements have been investigated further to improve the standard Laplace Adomian decomposition method [18], [11], [16].
In this paper, a novel and simple technique is introduced based on Bernstein polynomials.For finding the approximate solutions of BVPs, the technique expands the source function in terms of a set of continuous polynomials over a closed interval, i.e.Bernstein polynomials and then the standard Laplace decomposition method is adopted which provides tremendous results.

Basic Concepts of Bernstein Polynomials
Polynomials are the mathematical tools as these can be defined, calculated, differentiated and integrated easily.The Berstein basis polynomials are exercised to approximate the functions and curves.Bernstein polynomials are the better approximation to a function with a few terms.These polynomials are used in the fields of applied mathematics, physics and computer aided-geometric designs and are also combined with other methods like Galerkin and collocation method to solve some differential and integral equations [3], [20].
Following are some basic definitions and results [13]: Definition 1 (Bernstein basis polynomials).The Bernstein basis polynomials of degree m form a complete basis over the interval [0, 1] and are defined by where the binomial coefficient is

Definition 2 (Bernstein polynomials). A linear combination of Bernstein basis polynomials
is called the Bernstein polynomials of degree m, where β i are the Bernstein coefficients.
Definition 3.With f a real valued function defined and bounded on [0, 1], let B m (f ) be the polynomial on [0, 1], that assigns to x the value where B m (f ) is the mth Bernstein polynomials for f (x) [13].
With the help of these polynomials Weierstrass approximation theorem is proved.
Theorem 4. For all functions f in C[0, 1], the sequence of B m (f ) converges uniformly to f, where B m (f ) is defined by (2).

Description of Modified Technique
Let us consider the nonlinear singular boundary value problem subject to the conditions ) where N y is the nonlinear operator, g(x) is the given source function and a 0 , a 1 , . . .a r−1 , c 0 , c 1 . . .c n−r , b are the known constants with m ≤ r ≤ n, r ≥ 1.
Initially multiplying with x, then taking Laplace transform on both side of (3) and using the derivative property of Laplace transform, we get By substituting the conditions given in (4) and solving (5), gives Now according to standard Laplace Adomian decomposition method, we write the approximate solution as: and the nonlinear term is decomposed as follows: where A k 's are the Adomian polynomials Substituting these values in (6) and comparing the terms yields the iterative algorithm: In general, After this integrating and employing Laplace inverse transform on (10), we obtain The new modification in LADM is introduced here by using the Bernstein polynomial approximation to the given function g(x) in (13).After that the value of y 0 is used to compute A 0 and applying the same process to ( 11) and ( 12), gives Thus we obtain the components y 0 , y 1 , y 2 . . .successively.Hence the kth approximate solution of nonlinear boundary value problem is given by Now we consider the nonsingular boundary value problem of the form subject to the conditions where N y is the nonlinear operator, g(x) is the given source function and a 0 , a 1 , . . .a r−1 , c 0 are given constants.
Here were adapting the same procedure as stated above.In nonsingular BVPs, we obtain the solution components y 0 , y 1 , y 2 . . .successively as: Therefore, we attain the kth approximate solution of nonsingular boundary value problem given by ( 16).

Convergence Analysis
In this section, we study the convergence of above technique using the analysis in [14], [9].
where L(y) is the differential part, F (y) is the nonlinear term, q(x) be any function of x and g(x) is the source term.
For instance, here we develop the following theorem for second order nonlinear boundary value problem.
Proof.Now reformulating (22) into integral equation, for this integrate two times w.r.to x and taking limits from 0 to x and use the given boundary conditions, we have Here using the boundary condition y(π) = 0 in (23), gives the value of y ′ (0).Hence putting the value of y ′ (0) in ( 23), we attain the integral equation Analyzing the convergence, we suppose here that y n be the nth approximate solution of ( 22), therefore where B m (g(t)) is the Bernstein polynomial of function g(t).shows that expanding the source term in the Bernstein polynomials, the solution is very much close to the exact solution which proves that our method is well suited to this problem.

Discussion
Bernstein polynomials are exclusively used to modify the standard Laplace decomposition method.The proposed new technique is analyzed and applied to the nonlinear singular and nonsingular boundary value problems.The results shown in the figures exclude that the numerical approximations are in good agreement with the exact one.The convergence analysis exhibits the applicability of technique.
Let us consider that (C[I], ||.||) be the Banach space of all continuous function on I.Here we first convert the boundary value problem into the integral equation and discuss the convergence analysis.Then for the integral equation we also assume that the kernel given by the converted equation; i.e. |k(x, t)| ≤ S and the nonlinear function F (y) satisfy the Lipschitz condition such that |F (y) − F (z)| ≤ T |y − z|.For this we write the nonlinear boundary value problem in operator form

Figure 1 :
Figure 1: Comparison of Approximate solution with Exact solution

Figure 2 :
Figure 2: Comparison of Approximate solution with Exact solution