Green’s Function Solution of Non-Homogenous Regular Sturm-Liouville Problem

In this paper, we propose a new method called exp(−φ(ξ)) fractional expansion method to seek traveling wave solutions of the nonlinear fractional Sharma-Tasso-Olver equation. The result reveals that the method together with the new fractional ordinary differential equation is a very influential and effective tool for solving nonlinear fractional partial differential equations in mathematical physics and engineering. The obtained solutions have been articulated by the hyperbolic functions, trigonometric functions and rational functions with arbitrary constants. Citation: Hassana AA (2017) Green’s Function Solution of Non-Homogenous Regular Sturm-Liouville Problem. J Appl Computat Math 6: 362. doi: 10.4172/2168-9679.1000362


Introduction
The series solution of differential equation yields an infinite series which often converges slowly. Thus it is difficult to obtain an insight into over-all behavior of the solution [1,2]. The Green's function approach would allow us to have an integral representation of the solution instead of an infinite series.
To obtain the filed y, caused by distributed source we calculate the effect of each elementary portion of source and add (integral) them all. If G(r,r 0 ) is the field at the observers point r caused by a unit source at the source point r 0 , then the field at r caused by distribution f(r 0 ) is the integral of f(r 0 ) G(r,r 0 ) over the whole range of r 0 occupied by the course. The function G is called Green's function [3][4][5].

Sturm-Liouville problem
Consider a linear second order differential equation Where λ is a parameter to be determined by the boundary conditions? A(x) is positive continuous function, then by dividing every term by A(x), equation (1) can be written as Thus equation (3) can be written as (4) If the above equation is associated with the following boundary condition Where α 1 +α 2 ≠0 and β 1 +β 2 ≠0 The equation (4) and the boundary condition (7) are called regular Sturm -Liouville problem (RSLP).

(a) Singular Sturm-Liouville problem
Consider the equation L[y]+λr(x)y=0 a<x<b (8) Where L is defined by (6), p(x) is smooth and r(x) is positive, then the Sturm-Liouville problem is called singular if one of the following situations is occurred.

(b)Eigenvalue and eigenfunction
The Eigenvalue from equation (4) defining by a Sturm-Liouville operator can be expressed as The non-trival solutions that satisfy the equation and boundary conditions are called eigenfunctions. Therefore the eigenfunction of the Sturm-Liouville problem from complete sets of orthogonal bases for the function space is which the weight function is r(x).

The Dirac delta function
The delta function is defined as In fact the first operator where Dirac used the delta function is the integration Where f(x) is a continuous function, we have to find the value of the integration (12). Since δ (xζ) is zero for x≠ζ, the limit of integration may be change to ζ−ε and ζ+ε, where ε is a small positive number, f(x) is continuous at x−ζ, it's values within the interval (ζ−ε,ζ−ε) will not different much from f(ζ), approximately that: With the approximation improving as ε approaches zero.
From (11), we have From all values of ε, then by letting ε→0, we can exactly have Despite the delta function considered as fundamental role in electrical engineering and quantum mechanics, but no conventional could be found that satisfies (10) and (11), then the delta function sought to be view as the limit of the sequence of strongly peaked function δ n (x)such that We can write: Therefore, from (19) and (20), we conclude that δ(−x)=δ(x) We can write: The argument of this function goes to zero when x = a and x=−a, wherefore Only at the zero of the argument of the delta function that is: Near the two zeros x 2 −a 2 can be approximated as: In the limit as ε→0 the integral (23) becomes:

Green's Function The concept of Green's function
In the case of ordinary differential equation we can express this problem as Where L is a linear differential operation f (x) is known function and y(x) is desired solution. We will show that the solution y(x) is given by an integral involving that Green's function G(x,ξ).

Green's function for ordinary differential equation
Here we consider non-homogenous ordinary differential equation Where L is an ordinary linear differential operator that can be represented bySturm-Liouville operator, i.e.
And the Sturm-Liouville type is gives by Where λ is a parameter. Now consider the linear non homogenous ordinary differential equation of the form With the boundary condition With the boundary condition Now consider the region a≤x<ζ.
Let y 1 (x) be a nontrivial solution at x=a, i.e Then α 1 y 1 (a,ζ)+α 2 G′(a,ζ)=0 The wronskian of y 1 and G must vanish at x=a or Since G(x,ζ) and q(x) are continuous at x=ζ then we have The continuity condition of G and the Jump discontinuity of G′ at x=ζ from equation (33) Where w(ζ) is the wronskian of y 1 (x) and y 2 (x) at x=ζ Therefore ( ) ( ) ( ) ( ) ( ) 1 2 , Some properties of Green's function: The following properties followed Green's function

Property (v)
G(x,ζ) is symmetric in x and ζ

Poof
Let ( ) , Find the solution of the following problem by construction the Green's function The Green's function is given by , The solution is given by

Problem (2)
Solve the problem by construction the Green's function with the boundary condition

Solution
Let G(x,ζ) be the Green's function of the problem, then With the boundary condition G(0,ζ) +G(L,ζ)=0 The general solution to the homogenous equation is given by Applying to the above solution then, and y(0)=0 then c 2 =0, and Applying the boundary condition, then y(L)=0 The Wronskian is given by The Green's function is given by