Computa & tio d n e a li Hassana, J Appl Computat Math 2017, 6:3 l p M p a Journal of A t h DOI: 10.4172/2168-9679.1000362 f e o m l a a n t r ISSN: 2168-9679i c u s o J Applied & Computational Mathematics Research Article Open Access Green’s Function Solution of Non-Homogenous Regular Sturm-Liouville Problem Abdelgabar Adam Hassana* Department of Mathematics, College of Science and Arts at Tabrjal, AlJouf University, Kingdom of Saudi Arabia Abstract 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. Introduction d dy d22 ydp() x dy d y =+=+ So px() px() 22p()x p(x) bxz() The series solution of differential equation yields an infinite series dx dx dx dx dx dx which often converges slowly. Thus it is difficult to obtain an insight Thus equation (3) can be written as into over-all behavior of the solution [1,2]. The Green’s function d dy approach would allow us to have an integral representation of the px() ++ qxy()()λ rxy =0 (4) solution instead of an infinite series. dx dx To obtain the filed y, caused by distributed source we calculate the Where q(x)=p(x)c(x) and r(x)=p(x)d(x) effect of each elementary portion of source and add (integral) them all. Equation in form (4) is known as Sturm-Liouville equation. Satisfy If G(r,r0) is the field at the observers point r caused by a unit source the boundary conditions at the source point r0, then the field at r caused by distribution f(r0) is Regular Sturm-Liouville Problem the integral of f(r0) G(r,r0) over the whole range of r0 occupied by the course. The function G is called Green’s function [3-5]. In case p(a)≠0 and p(b)≠0, p(x), q(x), r(x) are continuous, the The Green’s function is powerful tool of mathematical method Sturm-Liouville equation (4) can be expressed as which used I solving linear non-homogenous differential equation L[y]=λr(x)y (5) (ordinary and partial) [6-9]. d dy Where L= px() + qx() (6) Preliminaries dx dx If the above equation is associated with the following boundary Sturm-Liouville problem condition Consider a linear second order differential equation α1y(a)+α2y (a)=0 d2 y dy + ++λ = (1) Ax() 2 Bx() C(x) y D(x) y 0 β1y(b)+β2y′(b)=0 (7) dx∂ x Where α +α ≠0 and β +β ≠0 Where λ is a parameter to be determined by the boundary 1 ′ 2 1 2 conditions? A(x) is positive continuous function, then by dividing The equation (4) and the boundary condition (7) are called regular every term by A(x), equation (1) can be written as Sturm -Liouville problem (RSLP). d2 y dy (a) Singular Sturm-Liouville problem +Bx() ++ C(x) yλ D(x) y = 0 (2) dx2 ∂ x Consider the equation Bx() Cx() Dx() Where b(x)= , c(x)= and d(x)= L[y]+λr(x)y=0 a<x<b (8) Ax() Ax() Ax() Let us define integrating factorp (x) by x *Corresponding author: Abdelgabar Adam Hassana, Department of Mathematics, Px()()= exp∫ bξξ d College of Science and Arts at Tabrjal, AlJouf University, Kingdom of Saudi Arabia, a Tel: +966 14 624 7493; E-mail: [email protected] Multiplying equation (2) by p(x), we have Received April 05, 2017; Accepted April 25, 2017; Published May 11, 2017 d2 y dy Citation: Hassana AA (2017) Green’s Function Solution of Non-Homogenous + ++λ = Regular Sturm-Liouville Problem. J Appl Computat Math 6: 362. doi: 10.4172/2168- px() 2 pxbx()() p(x)c()() x y pxd (x) y 0 (3) dx ∂x 9679.1000362 xx bd()ζζ bd() ζζ x Copyright: © 2017 Hassana AA. This is an open-access article distributed under dp() x dd∫∫ Since = eaa= e b(ζζ )d = pxbx()() the terms of the Creative Commons Attribution License, which permits unrestricted ∫ use, distribution, and reproduction in any medium, provided the original author and dx dx dx a source are credited. J Appl Computat Math, an open access journal Volume 6 • Issue 3 • 1000362 ISSN: 2168-9679 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 Page 2 of 6 Where L is defined by (6), p(x) is smooth and r(x) is positive, then n δ = the Sturm- Liouville problem is called singular if one of the following n ( x) 22 π 1+ nx situations is occurred. ( ) n −nx22 (i) If p(a)=0 or p(b)= 0 or both δn ( xe) = (17) π (ii)The interval (a,b) is infinite. sin2 (nx) δ = n ( x) 2 etc (b)Eigenvalue and eigenfunction nxπ The Eigenvalue from equation (4) defining by a Sturm-Liouville (c) Some important properties of Dirac delta function operator can be expressed as Property (1): Symmetry −1 d dy λ = px( ) + qx( ) (9) δ(−x)=δ(x) (18) r( x) dx dx The non-trival solutions that satisfy the equation and boundary Proof conditions are called eigenfunctions. Therefore the eigenfunction of Let ζ=−x, then dx=−dζ the Sturm-Liouville problem from complete sets of orthogonal bases for the function space is which the weight function is r(x). We can write: +∞ −∞ +∞ The Dirac delta function ∫∫∫fxxdxfdfdf( )δ(−=) (ζδζ) ( ) ζ ≈( ζδζ) ( ) ζ =(0) (19) −∞ +∞ −∞ The delta function is defined as +∞ 0 x ≠ξ But, f( x)δ ( x) dx= f (0) (20) (10) ∫ δξ( x −=) −∞ ∞=x ξ Therefore, from (19) and (20), we conclude that δ(−x)=δ(x) But such that the integral of δ (x−ζ) is normalized to unit +∞ Property (2): Scaling ∫ δξ( x−=) dx 1 (11) 1 −∞ δδ(ax)= ( x) (21) a In fact the first operator where Dirac used the delta function is the Proof integration 1 Let ζ=ax, then dx= dζ +∞ a ∫ f( x)δξ( x− ) dx (12) If a>0, then −∞ +∞ −∞ ζζ1 1+∞ 10 Where f(x) is a continuous function, we have to find the value of f( x)δ( ax) dx= f δζ( ) d ζ= f δζ( ) d ζ= f f (0) ∫∫a a a ∫ a aa 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) +∞ +∞ 11 1 is continuous at x−ζ, it’s values within the interval (ζ−ε,ζ−ε) will not Since ∫∫f(ζδ) ( x) dx= f( x)δ( x) dx= f (0) different much from f(ζ), approximately that: −∞ aa−∞ a +∞ ξξ+∞ +∞ Therefore: f xδξ x−= dx f x δξ x −≈ dx f ξδξ x − dx (13) ∫∫∫( ) ( ) ( ) ( ) ( ) ( ) 1 −∞ ξξ−∞ −∞ δδ(ax)= ( x) a With the approximation improving as ε approaches zero. Similarly for a<0, 1 From (11), we have δδ(ax)= ( x) −a +∞ ξ +∞ then I=∫∫δξ( x −=) dx δξ( x −=) dx 1 (14) −∞ ξ −∞ We can write: From all values of ε, then by letting ε→0, we can exactly have 1 δδ(ax)= ( x) +∞ a ∫ f( x)δξ( x−=) dx f ( ζ) (15) −∞ Property (3) 22 1 Despite the delta function considered as fundamental role in δ(x− a) = δδ( xa +) +( xa −) (22) electrical engineering and quantum mechanics, but no conventional 2a could be found that satisfies (10) and (11), then the delta function Proof sought to be view as the limit of the sequence of strongly peaked The argument of this function goes to zero when x = a and x=−a, function δn(x)such that wherefore +∞ +∞ 22 δδ( xx)= lim n ( ) (16) fζδ x− a dx = f x δ x + a + δ x − a dx n→∞ ∫∫( ) ( ) ( )( ) ( ) −∞ −∞ As Only at the zero of the argument of the delta function that is: J Appl Computat Math, an open access journal Volume 6 • Issue 3 • 1000362 ISSN: 2168-9679 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 Page 3 of 6 +∞ −ae + ae+ 22 22 22 αGa( ) += αζ G′( a,0) ∫∫f(ζδ) ( x−= a) dx f( x)δ( x −+ a) dx ∫ f( x)δ( x − a) dx (23) 12 (31) −∞ −ae − ae− β12Gb( )+= βζ G′( b,0) Near the two zeros x2−a2 can be approximated as: Now consider the region a≤x<ζ. 22 (−2,axa)( +) x →− a ( x− a) =−( xaxa)( +=) Let y1(x) be a nontrivial solution at x=a, i.e (+2,axa)( −) x →+ a α y (a)+α y (a)=0 (32) In the limit as ε→0 the integral (23) becomes: 1 1 2 +∞ −ae + ae+ Then α y (a,ζ)+α G (a,ζ)=0 (33) 22 1 1 ′ 2 ∫∫f(ζδ) ( x− a) dx = f(x)δ −( 2 a)( x + a) dx + ∫ f(x)δ ( 2 a)( x − a) dx −∞ −ae − ae− The wronskian of y1′ and G must vanish at x=a or 11ae++ae = f(x)δδ( x) ( x + a) + f(x)δ( x− a) dx ∫∫ y1(a)G (a,ζ)+y 1(a)G(a,ζ)=0 22aaae−−ae +∞ 1 So G(x,′ζ)= u1y1′(x) for a≤x<ζ (34) =∫ fx( ) δδ[ xa−+] ( xadx −) −∞ 2a Where u1 is an arbitrary constant.