Medical, Pharma, Engineering, Science, Technology and Business

Department of Mathematics, College of Science and Arts at Tabrjal, AlJouf University, Kingdom of Saudi Arabia

- *Corresponding Author:
- Abdelgabar Adam Hassana

Department of Mathematics

College of Science and Arts at Tabrjal

AlJouf University, Kingdom of Saudi Arabia

**Tel:**+966 14 624 7493

**E-mail:**[email protected]

**Received Date:** April 05, 2017; **Accepted Date:** April 25, 2017; **Published Date:** May 11, 2017

**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

**Copyright:** © 2017 Hassana AA. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.

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Journal of Applied & Computational Mathematics

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

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

The Green’s function is powerful tool of mathematical method which used I solving linear non-homogenous differential equation (ordinary and partial) [6-9].

**Sturm-Liouville problem**

Consider a linear second order differential equation

(1)

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

(2)

Where

Let us define integrating factor *p*(*x*) by

Multiplying equation (2) by *p*(*x*), we have

(3)

Thus equation (3) can be written as

(4)

Where *q*(*x*)=*p*(*x*)*c*(*x*) and *r*(*x*)=*p*(*x*)*d*(*x*)

Equation in form (4) is known as Sturm-Liouville equation. Satisfy the boundary conditions

Regular Sturm-Liouville Problem

In case *p*(*a*)≠0 and *p*(*b*)≠0, *p*(*x*), *q*(*x*), *r*(*x*) are continuous, the
Sturm-Liouville equation (4) can be expressed as

*L*[*y*]=λ*r*(*x*)*y* (5)

Where (6)

If the above equation is associated with the following boundary condition

(7)

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.

(i) If *p*(*a*)=0 or *p*(*b*)= 0 or both

(ii)The interval (a,b) is infinite.

**(b)Eigenvalue and eigenfunction**

The Eigenvalue from equation (4) defining by a Sturm-Liouville operator can be expressed as

(9)

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

(10)

But such that the integral of δ (x−ζ) is normalized to unit

(11)

In fact the first operator where Dirac used the delta function is the integration

(12)

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:

(13)

With the approximation improving as *ε* approaches zero.

From (11), we have

(14)

From all values of *ε*, then by letting *ε*→0, we can exactly have

(15)

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}*(

(16)

As

(17)

**(c) Some important properties of Dirac delta function**

**Property (1): Symmetry**

*δ*(−*x*)=*δ*(*x*) (18)

**Proof**

Let *ζ*=−*x*, then *dx*=−*dζ*

We can write:

(19)

But, (20)

Therefore, from (19) and (20), we conclude that *δ*(−*x*)=*δ*(*x*)

**Property (2): Scaling**

(21)

**Proof**

Let ζ=ax, then

If *a*>0, then

Since

Therefore:

Similarly for *a*<0,

then

We can write:

**Property (3)**

(22)

**Proof**

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:

(23)

Near the two zeros *x*^{2}−*a*^{2} can be approximated as:

In the limit as *ε*→0 the integral (23) becomes:

Therefore:

**The concept of Green’s function**

In the case of ordinary differential equation we can express this problem as

*L*[*y*]=*f* (24)

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

*L*[*y*]=*f* (25)

(26)

And the Sturm-Liouville type is gives by

(27)

Where λ is a parameter. Now consider the linear non homogenous ordinary differential equation of the form

(28)

With the boundary condition

(29)

where the constant are such that *α*_{1}+*α*_{2}0 and _{1}+*β*_{2}≠0 if λ=0 then equation
(27) and equation (28) are identical in the interval and *r*(*x*) are real and
positive in that interval.

Now we are seeking to determine the Green’s function *G* for the
equation satisfies the following

(30)

With the boundary condition

(31)

Now consider the region *a*≤*x*<*ζ*.

Let *y*_{1}(*x*) be a nontrivial solution at *x*=*a*, i.e

*α*_{1}*y*_{1}(*a*)+*α*_{2}*y*′(*a*)=0 (32)

Then *α*_{1}*y*_{1}(*a*,*ζ*)+*α*_{2}*G*′(*a*,*ζ*)=0 (33)

The wronskian of *y*_{1} and *G* must vanish at *x*=*a* or

*y*_{1}(*a*)*G*′(*a*,*ζ*)+*y*′_{1}(*a*)*G*(*a*,*ζ*)=0

So *G*(*x*,*ζ*)= *u*_{1}*y*_{1}(*x*) for *a*≤*x*<*ζ* (34)

Where *u*_{1} is an arbitrary constant. Similarly if the nontrivial
solution *y*_{2}(*x*) satisfies the homogeneous equation and the condition
at *x*=*b*, then

*G*(*x*,*ζ*)= *u*_{2}*y*_{2}(*x*) for *ζ*≤*x*<*b* (35)

Now by integrating equation (29) from *ζ*−*ε* to *ζ*+*ε* we obtain

(36)

Since *G*(*x*,*ζ*) and *q*(*x*) are continuous at *x*=*ζ* then we have

(37)

The continuity condition of G and the Jump discontinuity of *G*′ at *x*=*ζ* from equation (33) , (34) and equation (36) imply

(38)

we can solve equation (37) for *u*_{1} and *u*_{2} provided the wronskian *y*_{1} and *y*_{2} doesn’t varnish at *x*=*ζ* or

(39)

The system of equation (37) has the solution

(40)

Where *w*(*ζ*) is the wronskian of *y*_{1}(*x*) and *y*_{2}(*x*) at *x*=*ζ*

Therefore

(41)

(42)

Now from (42) the solution (27) can be expressed as

(43)

(44)

**Some properties of Green’s function:**

The following properties followed Green’s function

**Property (i)**

*G*(*x*,*ζ*) is exit because both *p*(*x*)≠0, *w*(*x*)≠0

**Proof**

From equation (32) and (33) we obtain

(45)

And from equation (35) at *x*=*ζ* we have

(46)

Substituting from (44) into (45) we obtain

(47)

Let

(48)

(49)

but *u*_{1} and *u*_{2} are arbitrary constant

∴ both *p*(*x*)≠ 0 and *w*(*x*)≠ 0

Property (ii)

*G*(*x*,*ζ*) satisfies the homogenous equation except at *x*=*ζ*

**Proof**

From equation (37) *δ*(*x*−*ζ*)=0 except at *x*=*ζ*

(50)

where *p*(*x*)≠0 , *p*′(*x*),*q*(*x*) are continuous on [a,b]

**Property (iii)**

*G*(*x*,*ζ*) is continuous at *x*=*ζ*

**Proof**

Therefore *G*(*x*,*ζ*) is continuous at *x*=*ζ*

**Property (iv)**

The first and second derivatives are continuous for all

*x*≠*ζ* in *a*≤*x*, *ζ*≤*b*

**Proof**

Differentiation equation (32) with respect to *x*

(51)

But G(x,ζ) is continuous everywhere, there we have

*G*(*x*, *x*^{+})=*G*(*x*, *x*^{−}) so that

Differentiation once more gives

(52)

The second and fourth terms on the right side will not cancel in this case to the contrary

(53)

We note that the term denotes a differentiation of *G*(*x*,*ζ*)with respect to *x* using the *x*>*ζ*, at *ζ*→*x*. Thus

For: *dG*(*x*, *x*^{+}) we use the *x*<*ζ* then

**Property (v)**

*G*(*x*,*ζ*) is symmetric in *x* and *ζ*

**Poof**

**Problem(1)**

Find the solution of the following problem by construction the Green’s function

(54)

Subject to the boundary conditions

*y*(0)+ *y*(*L*) =0 (55)

With *k*≠0

**Solution**

Let *G*(*x*,*ζ*) be the Green’s function of the problem, then

(56)

With *G*(0,*ζ*)+*G*(*L*,*ζ*)=0 (57)

The general solution to the homogenous equation is given by

*y*(*x*)=*c*_{1}*coskx*+*c*_{2}*sinkx* (58)

Now applying to the above solution

*y*(0)=0

*y*(0)=*c*_{1}*coskx*+ *c*_{2}(0)=0⇒*c*_{1}=0

∴*y*_{1}(*x*)=*c*_{2}*sinkx* (59)

and *y*(*L*)=0⇒0=*c*_{1}*cosL*+*c*_{2}sin *kL*

(60)

The Wronskian is given by

(61)

The Green’s function is given by

(62)

The solution is given by

(63)

**Problem (2)**

Solve the problem by construction the Green’s function

(64)

with the boundary condition

*y*(0)+*y*(*L*)=0 where *a*≠0 (65)

**Solution**

Let *G*(*x*,*ζ*) be the Green’s function of the problem, then

(66)

With the boundary condition *G*(0,*ζ*) +*G*(*L*,*ζ*)=0

The general solution to the homogenous equation is given by

(67)

Applying to the above solution

then, and *y*(0)=0 then *c*_{2}=0, and

(68)

Applying the boundary condition, then *y*(*L*)=0 , and (69)

The Wronskian is given by

(70)

The Green’s function is given by

Where we chose *c*_{1}=1, then

(71)

The solution is given by

(72)

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