Green’s Function Solution of Non-homogenous Singular Sturm-Liouville Problem

(1)

Available at _{www.ijsred.com}

Green’s Function Solution of Non-homogenous Singular

Sturm-Liouville Problem

Abdelgabar Adam Hassan

*a, b

a) Jouf University, College of Science and Arts at Tabrjal, Department of Mathematics, kingdom of Saudi Arabia. b) University of Nyala, Department of Mathematics, Sudan

Email : [email protected]

---

************************

---Abstract:

In this paper, the initial- boundary value problem consisting non-homogeneous ordinary differential equation are solved using Green’s function approach. The important tool for potential theory is provided by Green's function. We present a calculation for many differential equations to find general solutions by using Green’s Function Solution of non-homogenous singular Sturm-Liouville problem.

Keywords —Green’s function, Sturm-Liouville, differential equations.

---

************************

---I. INTRODUCTION

Since we shall only consider very simple cases we need not go into the general theory of characteristics. It is often easy to find the solution of a differential equation which is defined in all space and is equal to zero at points at infinity [13],but which has a singularity at one point, i.e., becomes infinite at this point. Whether or not this function has a physical meaning depends on the nature of the singularity. We con-structed Green's function for circular and spherical fundamental domains [13], [14].Green's function for the circle (or sphere) can use asa majorant for Green'sfunction

for an arbitrary bounded domainGin the following manner [13], [15].

In the remaining part we apply this theory to Sturm-Liouville problem and Dirac systems, studying mainly oscillation theory and absolute continuity of the spectrum. Most of the results can be found in the literature in some form; but there are also some new results, mainly connected with the problem of existence of self-adjoint realizations with separated boundary conditions

The first portion of this concise introduction to ordinary and partial differential equations acquaints the reader with equationsdescribing the

(2)

www.ijsred.com

ISSN : 2581-7175 ©IJSRED: All Rights are Reserved Page 736

moreimportant theories of classical physics [13].Theconcluding paper introduce some of the standard ways for solving those differential equations which have been derived: Eigen functions, Fourier series and integrals [14], Green's theorem, particular solutions in coordinates, asymptotic expansions, change of variables, conformal mapping, singularities, and transition to integral equations [13].

II.STURM-LIOUVILLE PROBLEM

Consider a linear second order differential equation

( )

22

( )

0

( )

1

d y dy

A x B x C x y D x y

dx + ∂x+ +λ =

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

( )

[1], equation (1) can be written as [1], [2].

( )

2

2 0 2

d y dy

b x c x y d x y

dx + ∂x+ +λ =

Where

( )

,

B x C x

b x c x

A x A x

= = and

( )

D x d x

A x

=

Let us define integrating factorp x

( )

by

( )

exp

{

( )

}

x

a

p x = ∫b ζ dζ

Multiplying equation (2) byp x

( )

, we have:

( )

( ) ( )

( )

2

0 3

d y dy

p x p x b x p x c x y

dx x

p x d x y

λ

+ +

∂

+ =

Since

( )

( ) ( )

( )

( ) ( )

x x

x

a a

a

b d b d

d d

b d p x b x

dx dx

dp x

e e

dx

ζ ζ ζ ζ

ζ ζ

∫ ∫

=

 

 

= _ _= ∫ 

 

So

( )

( ) ( )

2

dp x

d dy d y dy

p x p x

dx dx dx dx dx

d y dy

p x p x b x

dx dx

 

= +

 

 

= +

Thus equation (3) can be written as:

( )

0

( )

4

d dy

p x q x y r x y

dx dx λ

 

+ + =

 

 

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

III. Singular Sturm-Liouville Problem

In this case p x

( )

and r x

( )

the Sturm-Liouville equation to one or both endpoint, we call it singular, because consider Sturm-Liouville problem [1], [2]

(

py′ +

)

′ qy+

λ

r x y

( )

= f x

( )

5

Can be written as

( )

6

py′′+ p y′ ′+qy+

λ

r x y= f x

Or

( )

1 1

7

y p y qy r x y f x

p p λ

′′+ ′ ′+ + =

With the boundary conditions

( )

0 8

( )

n n n n

m m m m

y b y b y a y a

p b p a

y b y b y a y a

′ ′

− =

(3)

www.ijsred.com

If both p a

( )

and p b

( )

are zero. The boundary condition (8) is satisfied automatically. Ifp a

( )

=0 andp b

( )

≠0then the boundary (8)becomes

( )

0

( )

9

n n

m m

y b y b y b y

′ =

this condition will be met, if all solution of the equation(8) satisfy the boundary condition

( )

1y b 2y b 0 10

β

+

β

′ =

Where β β₁, ₂are constant and not both zeroi.e

( )

2 2

1 1 0 11

β +β ≠

In addition solution must be boundedatx=a

.

Similarly, ifp b

( )

=0andp a

( )

≠0, theny x

( )

must be bounded atx=b

And

α

₁y a

( )

+

α

₂y a′

( )

=0

( )

12 Whereα₁ andα₂are constant and not both zero i.e

( )

2 2

1 1 0 13

α +α ≠

Now we consider typical example of singular sturm-liouville problems.

IV. BESSEL’S EQUATION

( )

2

, 0 14

n

y y q r x y f x x

x λ

′′+ ′− + = >

Here both function If both

( )

p x =x andr x

( )

=xvanishes atx=0

SOLUTION OF NON- HOMOGENOUS BESSEL

EQUATION:

With the boundary condition

( )

0

lim and y 0, 15

x→ y x < ∞ L =

Wheren≥0and is an integer Let us find the green’s function for

(

)

( )

2

, 0 x L 16

n

xG G x G x

x

λ δ ζ

 

′′+ ′+ −  + = − − < <

 

the boundary conditions are

(

)

(

)

( )

0

lim , and , 0, 17

x→ G xζ < ∞ G Lζ =

The homogeneous solution that satisfy the boundary condition are [8]

( )

1 n 18

y x =J λx

And

( )

(

) (

)

( )

2 n n n 19

y x =J λx −J λL y λx

The wronskian is given by

( )

(

)

( )

(

) ( )

( )

(

)

(

) ( )

n n n n

J x J x J L y x

w x

J x J x J L y x

λ λ λ λ

λ λ λ λ λ λ

− =

′ _ ′ − ′ _

( )

n

(

)

n

( )

n

(

) ( )

n

w x =λJ λL _J′ λx −J λL y′ λx _

( )

2 _n

(

)

( )

20

w x J L

x λ

π

= −

The Green’s function is given by

(

)

( ) ( )

( )

1 2

2 1

, 0

, 21

,

y x y

x p x w x

G x

y x y

x L p x w x

ζ

ζ ζ

ζ

 

− ≤ <

 

= 

 

− < ≤

 

 

(4)

www.ijsred.com

(

)

( )

( ) ( )

( )

( ) ( )

( )

2

, , 0 22

n n n

n n n n n

J x J y L J x

p x w x

G x x

J x y L J L y x J

π λ λζ λ λ

ζ ζ

π λ λ λ λζ

− −

= ≤ <

−

 _ _ 

 

 

 _ _ 

 

 

 

EXAMPLE(1)

Consider the equation

( )

2

0 23

xy′′+y′+k xy= −f x < <x L

( )

0

( )

0

( )

24

y′ =y L′ =

Wherec J₁ _n

( )

k +c Y k_{2 1}

( )

=0and ₂ 2

c =π

SOLUTION

Let G be the Green’s function, then

(

)

( )

2

0 , 1 25

xG′′+G′+k xG= −

δ

x−

ζ

<x

ζ

<

(

0,

)

(

,

)

0

( )

26

G′

ζ

=G L′

ζ

=

The homogenous solution is given by

( )

1 1 0 2 0

2 2 0

y x C J kx C Y kx

y x C J kx

= +

=

( )

27

The wronskianw x

( )

is

( )

1 2

y x y x

w x

y x y x

=

′ ′

( )

1 0 2 0 3 0

C J kx C Y kx C J kx

w x

C kJ kx C kY kx C J kx

+ =

′ + ′ ′

( )

1 3 0

( ) ( )

0 0

( ) ( )

0 w x =C C k J_ ′ kx Y kx −J kx y′ kx _

( )

2 3

( )

2

28

w x C C

x

π

= −

at C₂ 2

π

= and _{w x}

( )

C3 x

= − the form of Green’s function is

(

)

( ) ( )

1 2

2 1

, 0 ,

,

y x y

x p x w x

G x

y x y

x L p x w x

ζ

ζ ζ

ζ



− ≤ <

 

= 

− < ≤

 

(

)

( )

1 0 2 0 0

0 0 0

, 0

29 , 0

,

C J kx C Y kx J k x

J kx aJ k bY k x

G x

ζ ζ

ζ ζ ζ

ζ

 +  ≤ <

+ < ≤

 

  



=

( )

(

,

) ( )

b

a

y x =∫G x

ζ

f

ζ

d

ζ

EXAMPLE(2)

Find the green’s function of the singular differential

equation and find the solution

( )

2 2 2

2 0 31

a x y′′+ a y′− = −y f x < < ∞x

( )

0 0, lim

( )

32

x

y y L

→∞

= < ∞

Wherea>0

SOLUTION

Let G be the Green’s function, then

(

)

( )

2 2 2

2 0 , 33

a x G′′+ a xG′−G= −

δ

x−

ζ

<x

ζ

< ∞

(

0,

)

0, lim

(

,

)

( )

34

x

G

ζ

G x

ζ

→∞

= < ∞

The homogenous solution is given by

( )

m

( )

35

y x =Ax

So

( )

(

)

( )

1

2 36

1

m

y x mAx

y x m m Ax

−

′ = 



′′ = − _

Substituting from (37), (38) into (33) we have

(

)

2 2

2 2 2

2

1 2 0

1 0

1

m m m

Aa m m x a Amx Ax a m a m

m m a

− + + =

+ + =

(5)

www.ijsred.com

So

1 2 1 2 2 1 2

2

2 2 2

1

1 , , 1

1 0

r r r r r r

a r r a + = = − = − ∴ − − = 2 2 2 1 4 4

1 1 1 1

, 2 2 a a r r ± + ± + = =

( )

2 2 4 4

1 1 1 1

2 2

1 1

,

2 2 37

a a

y x A x

y

x A x

± + ± +

= =

∴

The wornskian is given by

( )

2 2 2 2 2 2 2 4 4

1 1 1 1

2 2

1 4 1 4

1 1 1 1 1 1

2 2

1 2

1

4 4

1 1 1 1

1 1

2 2 2 2

a a

w x

a a

A A x

Ax A x

x     ± + ± +                             ± + − ± + −             = ± + ± +                        

There Green’s function is given by

(

)

( ) ( )

1 2 2 1 , 0 , ,

y x y

x p x w x

G x

y x y

x L p x w x

ζ ζ ζ ζ ζ 

− ≤ <

 

= 

− < ≤

 

(

)

( )

2 2

1 4 1 4

1 1 1 1

2 2

1 4 1 4

1 1 1 1

2 2 2 2 2 2 0 4 1 , 39 4 1

,

a a a a x x a x a G x x x a

a x

ζ ζ ζ ζ ζ     ± + ± +                 ± + ± +               < <     +       =  

< < ∞

 _ _

 _ ₊ _

 _ _



Therefore the solution is given by

( )

2 2

1 4 1 4

1 1 1 1

2 2

1 4 1 4

1 1 1 1

2 2 0 40 a a a a x b x

y x x f d

x f d

ζ ζ ζ

    ± + ± +                 ± + ± +             =

_∫

+

∫

V.LEGENDRE’S EQUATION

(

2

)

(

)

( )

1−x y′′−2xy′+n n−1 y= f x , 1− < <x 1 41

Where and the function

( )

₍

2

₎

1

p x = −x vanishes at the endpointsx= ±1

(

)

( )

,

( )

42

xy′′− c−x y′+ny= f x

Which is transformed to the

(

)

( )

,

( )

43

x x x x

xe y− ′′− c−x e y− ′+ne y− =e− f x

By multiplication e−x

,

Here P x

( )

= p x

( )

=xe−x

vanishes atx=0

SOLUTION OF NON- HOMOGENOUS LEGENDRE

EQUATION

Considers the Legendre equation

(

₁₋_x2

)

_y_′′₋_xy_′₊_λ_y_{= −}_{f x}

( )

_{, 0}_{< <}_x _L

( )

₄₄ With the boundary condition

( )

0 lim x y x →

< ∞andy L

( )

=0

( )

45 Let us find the Green’s function for

(

2

)

(

)

( )

1−x G′′−xG′+λG= −δ x−ζ , 0< <x L 46 The homogenous solution of equation

(

2

)

( )

1−x y′′−xy′+λy=0 47

Let

φ

_n

( )

x

and

φ

_m

( )

x

are two solution of equation (47)

Let us divided the equation (47) by

(

1−x2

)

(

1 )

n n

λ

=

−

( )

1 2 2

( )

4

1 38

w x A A

a

(6)

www.ijsred.com

(

2

)

(

2

)

0

( )

48

1 1

x

y y y

x x

λ

′′− ′+ =

− −

( )

exp ₁ 2

( )

49

x

p x

ζ

d

ζ

∞

 

= −∫ 

−

 

To evaluation integral

(

2

)

( )

1

ln 1 50

2 ₂

1

x

d x

ζ

∞

= −

∫

−

Then

( )

(

) (

)

( )

1

2 2 ₂

1

exp ln 1 1 51

2

p x = _ −x _= −x

 

Multiplying equation (48) by

(

)

1 2 ₂

1−x we have

(

₁₋_x2

)

21 _y_′′₋

(

₁₋_x2

)

21_xy_′₊_λ

(

₁₋_x2

)

21 _y₌₀

( )

₅₂ Since

(

₁ _x2

)

12 _y

(

₁ _x2

)

21 _y

(

₁ _x2

)

−12 _xy

( )

₅₃

′

 

′ ′′ ′

− = − − −

 

 

The equation can be written as

(

)

(

)

( )

1 1

2 ₂ 2 ₂

1 x y λ 1 x − y 0 54

′

 

′

− + − =

 

 

This isin the form of Sturm-Liouville equation with

( )

(

)

( )

(

)

1 1

2 2 2 2

1 , 0 , 1

p x x q x r x x

−

= − = = −

this is a singular Sturm-Liouville problem we can

define

( )

1 2

55 ,

n n

n m

y x

y y

φ =

So that

( ) ( ) ( )

( )

56

b

n m m n mn

a

x x w x dx

φ φ =∫φ φ =δ

( )

1

57

n n n

y x ∞ cφ x

=

=∑

Where

( )

1

2 1

58 2

n n

n

C φ x dx

−

+

= ∫

Substitute equation (57) into

(

)

(

)

( )

1 1

2 ₂ 2 ₂

1 1 59

d

x k x y f x dx

−

 

− + − =

 

 

Then

(

2

)

1₂

( )

(

2

)

₂1

( )

( ) ( )

1 1

1 1 60

n n n

n n

d d

c x x k x c x f x

dx dx φ φ

−

∞ ∞

= =

  

− + − =

∑    ∑

 

Since from equation (55) we obtain

(

2

)

1₂

( )

₍

2

₎

₂1

( ) ( )

2

1 _n 1 61

d d

x x x x

dx dx φ λ φ

−

 

− = − −

 

 

So

(

)

(

)

( )

( ) ( )

1 2 ₂ 1

1 62

n n n

n

d

c k x x f x

dx λ φ

∞ =

+ − =

∑

Then 1

( ) ( )

( )

63

b

n n

a n

C f x x dx

k λ φ

= ∫

−

Whereλ_n =n n

(

+1

)

(

)

( ) ( )

(

)

( )

1

, 64

1

n n

n

x G x

k n n

φ φ ζ

ζ ∞

=

= ∑

− +

Hence the solution is given by

(

,

) ( )

( )

65

b

a

y=∫G x ζ f ζ dζ

PROBLEM(3)

Express the boundary solution of the following

inhomogeneous differential equation

(

2

)

( )

1−x y′′−2xy′+ky= f x , 1− < <x 1 66 In terms of Legendre polynomials with the Green’s function

(7)

www.ijsred.com

Let

G x

(

,

ζ

)

be the Green’s function forthe problem, then

(

₁ 2

)

₂

(

)

_{, 1} ₁

( )

₆₇

x G′′ xG′ kG δ x ζ x

− − + = − − < <

The corresponding homogeneous equation is

(

2

)

( )

1−x y′′−2xy′+ky=0 68 The homogenous solution

( )

0

69

k k

y x ∞ C P x

=

= ∑

Then

( )

_n _n

( )

70

y x =C P x

And y x

( )

=C P_m _m

( )

x

( )

71

Substituting these solutions into the homogenous

solution, we have

(

)

( )

(

)

( )

2

1 2 0

1 2 0 72

n n n n n n

n n n

x C P x xC P kC P

x P x xP kP

′′ ′

− − + =

′′ ′

− − + =

And

(

2

)

( )

1−x P_m′′ x −2xP_m′+λP_m =0 73 Multiply (72) by

P x

_m

( )

and

(73)

by P x_n

( )

and

subtract

(

₁ 2

)

₂

(

)

(

)

₀

n m m n n m m n m n

x P P′′ P P′′ x P P′ P P′

λ

k P P

− _ − _− − + − =

 

But

(

n m m n

)

n m n m m n m n n m m n

d

PP P P PP P P P P P P PP P P

dx

′′ ′′ ′′ ′′

′₋ ′ ₌ ₊ ′ ′₋ ₋ ′ ′₌ ₋

Therefore, the above expression reduce to

(

₁ 2

)

(

)

₂

(

)

(

)

₀

n m m n n m n m m n

d

x PP P P x PP P P k P P

dx λ

′′ ′′

′ ′ ′

− − − + + − =

(

₁ 2

)

(

)

₂

(

)

(

)

n m m n n m n m m n

d

x P P P P x P P P P k P P

dx λ

′′ ′′ ′ ′ ′

− − − + = − −

So

{

(

2

)

(

)

}

(

)

1 _n _m _m _n _m _n

d

x P P P P k P P

dx λ

′ ′

− − = −

Integrate both sides fromx= −1tox=1

,

we have

(

2

)

(

)

1

(

)

1

( )

1 1

1 x P P_{n m} P P_{m n} 0 k λ P P x dx_{m n} 74

− −

′ ′

− − = = − ∫

But

( )

1 1

2

1

n m mn

P x P

x dx

n

δ

−

=

+

∫

where

( )

,

( )

n m

P x P x are odd.

IfP x Pn

( )

, m

( )

x are even, then

(

)

(

)

( )

2 2 1

1 75

2 1 mn mn 2

n k

λ δ δ

λ

+

− = ⇒ =

+ −

(

)

(

)

( ) ( )

( )

0

2 1

, 76

2

n

G x P x P

k

ζ ζ

λ

∞ =

+ = ∑

−

Therefore

( )

(

) ( )

1

,

y x G x

ζ

f

ζ

d

ζ

−

= ∫

( )

(

)

( ) ( ) ( )

( )

1

0 1

0

2 1 1 2

77

n n

n _n

n n n

n

y x P x P f d

k

y x a P x

ζ

λ

∞ = −

∞ =

+ = ∫ ∑

−

=∑

Where

( ) ( ) ( )

1

2 1 1 2

n n

n

a f P f d

k λ − ζ ζ ζ ζ

+

= ∫

−

VI. CONCLUSIONS

We can find Solution of non- homogenous Bessel

equation for example(2.2.1.1.) and

example(2.2.1.2.), and also Solution of non-

homogenous Legendre equation see problem

(2.2.2.1.) by using Green’s Function Solution of

(8)

www.ijsred.com

Solution of non- homogenous Bessel equation,

Solution of non- homogenous Legendre equation.

ACKNOWLEDGMENT

I’m forever indebted my family for their endless

patience and encouragement, also Iwant to

recognize and express my thank to any one helped

me.

REFERENCES

[1] Abdelgabar Adam Hassan, Green’s Function Solution of Non- Homogenous Regular Sturm-Liouville Problem, Journal of Applied & ComputationalMathematics, 2017.

[2] Abdelgabar Adam Hassan, Green’s Function for the Heat Equation, Fluid Mechanics:Open Access, 2017.

[3] Al-Gwaiz.M.A., Sturm-Liouville theory& its Application, Verlag London limited, 2008.

[4] Blakledge. G. Evans. J. P. Yardley, Analytic methods for Partial differential Equations, Sprenger London Limited, 2000.

[5] Kwong-tintang, Mathematical methods for Engineers and Scientists 3, VerlagBerlin Heidelbarg, 2007.

[6] Holland Gerald B., Fourier analysis and Application, Wodsworth. Inc. BelmonCalifornia, 1992.

[7] Kwong-tintang, Mathematical methods for Engineers and Scientists 2, VerlagBerlin Heidelbarg, 2007.

[8] Neta. B, Partial differential equations, lecture notes, Naval postgraduate School, California, 2002.

[9] Pinchover.Y and Rubinstein.j, An introduction to Partial differential equations, Cambridge University, 2005.

[10] Roach G.F., Green’s function introductory theory with Application, New York Toronto Melbourne, 1970.

[11]Wesley Addison, Boundary Value problem, Monterey California, 1984. [12] Hopf, Introduction To The Differential Equations Of Physics, New York, 1948 .

[13] G. Evans, J. Blackledge and P. Yardley , Analytic Methods for Partial Differential Equations, Springer-Verlag Berlin Heidelberg New

York,1999.

[14] Zauderer E. Partial Differential Equations of applied mathematics.2ed, New York,1998.

[15] Joachim Weidmann , Spectral Theory of Ordinary Differential Operators,