Scientific Research Journal (SCIRJ), Volume I, Issue 2, September 2013 ISSN 2201-2796
26
CERTAIN QUADRUPLE SERIES EQUATIONS Kuldeep Narain School of Quantitative Sciences College of Arts and Sciences Universiti Utara Malaysia 06010 UUM, Sintok, Kedah Darul Aman, Malaysia.
Abstract- In this paper, we have obtained the solution of the quadruple series equations involving Jacobi polynomials as kernel. These equations are finally reduced to Fredholm integral equations of second kind. Index Terms— Integral Equation, Series Equation, Jacobi polynomial
I. INTRODUCTION In this paper, the solution of the following quadruple series equations have been derived ∑
(
( )
∑
(
)
(
( )
(
)
( ) )
)
( )
(
) (
)
)
( )
(
) (
∑
)
( (
∑
(
)
)
( )
(
)
( )
(
)
( )
(
)
( )
(
)
( )
(
)
Where f1 (x) to f4 (x) are prescribed functions, sequences {An} is to be determined and the parameters α, , , , conditions - - > o, α> - 1, > - 1, > - 1 and > - 1. II. SOME USEFUL RESULTS Here we listed some results for ready reference: (i) The orthogonality relation for Jacobi polynomials is (
∫
)
(
( )
( ( where,
m,n is
) (
)
( )(
) (
) ( ) (
)
)
(
)
kronecker’s delta and α > - 1, >-1.
(ii) The series (
)
∑
(
)
( (
)( )(
)(
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) ( )
)
)
satisfy the
Scientific Research Journal (SCIRJ), Volume I, Issue 2, September 2013 ISSN 2201-2796
27
( (
) ) (
)
( Where and
( )
(
( )
) ∫
(
) ( ) (
(
)
(
-
(1-y)-
)
( )
(
( )( (
)
n(y) = (1+y) -
w = min (x, y), (
)
(
)
(
)
)
)
(
)
-
)(
) ( ) (
) ) (
)
It being assumed that the parameters are so constrained that Is independent of (iii) It f(x) and f1(x) are continuous in a x b and if o < < 1 then the solution of Able integral equations ( )
∫
(
( ) )
(
)
( ) ( )
(
)
and ( )
∫
are given by
( )
( )
( )
∫
(
)
(
)
)
(
)
and ( )
( )
( )
∫
(
respectively
III. SOLUTION OF EQUATIONS Let us suppose (
∑
(
)
)
( )
(
( ) ( )
)
(
)
Making use of the orthogonality relation (2.1), we get from (1.2), (1.4) and (3.1), (
) ( (
) ) (
)
∫(
) (
)
∫ (
) (
)
[∫ (
) ( (
( ) ( )
)
(
(
) ( )
)
( )
∫ (
) (
)
( )
(
)
( )
( ) )
( )
]
(
)
Substituting this expression for An from (3.2) in (1.1) and (1.3) and interchanging the order of integration and summation, we obtain the equations ∫ (
) (
)
( )
(
)
(
)
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Scientific Research Journal (SCIRJ), Volume I, Issue 2, September 2013 ISSN 2201-2796
∫(
) (
)
∫ (
∫(
( )
) (
) (
28
(
)
)
( )
)
(
)
( )
(
)
)
(
)
(
( )
(
(
)
)
( )
(
)
Where, ( )
( )
( )
∫ (
) (
)
( )
(
)
(
)
∫(
) (
)
( )
(
)
(
)
(
)
∫ (
) (
)
( )
(
)
(
)
(
)
(
)
( )
S (r, x) is same as detained by (2.2) and
(
)
(
∑
( (
(
)
)( )(
(
) )
(
)
( )
(
)
)( )
)(
)
( )
It is assumed that the form of Hn is such that T (r,x) converges. Equation (3.3) and (3.4) can be written as with the help of (2.3) ∫ ∫
( ) )
(
∫
( ) (
)
( (
)
(
( )
) (
{∫ (
) ( )
∫ (
) ) (
)
) ( )
( ) ∫ (
) (
) (
)
(
)
With the help of (2.6), we find from (3.8) ∫
(
( )
) (
)
( ) ( )
(
)
∫
(
( )
) (
)
Where, ( )
∫
(
) (
( ) )
Using (2.7), we obtain from (3.9)
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(
)
( )} (
)
Scientific Research Journal (SCIRJ), Volume I, Issue 2, September 2013 ISSN 2201-2796
( (
)
(
( )
)
∫
(
)
)
) (
(
)
(
( )
) (
(
)
) (
)
( ) ( )( )
∫
(
∫
Now, using the result ∫
29
)
(
)
(
)
(
)
(
)
We find that ( )
(
( )
Where,
(
( )
) ( ( )
Using (3.4) and (2.3), we find ( ) ( ) ∫ {∫ ( ( )
(
)
( )
∫ (
)
( ) )
(
(
) ( (
∫ ( ( )
)
) (
)
( )} (
)
(
)
( )
∫
(
∫
) ( )
( ) ( )( )
∫
)
) (
∫
) ∫ )
( )
(
)
(
)
( ) )
(
)
(
)
(
)
(
)
(
)
(
)
Where, ( )
∫
(
) (
( ) )
From (3.16) and (2.7), we find (
(
) ( )
)
∫
( ) (
)
The equation (3.15) has the form (2.4), then using (2.6), it can be written in the form, n (y) G (y) = N1 (y) + 1 +2, b<x<c
(3.18)
Where, (
( )
)
∫
(
) (
)
( ∫ (
)
( )
)
(
() )
∫
∫
∫
(
)
() (
)
( ∫
( )
( )
(
) (
∫
)
(
)
( ) )
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( )
Scientific Research Journal (SCIRJ), Volume I, Issue 2, September 2013 ISSN 2201-2796
After some manipulation, we get ( (
)
(
∫
)
30
) (
()
∫
)
(
) ( ) )
(
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
Putting the value of the last integral in the above equatin from (3.9), we get (
( )
) (
∫(
) ( (
∫
) )(
)
( )
() )
Where, (
( )
) (
( )( (
∫
)
) )
After some manipulaton the equation (3.21) can be written in the form. ( ) ) ( ) ∫ ( ∫ ( )
( (
) ) (
() )
Hence, (
( ) Where,
(
)
( )( (
∫
) )
( ) ( ) ( ) Using equations (3.23) and (3.24) the equations (3.18) can be written in the form ( )
)
) (
( )
(
) ∫
( )
( )
∫
( )
(
)
(
)
Where, (
)
(
) (
( )
)
∫
( )( ) ( )(
)
Equation (3.27) is a Fredholm integral equation of second kind. From equation (3.27), we can determine G(y), then g(u) and h(u) can be easily obtained from equation (3.18) and equation (3.13) and hence the coefficients can be found.
REFERENCES [1] Cooke, J.C; Tripal integral equations, Quart. J. Mech. Appl. Math; 16, pp. 193-203 (1963). [2] Cooke, J.C; The solution of triple and quadruple integral equations and Fourier-Bessel Series, Quart. J. Mech. Appl. Math. 25, pp.247-263 (1972). [3] Rainville, E.D; Special Functions, Macmillan (1967). [4] Spence, P.A; On dual series relations involving Laguerre Polynomials, Pacific. J. Maths ; 19, pp.529-533(1966b).
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