This academic article discusses solving simultaneous triple series equations associated with Laguerre polynomials with matrix arguments. It presents the following simultaneous triple series equations that arise in crack problems in fracture mechanics. It then provides solutions for the unknown function Cnj by using integral equations, orthogonality relations of Laguerre polynomials, and differential formulas. The solutions are determined for specific conditions on the parameters a, b, and m.
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optimal solution method of integro-differential equaitions under laplace tran...INFOGAIN PUBLICATION
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Lattice points on the homogeneous cone 5(x2+y2) 9xy=23z2eSAT Journals
Seven different method s of the non-zero non-negative solutions of homogeneous Diophantine equation 5(x2 + y2) – 9xy = 23z2 are obtained. Introducing the linear transformation x =u + v, y= u – v, u v0 in 5(x2+y2) -9xy = 23z2, it reduces to u2 + 19v2 =23z2. We are solved the above equation through various choices and are obtained seven different methods of solutions which are satisfied it. Some interesting relations among the special numbers and the solutions are exposed.
optimal solution method of integro-differential equaitions under laplace tran...INFOGAIN PUBLICATION
In this paper, Laplace Transform method is developed to solve partial Integro-differential equations. Partial Integro-differential equations (PIDE) occur naturally in various fields of science. Engineering and Social Science. We propose a max general form of linear PIDE with a convolution Kernal. We convert the proposed PIDE to an ordinary differential equation (ODE) using the LT method. We applying inverse LT as exact solution of the problems obtained. It is observed that the LT is a simple and reliable technique for solving such equations. The proposed model illustrated by numerical examples.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Lattice points on the homogeneous cone 5(x2+y2) 9xy=23z2eSAT Journals
Abstract
Seven different method s of the non-zero non-negative solutions of homogeneous Diophantine equation 5(x2 + y2) – 9xy = 23z2 are obtained. Introducing the linear transformation x =u + v, y= u – v, u v0 in 5(x2+y2) -9xy = 23z2, it reduces to u2 + 19v2 =23z2. We are solved the above equation through various choices and are obtained seven different methods of solutions which are satisfied it. Some interesting relations among the special numbers and the solutions are exposed.
DISTANCE TWO LABELING FOR MULTI-STOREY GRAPHSgraphhoc
An L (2, 1)-labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k }such that |f(x)-f(y)| ≥2 if d(x, y) =1 and | f(x)- f(y)| ≥1 if d(x, y) =2. The L (2, 1)-labeling number λ (G) or span of G is the smallest k such that there is a f with
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aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of
triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A
comparison has been made betweenapproximation of triangular fuzzy systems and the corresponding fuzzy
triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular
fuzzy numbers are also found.
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
In this article we assume that experts express their view points by way of approximation of Triangular
fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to
aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of
triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A
comparison has been made betweenapproximation of triangular fuzzy systems and the corresponding fuzzy
triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular
fuzzy numbers are also found.
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
In this article we assume that experts express their view points by way of approximation of Triangular
fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to
aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of
triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A
comparison has been made betweenapproximation of triangular fuzzy systems and the corresponding fuzzy
triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular
fuzzy numbers are also found.
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
In this article we assume that experts express their view points by way of approximation of Triangular
fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to
aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of
triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A
comparison has been made betweenapproximation of triangular fuzzy systems and the corresponding fuzzy
triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular
fuzzy numbers are also found.
Radix-3 Algorithm for Realization of Type-II Discrete Sine TransformIJERA Editor
In this paper, radix-3 algorithm for computation of type-II discrete sine transform (DST-II) of length N =
3푚 (푚 = 1,2, … . ) is presented. The DST-II of length N can be realized from three DST-II sequences, each of
length N/3. A block diagram of for computation of the radix-3 DST-II algorithm is given. Signal flow graph for
DST-II of length 푁 = 32 is shown to clarify the proposed algorithm.
Radix-3 Algorithm for Realization of Type-II Discrete Sine TransformIJERA Editor
In this paper, radix-3 algorithm for computation of type-II discrete sine transform (DST-II) of length N =
3𝑚 (𝑚 = 1,2, … . ) is presented. The DST-II of length N can be realized from three DST-II sequences, each of
length N/3. A block diagram of for computation of the radix-3 DST-II algorithm is given. Signal flow graph for
DST-II of length 𝑁 = 32 is shown to clarify the proposed algorithm.
On Some Notable Properties of Zero Divisors in the Ring of Integers Modulo m ...inventionjournals
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Simultaneous triple series equations
1. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
www.iiste.org
Simultaneous Triple Series Equations
Associated With Laguerre Polynomials With Matrix Argument
Kuldeep Narain
School of Quantitative Sciences, UUM College of Arts and Sciences, University Utara Malaysia, Sintok –
06010, Malaysia
E-mail: kuldeep@uum.edu.my
Abstract
Integral and Series equations are very useful in the theory of elasticity, elastostatics , diffraction theory and
acoustics. Particularly these equations are very much useful in finding the solution of crack problems of fracture
mechanics. In this paper solution of simultaneous triple series equations associated with Laguerre polynomials
with matrix argument has been obtained , which arises in the Crack problems of Fracture Machanics.
Keywords: Integral equations, Series equations , Laguerre polynomials, Matrix argument.
1. Introduction
In the present paper, we have considered the following simultaneous triple series equations of the form
¥
s
n=0
j =1
¥
s
n=0
j =1
S Sa
ij
S S
bij cnj .
Lni (a + b ¥
s
n=0
j =1
S S
for
G m (a + b + ni)
cnj
Lni (a : x) = fi (x), 0 £ x £ D,
m +1
)G m (a + b + ni)
2
II m (a + b + ni)
G m (a + ni +
m +1
, y ) = gi ( x), D £´£ E ,
2
(1.2)
cnj G m (a + b - m + 1) = Lni (a , x) = hi ( x), E £´£ ¥ ,
a +b >
2
m +1
- 1,0 £ b £1
2
where,
Ln (a , x) =
(1.1)
II m (a + n)
m +1
1F1 (-n, a +
), x)
P m (a )
2
88
(1.3)
2. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
www.iiste.org
is Laguerre polynomial of matrix argument ,
for
R (a ) > - 1,
R (n + a ) > 1, and II m (a) = G m (a +
m
G m (a) = p m ( m-1)/4 Õ (a i =1
m +1
)
2
i -1
)
2
n =0,1,2,……………………., J = 1,2,3,……………s ,
f(x), g(x) h(x) are known functions of non – singular matrix x of order m ; aij , bij and cij are known
constant .By using multiplying factor technique [ Srivastava], the unknown function C nj is determined .
2 . Some Useful Results
(i) The following integrals are required from Erdelyi et al with matrix argument
y
ò0
y
a
y-x
B -(m +1)/2
Ln (a , x)dx
m +1
)
a +b
2
=
y
Ln (a + b ; y )
m +1
G m (a + b + n +
)
2
G m ( b ) G m (a + n +
for
a > - 1, b >
¥
ò x- y
y
= Gm (
for
-b
m +1
-1
2
(2.1)
and
etr (- x) Ln (a , x)dx
m +1
m +1
- b ) etr (- y) Ln (a + b ; y)
2
2
b<
m +1
m +1
,a + b >
-1 .
2
2
(ii) The orthogonality relation for Laguerre polynomial with matrix argument
89
(2.2)
3. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
www.iiste.org
a
ò x >0 x etr (- x) Lp (a ; x) Lq (a ; x)
m +1
)
2 d
pq
m +1
)
G m (p+
2
G m (a + b +
=
for a > - 1 and
d pq
(2.3)
being the kronecker delta .
(iii) The differential formula with matrix argument due to Erdelyi et al
a
Dx [ x Ln (a ; x)] = x
a-
m +1
2
Ln (a -
m +1
; x)
2
(2.4)
3. Solution
Multiply eq. (1.1) by
y-x
b - ( m+1) 2
and eq. (1.2) by
x- y
-b
etr (- x) and then integrating w.r.t. x
m +1
) G m (a + b + ni)
2
Lni (a + b ; y ) =
II m (a + b + ni )
G m (a + ni +
s
åå
a
(0, y) and ( y, ¥) respectively, on using the result (2.1) and (2.2), we obtain
over the range
¥
x
aij cnj
n = 0 j =1
-a - b
y
a
b - ( m +1)/2
y
ò0 x y - x
fi ( x)dx ,
Gm (b )
b>
for
(3.1)
m +1
, a > -1, 0 < y < D
2
and
¥
s
S S bij cnj
n = 0 j =1
=
m +1
) G m (a + b + ni)
m +1
2
Lni (a + b ; y)
II m (a + b + ni)
2
G m (a + ni +
etr ( y )
-b
ò ¥ etr (- x) x - y gi ( x)dx ,
y
m +1
Gm ( 2 - b )
for b <
(3.2)
m +1
m +1
, a +b >
- 1, D < y < ¥ .
2
2
90
4. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
If we new multiply eq.(3.1) by by y
¥
s
S S bij cnj
for
a
, differentiating w.r.t. Y and using the formula (2.4), we find
m +1
m +1
(
) -a - b
) G m (a + b + ni)
s
y 2
m +1
2
. Lni (a + b .
; y ) = å eij
2
Gm (b )
II m (a + b + ni)
j =1
G m (a + ni +
n = 0 j =1
y
Dy ò 0 x
a +b
www.iiste.org
y-x
b -(m +1)/2,
0 < y < D, b >
fi ( x)dx ,
(3.3)
m +1
- 1, a > -1, and eij are the element of the matrix [bij ][aij ]-1 and
2
i = 1, 2,.....s.
The left- hand side of eq.. (3.2), (3.3) and (1.2) are identical and hence on using the orthogonality relation (2.3),
we find the solution of eq. (1.1), (1.2) and (1.3) for
a +b >
m +1
- 1, 0 < b < 1
2
m +1
) G m (a + b + ni )
2
Cnj = S dij
Bni (a + b ; D) (3.4)
j =1
m +1
2
)[G m (a + b + ni)]
G m (a + ni +
2
G m (ni +
s
Where ,
n = 0,1, 2.......; j = 1, 2,3,.......,s and d ij are the element of the matrix [bij ]-1 and
s
Bni (a , b ; D) = å eij
j =1
D
1
ò etr (- y)Lni (a + b - m2+1 ; y) .
Gm (b ) 0
m +1
, y )Gi ( y )dy
2
¥
m +1
1
a + b - ( m +1)/2
+
ò y
Ln (a + b ; y ) H ( y )dy
m +1
E
2
Gm (
- b)
2
E
Fi ( y )dy + ò etr (- y ) y
a + b - (m +1)/2
D
y
Fi ( y ) = Dy ò x
a
y-x
-b
etr (- x)hi ( x)dx
0
b - ( m +1)/2
Ln (a + b -
fi ( x)dx
(3.6)
Gi ( y) = gi ( y)
¥
H i ( y) = ò x - y
y
(3.5)
91
5. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
www.iiste.org
References
T.W. Anderson ; (1958) , An Introduction to Multivariate statistical Analysis, John Wiley and Sons, New York .
A. Erdelyi , et al ; (1953) , Higher Transcendental functions, Vol, II, Mc Graw Hill Book co, Inc; New York.
A. Erdelyi , et al ; (1954) ,Tables of Integral Transforms , Vol, II, Mc Graw Hill Book co, Inc; New York .
A.M. Mathoi and R.K. Saxena; (1978) , The H-function with Applications in Statistics and other Disciplines,
Wiley Eastern Limited, New Delhi , India, 96-132 .
H.M. Srivastava,; (1969) , Notices Am. Math. Soc. 16 , 568, (See also p. 517) .
H. M. Srivastava ; (1969) , Pacific J. Math , 30 ( 1969) , 525 -27 .
H.M. Srivastava ; (1970) , J. Math . Anal . Appl. 31, 587-94 (see also p.587) .
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