Boundary Value Problem and its application in I-Function of
Multivariable
*KIRTI VERMA* S.S.L.Jain P.G.Coll.vidisha,M.P.,India ([email protected]) ** RAJESH SHRIVASTAVA** Govt.Sci&Comm. coll. Benazir Bpl ,M.P.,India([email protected]) *** A.K.RONGHE *** S.S.L.JaiP.G.Coll.vidisha,M.P.,India
Abstract
In this research paper, we make a model of a boundary value problem and its application in I-function of multivariable. Some particular cases have also been derived at the end of the paper.
Keywords: Boundary value problem, I-Function of multivariable, General class of polynomials. Mathematics subject classification: 2011, 33c, secondary 33c50.
1. Introduction: Y.N. Prasad [5] is defined. I-Function of multivariable are as follows:
[
]
, , ( ) ( ) 2 3 : , , ( ) ( ) 2 2 3 3 , : , :...: , ( , );...;( , ) 1...,
, : , :....: , :( , );...;( , ) r r r r r r r o n o n o n m n m n r p q p q p q p q p qZ
Z =
∑
∑
' '' ' ( ) ' ' ( ) ( ) ( ) 2 2 2 2 1 2 ' '' ' ( ) ' ' ( ) ( ) ( ) 2 2 2 2 ( , ; )1, ...( ; ;... )1, ( ; )1, ...( ; )1, , ,... ( , ; )1, ...( ; ;... )1, ( ; )1, ...( ; )1, r r r r j j j rj rj rj r j j j j r r r r r j j j rj rj rj r j j j j a p a p a p a p Z Z Z b q b q b q b q α α α α α α β β β β β β = 1 1 1 1 1 1 11
...
( )... ( ) ( .... )
...
...
(2
)
s s r r r r r r L Lrs
s
s
s Z
Z ds
ds
φ
φ
ψ
πω
∫
∫
(1.1) Where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 ( ) ( ) ( ) ( ) 1 1(
)
1 (
)
. .{1, 2,... }
(1
)
(
)
i i i i i i m n i i i i j j i j j i j j i i q p i i i i j j i j j i j m j nb
s
a
s
s
i e
r
b
s
a
s
β
α
φ
β
α
= = = + = +−
−
+
=
∀
−
−
−
∏
∏
∏
∏
and 2 2 2 2 ( ) 2 2 1 1 1 2 2 ( ) 2 2 1 1(1
)(
)
( ,
,.... )
(
)(
)
n i j j i i j r p i j j i i j na
s
s s
s
a
s
α
ψ
α
= = = = +−
+
=
×
−
∑
∏
∑
∏
2 2 ( ) ( ) 2 2 1 1 1 11
(1
)(
)...
(1
)(
)
r q q r i i j j i rj rj i i i j jb
β
s
b
β
s
= = = =−
+
∑
−
+
∑
∏
∏
The operational techniques are important tools to compute various problem in various fileds of sciences. Which are used in works of Kumar [3] to find out several results in various problem in different field o sciences and thus motivating by this work, we construct a model problem for temperature distribution in a rectangular plate under prescribed boundary conditions and then evaluate it solution involving I-Function of multivariable with products of general class of polynomials
[
]
1 1 1 1 1 1 [ / ] [ / ] ..., 1 1 1 ..., 1 1 1 1 0 0 1(
)
(
)
( ,...,
)
...
...
, ,...
...
!
!
r r r r r r n m n m m m r r r k k n n r r r r k k rn m k
n m k
S
x
x
F n k
n k x
x
k
k
= =−
−
=
∑
∑
. (1.2)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications where,
m
1...,
m
r are arbitary positive integers and the coefficientsF n k
[
1, ,...
1n k
r r]
are arbitary constants real or complex. Finally, we derive some new particular cases & find their applications also. 2.A Boundary value problem the boundary value conditions are,2 2 2 2
, 0
, 0
2
2
u
u
a
b
a
x
y
x
y
∂
∂
+
=
< <
< <
∂
∂
(2.1)0, 0
2
2
0
u
u
a
b
x
y
x
x
x
∂
=
∂
=
=
< <
∂
∂
=
(2.2)( , 0)
0, 0
2
a
U x
=
< <
x
(2.3) 1 1 1 2 2 ... ,... 1( , )
( )
cos
cos
,...
cos
2
r r r s s m m n n rb
x
x
x
U x
f x
S
y
y
a
a
a
π
π
π
=
=
' ' ( ) ( ) ' ' ( ) ( ) 0, [ , ];...[ , ] , :[ , ];...[ , ] n n n n U V U V A c B D B D λ∑
1 26 ' ( ) ' ' ( ) ( ) 1 ' ( ) ' 1 ( ) ( )[( ) :
,...,
] :[( ); ];...;[(
);
];
cos
,...
cos
26
[( ) :
,...
] :[( ); ];...,[(
);
];
n n n n r n n nx
x
a
b
b
Z
Z
a
a
c
d
s
d
π
π
θ
θ
φ
φ
ψ
ψ
δ
where,0
2
a
x
< <
Provided thatRe( ) 1
η
−
, allσ
i(
i
=
1, 2,..., )
n
are positive real numbers, 1 1 ,..., ,....,[ ]
r r m m n nS
x
isdue to (1.1) and I-Function of multivariable. u
U x y
( , )
is the temperature distribution in the rectangular plate at point.3.Main Integral:In our investigations, we make an appeal to the modified formula
/ 2 0 1
2
(
1)
cos
cos
2
1
1
2
2
ax
m x
a
dx
a
a
m
m
η ηπ
π
η
η
η
++
=
+ +
− +
∫
(3.1) 1 1 1 2 2 / 2 ,..., ,..., 1 02
cos
cos
cos
,...,
cos
r r r s s a m m n n r
x
m x
x
x
S
y
y
a
a
a
a
ηπ
π
π
π
∫
' ' ( ) ( ) ' ' ( ) ( ) 0, :[ , ];...;[ , ] , :[ , ];...;[ , ] n n n n u v U V A c B D B D λ∑
' ( ) ' ' ( ) ( ) ' ( ) ' ' ( ) ( )[( ) :
,....,
] :[( ); ];...;[(
),
];
[( ):
,....,
] :[( );
];...;[(
),
];
n n n n n na
b
b
c
d
d
θ
θ
φ
φ
ψ
ψ
δ
δ
1 26 26 1cos
,...,
cos
n nx
x
Z
Z
dx
a
a
π
π
1 1 1 [ / ] [ / ] 1 1 1 1 1 1 0 0 1(
)
(
)
...
...
[ , ,... ,
]
2
!
!
r r r n m n m r r r r r n k k rn m k
n m k
a
F n k
n k
k
k
+ = =−
−
×
∑
∑
(
)
1 1 1 1,....,
...
4
4
r k k r r ry
y
k
k
p
p
∑
(3.2) where ' ' ( ) ( ) ' ' ( ) ( ) 0, 1:[ , ];...;[ , ] 1 1, 2:[ , ];...,[ ]( ,....,
)
n nn n U V U V r A c B D D Dk
k
=
+λ ++∑
∑
' ( ) 1 1 1 ' ( ) 1 1 1 1 1 1[( ) :
,...,
],[
2
... 2
: 2
,...2
],[______];
[( ) :
,...
],[
/ 2
...
:
,...,
],[
/ 2
...
,....
]
n r r n n r r n r r na
p k
p k
c
m
s k
s k
m
s k
s k
θ
θ
η
σ
σ
ψ
ψ
η
σ
σ
η
σ
σ
− −
−
−
− −
−
−
+ −
−
0 1 ' ' ( ) ( ) 1 ' ' ( ) ( )[ ];
;...; (
) :
;
,...
4
4
( );
;....; (
) :
;
n n n q n n nb
b
Z
Z
d
d
σ σφ
φ
δ
δ
(3.3)Provided that
F n k
( , ,... ,
1 1n k
r r)
are arbitary functions ofn k
1 1,...
n k
r r,
real or complex independent of,
j,
j,
1, 2,.... ,
x y p
j
=
r
the condition of (2.4) and (3.1) are satisfied and,( ) ( ) ( ) ( ) ( ) ( ) 1
/
1, , , ,
,
,
,
n i j i i i i e i i j iR
η
σ
d
S
λ
A C U
V
B
D
=
+
> −
∑
are such that,( ) ( ) ( ) ( )
0,
0,
i i0,
i i0
A
≥ ≥
λ
C
≥
D
≥
U
≥
B
≥
V
≥
and ( ) ( ) ( ) ( ) ( ) ( ),
1, 2,...., ;
,
1, 2,...
,
,
1, 2,.... ,
,
1, 2,...,
i i i i i i jj
A
jj
B
jj
C S
jj
D
θ
=
φ
=
ψ
=
=
are positive realnumbers,
arg( )
z
i≤ ∆
iy
2 and( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1
0
i i i i i i A V B C U D i i i i i i i j k j j j j j j j V j j jS
S
λφ
φ
φ
ψ
= + = = + = = = +∆ = −
∑
+
∑
−
∑
−
∑
+
∑
−
∑
>
U1, 2,...,
i
=
n
4. Solution of Boundary Value Problem:
In this section, we obtain the solution of the boundary value problem stated in the section (2) as using (2.1), (2.2) and (2.3) with the help of the techniques referred to Zill [1] as,
0 1
2
2
( , )
sinh
cos
, 0
, 0
2
2
p pp y
p x
a
b
U x y
A y
A
x
y
a
a
π
π
=−
+
∑
< <
< <
(4.1) for,
2
b
y
=
we find that 0 12
,
( )
sinh
cos
, 0
2
2
p p2
A b
b
p b
p x
a
U x
f x
A
x
a
a
π
π
∞ =
=
=
+
< <
∑
(4.2)Now making an appeal to (2.4) & (4.2) and then integrating both sides with respect to
x from
0
to a
/ 2
, We derive, 1 1 1 1 1 ( / ) ( / ) 0 1 1 1 0 02
...
(
)
(
)
( , ,...
)
r r r n m n m m k r r r r r k kA
n
n m k F n k
n k
b
π
= ==
∑
∑
−
−
11 1 1(
, ....
)
...
!
!
r k k r r ry
y
k
k
k
k
∑
(4.3) where ' ' ( ) ( ) ' ' ( ) ( ) 0, 1:[ , ];...;[ , ] 1 1, 1:[ , ];....;[ , ]( ,... )
n n n n V V r A C B D B Dk
k
=
+λ + +∑
∑
U U ' ( ) 1 1 1 1 ( ) 1 1 1[( ) :
....,
],[1 / 2
/ 2
...
:
,....,
]
[( ) :
,...,
,[
/ 2
...
:
,....,
]
n r r n n r r na
n
s k
s k
c
s k
s k
θ
θ
σ
σ
ψ
ψ
η
σ
σ
−
−
−
−
−
−
' ' ( ) ( ) 1 ' ' ( ) ( )[( ) :
],...,[(
) :
],
,....,
[( ) :
],...,[(
);
]
n n n n nb
b
z
z
d
d
φ
φ
δ
δ
(4.4) Where all conditions of (2.4), (3.1) and (3.3) are satisfied. Again making an appeal to (2.4) and (4.2) and then multiplying bycos 2
m x a
π
/
both sides and integrate w.r.tx
from 0 to a/2, we find,1 1 1 [ / ] [ / ] 1 1 1 1 1 1 0 0 1
(
)
(
)
1
...
...
[
,....,
]
!
!
2
sinh
r r r n m n m r r r m r r k k rn m k
n m k
A
F n k
n k
p b
k
k
a
η−π
= =−
−
=
∑
∑
(
)
1 1 1 1,...,
...
4
4
r k k r r ry
y
k
k
p
s
∑
(4.5)Provided that all condition of (2.4), (3.1), and (3.3) are satisfied. finally making an appeal to the result (4.1), (4.3), and (4.5), we drive the required solution of the boundary value problem,
1 1 1 ( / ) ( / ) 0 0 1
2
( , )
...
(
)
j r r r k n m n m r j j j j k k s jy
y
U x y
n m k
k
b
π
= = =
=
−
∑
∑ ∏
(4.6)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
[
1 1]
(
1)
1 1(
1)
11
, ,....,
.
,....
(
)
[ , ,...,
]
,...
4
!
j k r j r r r j j m r r r m j jy
F n k
n k
k
k
n m k
F n k
n k
k
k
s
k
=
+
−
∑
∑
∑
Where,
0
< <
x
a
/ 2
< < <
0
y
b
/ 2
, provided that all conditions f (2.4), (3.1) and (3.3) are satisfied. 5. Expansion Formula:With the aid of (2.4) and (4.6) and then setting
y
=
( / 2)
b
we evaluate the expansion formula, 1 1 1 2 2 ,..., ,..., 1cos
cos
,...,
cos
r r r n s s m m n n r
x
x
x
S
y
y
a
a
a
π
π
π
' ' ( ) ( ) ' ' ( ) ( ) , :[ ];...;[ ] , :[ ];...;[ ] n n n n o u v u v A C B D B D λ∑
' ( ) ' ( ) ( ) ' ( ) ' ' ( ) ( ) [( ): ,... ]:[( ): ];...;[( ) : ] [ : ,...., ]:[( ): ];...;[( ) ]: n n n n n n a b b c d d θ θ φ φ ϕ ϕ δ δ
1 2 2 1cos
,...,
cos
r s s rx
x
Z
Z
a
a
π
π
= 1 1 1 ( / ) ( / ) 1 1 0 0 11
...
(
)
[ , ,....,
,
].
r r r kj n m n m r j j j j r r k k j jy
n m k
F n k
n k
k
π
= = =
−
∑
∑ ∏
(
)
1 1 1 (( / ) ( / ) 1 1 1 0 0 1cos 2
1
,....,
...
(
)
2
4
j r r r k n m n m r j r n j j j m k k j i jy
m x
k
k
n m k
p
k
π
∞ − = = = =
+
−
∑
∑
∑
∑ ∏
1 1 1[ , ,...,
r,
r].
( ,... )
rF n k
n k
∑
k
k
.Where
0
< <
x
( / 2)
a
, provided that all conditions of (2.4), (3.1) and (3.3) are satisfied. 6. Particular cases and Applications:In this section, we do some setting of different parameters of our results and then drive some particular cases as stated here as taking
m
1=
m
r=
γ
and,1 ... 1 1 1 1
1
[ , ,....,
,
]
(
)
(1
.... ) / (
....
)
r k k r r r rh
F n k
n k
v
γp
n
n
k
k
+ +
=
−
+ −
+
+
in (1.1) We get,[
]
1,..., / 1 1 1 1 ,.... / ...( ) ,.... 1 1 ...(
)
,...
( )
(
)
r nr r r r r n n y n x n n r n nV
S
x
x
x
P
γ γ − γ + +−
=
−
, 1,...( , , , )
1/
1/
1
( ) ,...,( )
nrh v p
n
x
x
r
γ
−
γ
−
γ
∑
(6.1)and thus, we obtain an integral for product of a class of polynomials of several variables
1 1 1,... 1 / 2 / 2 1 0 ...
2
(
)
cos
cos
cos
...
(
)
r r n s n n a n nx
x
V
x
y
a
a
P
a
γ ηπ
π
π
+ +
−
−
∫
1 1 / 1/ 1/ 2 2 2 ( , , , ) ,... 1cos
cos
,...,
cos
r r r r n s s s h v p r n n r
x
x
x
y
y
a
a
a
γ γ γ γπ
π
π
γ
∑
' ' ( ) ( ) ' ' ( ) ( ) ' ( ) ' ' ( ) ( ) , :[ , ];...;[ , ] , :[ , ];...;[ , ] ' ( ) ' ' ( ) ( )[( ) :
,....,
] : ( ); ( );....;[( ) ;
]
[( ) :
,....,
] : ( ); ( );....;[( ) ;
]
n n n n n n n o u v u v A C B D B V n n na
b
b
c
d
λθ
θ
φ
φ
ϕ
ϕ
δ
δ
δ
∑
1 2 2 1cos
,...,
cos
n nx
x
Z
Z
dx
a
a
σ σπ
π
= 1 1 1 1 .... ( / ) ( / ) 1 1 1 1 0 0 1 1 ( ... ) 1 1 1 ... ( ) ...( ) ... ( ,.... ) 2 ( ) ! ! (1 ,.... ) r r r r k k n n r r n k k r r k k a h n k n k k k V k k P n n γ γ γγ
γ
γ
+ + + = = + − − − + − ∑
∑
∑
1 1 1
...
4
4
r k k r ry
y
s
s
(6.2)Provided that all condition one of (2.4), (3.1) and (3.2) are satisfied. The solution of the given problem is:
1 1 [ / ] [ / ] 0 0 1
2
1
( , )
...
(
)
(
)
!
j r r k n n r j j j k k jhy
y
U x y
n
k
v
k
b
γ γ γγ
π
= = ∂=
=
−
−
∑
∑ ∏
1 1 1 [ / ] 1 1 1 0 1 ( .... )2
2
sinh
cos
1
( ,...,
)
...
(1
....
)
2
sinh
r n r m k r k km y
m x
a
a
k
k
m b
p
n
n
a
γ ηπ
π
π
γ
∞ − = = + ++
+ −
−
∑
∑
∑
1 [ / ] 1 0 1 1 ( ... )1
1
(
)
.
( ,....,
)
(
) 4
! (1
...
) /
j r r r k n r j j kj r k j j j r k khy
n
k
k
v
s
k
p n
n
γγ
γ
γ
= = + +
−
−
+ − −
−
∑
∏
∑
(6.3) When0
< <
x
( / 2), 0
a
< <
y
( / 2),
b
provided that all condition of (2.4), (3.1) and (3.3) are satisfied, The expansion formula is,1 1 1... 1 / / 2 2 1 ...
(
)
cos
cos
....
cos
(
)
r r r r n n s s n n r n nV
x
x
x
y
y
P
a
a
a
γ γ ηπ
π
π
− + +
−
−
1 1 1/ 1/ 2 2 ( , , , ) ,... 1cos
,...,
cos
r r s s h y v p n n rx
x
y
y
a
a
γ γπ
−π
−
∑
' ' ( ) ( ) ' ' ( ) ( ) ' ( ) ' ' ( ) ( ) 0, :[ , ];...;[ , ] , :[ , ];....;[ , ] ' ( ) ' ' ( ) ( )[( ) :
,...,
] :[( ); ];...;[(
);
]
[( );
,...,
] :[( );
];...;[(
);
]
n n n n n n n u v u v A c B D B D n n na
b
b
c
d
d
λθ
θ
φ
φ
ϕ
ϕ
δ
δ
∑
1 2 2 1cos
,....,
cos
x nx
x
Z
Z
a
a
σ σπ
π
= 1 () 0 0 0 11
1
...
(
)
[
] 4
!
r j j j j k k j jhy
n
k
k
v
k
γ
γ
π
= = =
−
−
∑
∑ ∏
1 1 1 [ / ] [ / ] 1 1 1 0 0 1 1 ( .... )2
cos
1
1
( ,...,
)
...
(
)
(1
...
)
2
(
) 4
!
j r r r k n n r j r j j m k k j r k k j jm x
hy
a
k
k
n
k
p n
n
v
s
k
γ γ η γπ
γ
γ
∞ − = = = = + +
+
−
+ − −
−
−
∑
∑
∑
∑ ∏
1 1 ( .... ) 11
( ,..,
)
(1
...
)
k kr r rk
k
p
n
n
γ
+ ++ − −
−
∑
(6.4)When
0
< <
x
a
/ 2,
provided that all conditions of (2.4), (3.1) and (3.3) are satisfied1 1 1 ( , ,1, ) 1 ( , ,1/ , ) ,...., r
,....,
,... r( ,...,
1)
( , )...
1 r( , )
h p r h p p n n n n r n n rx
x
x
x
g
x h
g
x h
p
p
γ
=
γ=
γ γ
∑
∑
(6.5)to the results, we evaluate the another results of Hermite polynomials by same techniques. Reference:
1. Dennis, G. Zill : A first course in differential equations with applications, IIed. Prindle, Weber and Boston, 1982.
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
2. H.W. Gould and A.T. Hopper : Operational formulas connected with two generalizations of Hermite
polynomials. Duke Math. 29 (1962), p-51-64.
3. H. Kumar: Special functions and their applications in modern science and technology, Ph.D. thesis, Barkatullah University, Bhopal, M.P. India, (1992-1993).
4. H.M. Srivastava : Pacific J. Math. 117 (1985), p-183-191
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