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Boundary Value Problem and its application in I-Function of
Multivariable
*KIRTI VERMA*
S.S.L.Jain P.G.Coll.vidisha,M.P.,India
(kirtivrm3@gmail.com)
** RAJESH SHRIVASTAVA**
Govt.Sci&Comm. coll. Benazir Bpl ,M.P.,India
(rajeshraju0101@reddifmail.com)
*** A.K.RONGHE ***
S.S.L.JaiP.G.Coll.vidisha,M.P.,India
(dr.ashokrao@yahoo.co.in)
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 q
Z 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
1
1
....... ( )....... ( ) ( .... ) ..... .....
(2 )
s s
r r r r rr
L Lr
s s s s Z Z ds dsφ 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 n
b s a s
s i e r
b s a s
β α
φ
β α
= =
= + = +
− − +
= ∀
− − −
∏ ∏
∏ ∏
and
2
2
2
2
( )
2 2
11
1 2 2
( )
2 2
11
(1 )( )
( , ,.... )
( )( )
n
i
j j i
ij
r p
i
j j i
ij n
a s
s s s
a s
α
ψ
α
==
== +
− +
= ×
−
∑∏
∑∏
2 2
( ) ( )
2 2
1 11 1
1
(1 )( ).............. (1 )( )
rq q r
i i
j j i rj rj i
i ij j
b s b sβ β
= == =
− + − +∑ ∑∏ ∏
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 k kr r r
n n r r r r
k k r
n m k n m k
S x x F n k n k x x
k k= =
− −
= ∑ ∑ . (1.2)

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where, 1......., rm m are arbitary positive integers and the coefficients [ ]1 1, ,...... r rF n k n k 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 yx
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 r
b 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
∑
126
' ( ) ' ' ( ) ( )
1
' ( ) ' 1 ( ) ( )
[( ): ,......, ]:[( ); ];.......;[( ); ]; cos ,....... cos 26
[( ) : ,....... ]:[( ); ];........,[( ); ];
n n n
n r
n n n
x 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 that Re( ) 1η − , all ( 1,2,....., )i i nσ = are positive real numbers, 1
1
,...,
,...., [ ]r
r
m m
n nS x is
due 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
a x m x a
dx
a a
m m
η
η
π π η
η η+
+ 
= 
    + + − +  
  
∫ (3.1)
1
1
1
2 2
/2
,......,
,......., 10
2
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
η
π π π π      
      
       
∫
' ' ( ) ( )
' ' ( ) ( )
0, :[ , ];.......;[ , ]
, :[ , ];......;[ , ]
n n
n n
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
ψ ψ δ δ
 
 
 
126 26
1 cos ,......, cos
n
n
x x
Z Z dx
a a
π π   
   
   
1 1
1
[ / ] [ / ]
1 1 1
1 11
0 0 1
( ) ( )
......... ........ [ , ,....... , ]
2 ! !
r r
r
n m n m
r r r
r rn
k k r
n m k n m ka
F n k n k
k k+
= =
− −
×∑ ∑
( )
1
1
1
1
,...., ........
4 4
rk k
r
r
r
y y
k k
p p
   
   
   
∑ (3.2)
where
' ' ( ) ( )
' ' ( ) ( )
0, 1:[ , ];......;[ , ]
1 1, 2:[ , ];.......,[ ]
( ,...., )
n n
n n
U V U V
r A c B D D D
k 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 n
a 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 n
b b ZZ
d d
σσ
φ φ
δ δ
       
       
(3.3)

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Provided that 1 1( , ,...... , )r rF n k n k are arbitary functions of 1 1,....... ,r rn k n k real or complex independent of
, , , 1,2,.... ,j jx 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
i
R d S A C U V B Dη σ λ
=
 
+ >− 
 
∑ are such that,
( ) ( ) ( ) ( )
0, 0, 0, 0i i i i
A C D U B Vλ≥ ≥ ≥ ≥ ≥ ≥ ≥ and
( ) ( ) ( ) ( ) ( ) ( )
, 1,2,...., ; , 1,2,..... , , 1,2,.... , , 1,2,.....,i i i i i i
j j j jj A j B j C S j Dθ φ ψ= = = = are positive real
numbers, 2arg( )i iz y≤ ∆ and
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
1 1 1 11 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 jj V j
S S
λ
φ φ φ ψ
= + = = == + = +
∆ =− + − − + − >∑ ∑ ∑ ∑ ∑ ∑
U
1,2,......,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
p
p 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
1
2
, ( ) sinh cos ,0
2 2 2
p
p
A bb 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 0 / 2x from to a ,
We derive,
1 1
1 1
1
( / ) ( / )
0 1 1 1
0 0
2
....... ( ) ( ) ( , ,..... )
r r
r
n m n m
m k r r r r r
k k
A n n m k F n k n k
b π = =
= − −∑ ∑
1
1
1
1
( ,.... ) .........
! !
rk k
r
r
r
y 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 D
k k λ+
+ +
=∑ ∑ U U
' ( )
1 1 1
1 ( )
1 1 1
[( ) : ...., ],[1/ 2 / 2 ...... : ,...., ]
[( ) : ,..., ,[ / 2 ....... : ,...., ]
n
r r n
n
r r n
a n s k s k
c s k s k
θ θ σ σ
ψ ψ η σ σ
 − − −

− − −
' ' ( ) ( )
1' ' ( ) ( )
[( ) : ],......,[( ) : ],
,....,
[( ) : ],......,[( ); ]
n n
nn n
b 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 by cos2 /m x aπ both sides and integrate w.r.t x 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 r
n m k n m k
A F n k n k
p b k k
a
η π− = =
− −
= ∑ ∑
( )
1
1
1
1
,....., .......
4 4
rk k
r
r
r
y 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
kn m n m r
j
j j j
k k s j
yy
U x y n m k
kb π = = =
  
 = − 
    
∑ ∑ ∏ (4.6)

4. Mathematical Theory and Modeling www.iiste.org
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[ ] ( ) ( )1 1 1 1 1 1
1
1
, ,...., . ,.... ( ) [ , ,..., ] ,...
4 !
jk
r
j
r r r j j m r r r
m j j
y
F n k n k k k n m k F n k n k k k
s k=
  
 + −      
∑ ∑ ∑
Where, 0 / 2 0 / 2x a y b< < < < < , 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 ( / 2)y 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


12 2
1 cos ,....., cos
rs s
r
x x
Z Z
a a
π π   
   
   
=
1 1
1
( / ) ( / )
1 1
0 0 1
1
........... ( ) [ , ,...., , ].
r r
r
kjn m n m r
j
j j j r r
k k j j
y
n m k F n k n k
kπ = = =
  
−      
∑ ∑ ∏
( )
1 1
1
(( / ) ( / )
1 1
1 0 0 1
cos2 1
,...., ............ ( )
2 4
j
r r
r
k
n m n m r
j
r j j jn
m k k j i j
ym x
k k n m k
p k
π∞
−
= = = =
   
  + −       
∑ ∑ ∑ ∑ ∏
1 1 1[ , ,........, , ]. ( ,..... )r r rF n k n k k k∑ .
Where 0 ( / 2)x 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 1 rm m γ= = and,
1 ......
1 1
1 1
1
[ , ,...., , ]
( ) (1 .... ) / ( .... )
rk k
r r
r r
h
F n k n k
v p n n k kγ
+ +
 
=  − + − + + 
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 n
V
S x x x
P
γ
γ γ
−
+ +
−
=
−
, 1,...........
( , , , ) 1/ 1/
1( ) ,.....,( )nr
h vp
n rx xγ γ γ− −
  ∑ (6.1)
and thus, we obtain an integral for product of a class of polynomials of several variables
1
11,.........
1
/2
/2
1
0
......
2 ( )
cos cos cos .......
( )
r
r
nsn n
a
n n
x x V x
y
a a P a
γη
π π π
+ +
 −   
    
−     
∫
1
1
/ 1/ 1/2 2 2
( , , , )
,..... 1cos cos ,......, cos
r
r r
r
ns 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 n
a b b
c d
λ θ θ φ φ
ϕ ϕ δ δ δ



∑
12 2
1 cos ,......, cos
n
n
x x
Z Z dx
a a
σ σ
π π   
   
   
=
1
1
1 1
....( / ) ( / )
1 1 11
0 0 1 1 ( ... )
1 1 1
...... ( ) ........( ) ...... ( ,.... )
2 ( ) ! ! (1 ,.... )
r
r
r r
k kn n
r rn
k k r r k k
a h
n k n k k k
V k k P n n
γ γ
γ
γ γ
γ
+ +
+
= = +
 
− −  − + − 
∑ ∑ ∑

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1
1
1
.....
4 4
rk k
r
r
y 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
kn n r
j
j j
k k j
hyy
U x y n k
v kb
γ γ
γ
γ
π = = ∂=
    
 = −  
−     
∑ ∑ ∏
1
11
[ / ]
1
11 01 ( .... )
2 2
sinh cos
1
( ,..., ) ...........
(1 .... ) 2 sinhr
n
r
m kr k k
m y m x
a ak k
m bp n n
a
γ
η
π π
πγ
∞
−= =+ +
+
+ − −
∑ ∑ ∑
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 k
hy
n k k
v s k p n n
γ
γ
γ γ= = + +
    −  
− + − − −    
∑ ∑∏
(6.3)
When 0 ( / 2),0 ( / 2),x a y b< < < < provided that all condition of (2.4), (3.1) and (3.3) are satisfied,
The expansion formula is,
1
11......
1
/ /2 2
1
.....
( )
cos cos .... cos
( )
r
rr
r
n ns sn n
r
n n
V x x x
y y
P a a a
γ γη
π π π−
+ +
   −      
        
−            
1
1
1/ 1/2 2
( , , , )
,........ 1 cos ,....., cos
r
r
s s
h y v p
n n r
x x
y y
a a
γ γ
π π
− −
                 
           
∑
' ' ( ) ( )
' ' ( ) ( )
' ( ) ' ' ( ) ( )
0, :[ , ];.....;[ , ]
, :[ , ];....;[ , ] ' ( ) ' ' ( ) ( )
[( ) : ,..., ]:[( ); ];.....;[( ); ]
[( ); ,..., ]:[( ); ];.....;[( ); ]
n n
n n
n n n
u v u v
A c B D B D n n n
a b b
c d d
λ θ θ φ φ
ϕ ϕ δ δ



∑
12 2
1 cos ,...., cos
x
n
x x
Z Z
a a
σ σ
π π    
   
    
=
1
() 0
0 0 1
1 1
......... ( )
[ ] 4 !r
j
j j j
k k j j
hy
n k k
v k
γ
γπ = = =
   
−    −    
∑ ∑ ∏
1
11
[ / ] [ / ]
1 1
1 0 0 11 ( .... )
2
cos
1 1
( ,..., ) ........ ( )
(1 ..... ) 2 ( ) 4 !
j
r
rr
k
n n r
j
r j j
m k k jr k k j j
m x
hyak k n k
p n n v s k
γ γ
η γ
π
γ
γ
∞
−
= = = =+ +
    + −  
+ − − − −    
∑ ∑ ∑ ∑ ∏
1
1( .... )
1
1
( ,.., )
(1 ..... ) r
rk k
r
k k
p n n γ + +
+ − − −
∑ (6.4)
When 0 / 2,x a< < provided that all conditions of (2.4), (3.1) and (3.3) are satisfied
1 1 1
( , ,1, ) ( , ,1/ , )1
,...., ,... 1 1,...., ( ,..., ) ( , )...... ( , )r r r
h p h p pr
n n n n r n n r
x 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.

6. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
132
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
5. Prasad, Y.N.: on multivariable I-function. Vigyan anusandhan patrika, 26, (4), 1986, p-231-235.

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