Application of I Function of Several Variables in Problem ...

−

−

−

(1 + x) − Pl(x)I

.

2

{ ·· · · ·· } ∀ ∈ { · · · · · · }

r ∂r2

(rV ) + sin θ ∂θ

sin θ ∂θ

Application of I Function of Several Variables in Problem on Electrostatic Potential in Spherical

Region

Mithilesh kumar Mishra

Dept. of mathematics, Govt. S.N.S. college, Shahdol,

(M.P) Rajeev Shrivastava

Dept. of Mathematics, Govt. Indira Gandhi Girls College Shahdol (M.P)

Abstract: In this paper firstly we evaluate an integral involving I function of

several variables to solve a boundary value problem associated with

electrostatic potential in spherical region.

1 Introduction

Various problems in science and technology be formulated mathematically the

classical method in obtaining solution of the boundary value problems of mathe-

matics can be derived by I function of several variables. The theory

developed by Heaviside and Deotsch and others have unified the latter

investigation by Bromwhich and Carson in the recent work on the laplace

transformation. Al- though the laplace transform has been extensively (and

intensively) employed,it is particularly useful for problem associated with

ordinary differential equations

as well as for problems involving electrostatics potential in spherical regions. the integral involving the I− function of several variables to be established as

1 α 1 0,n:(m1,n1);······ ;(mr,nr)

pi,qi;R:[pi(r),qi(r):R(r)] −1

z12λ

. dx

. zr

= 2αI0,n:(m1,n1);······ ;(mr,nr) z12λ

······ ,(1−α,λ),(1−α,λ),······ , (1.1)

pi,qi;R:[pi(r),qi(r):R(r)] . ······ ,(−α−l,λ),(1+l−α,λ);······ zr

where Re(α) > 0, λ ≤ 0, (α − l) is not a negative integer and |arg(zk)| < 1 Vkπ, ∀k ∈ [1, · · · , r] where Vk be given by

n pi nk pi(k) qi mk qi(k)

Vk = α(k)− α(k)+ γ(k)− γ(k) − β(k)− δ(k)− δ(k) > 0

j

j=1

ji j

j=n+1 j=1 j=nk+s

ji(k) ji j

j=1 j=1 j=mk+1

ji(k)

(1.2)

∀i ∈ {1, · · · · · · , R}, ∀i(k) ∈ {1, · · · · · · , Rk}, pi, qi, pi(k), qi(k), n, ni(k) and mi(k), ∀i ∈

Problem: Let us consider the Harmonic function V represented the electrostatic potential in the domain r < c such that V be a special value of F (θ) on the spherical surface r = c, where r, φ and θ be the spherical polar coordinates and V be independent or φ.

Hence V will satisfies the Laplace equation

∂2 1 ∂ (

∂V

r

(k) k1, , R , i 1, , R ,

= 0. (1.3)

Journal of Shanghai Jiaotong University ISSN: 1007-1172

Volume 17, Issue 2, February - 2021 Page No: 87

≤ ≤ ≤ ≤

−

−f (x) = (1 + x) I

2

∀ ∈ − ≤ ≤

≥ | | ∀ ∈ · · · · · ·

∞

s c

s

In the domain r < c, 0 < θ < π under the condition

lim V = F (θ), (0 < θ < π, r < c), (1.4) r→c

where V and its partial derivatives of first and second orders are considered as continuous throughout the interior of the sphere o r < c, 0 θ π

The function V represent steady temperature in a solid sphere r c, where surface temperature depends upon only θ.

In this way the surface temperature be uniform over circle

θ = θ0, r = c.

Thus V denotes electrostatic potential in the space r < c, free of charges, for V = F (θ) on the boundary r = c.

If we taking cos θ = x, then the equation (1.3) reduces to

∂2 ∂ 2 ∂V

r ∂r2

(rV ) + ∂x

(1 − x ) ∂x

= 0, r < c; −1 < x < 1. (1.5)

If we consider

F (θ) = f (cos θ) = f (x),

and then V (r, x) satisfies the equations (1.5) as

lim V (r, x) = f (x), (r < c, 1 < x < 1). (1.6) r→c

Here (1.5) may be written as

α 1 0,n:(m1,n1);······ ;(mr,nr)

pi,qi;R:[pi(r),qi(r):R(r)]

z1(1+x)λ

.

. zr

, (1.7)

where Re(α) > 0, λ 0 and arg(zk) < 1 Vkπ, k [1, , r]. where Vk be considered in (1.2) and f (x), f t(x) be considered as sectionally

continuous over the interval (−1, 1) and

lim W (r, x) = 0 (1.8) r→∞

Here W be the harmonic function in the unbounded domain r > c, in exterior to the spherical surface and rW is bounded for large value of r and x [ 1 x

1]. Now solution of the problem connecting electrostatic potential in spherical region. Case-I: Solution for V (r, x) when r < c (interior to the sphere) In this case solution of the problem is given by

V (r, x) =

R ( r \s

P (x), (r < c). (1.9)

Making an appeal of (1.6), the above relation (1.9) reduces to

∞

f (x) = RsPs(x). (1.10)

s=0

s=0 s=0



1

∞

≥

∞

1 1 r r . ···(1−α,λ),(1−α,λ)··· .

(2l + 1)2α−1I 1 1 r r . 1

(2l + 1)2α−1I 1 1 r r . 1

Now making an appeal to orthogonal property of Legendre polynomial Erdelyi[2] and integral (1.1), from (1.7)and (1.10) we obtain

ν,η:(m ,n +2);···(m ,n ) (z12λ

1 l

Here all the condition (1.1) are satisfied . Making an appeal to (1.11) from (1.9), we obtain the required relation

0,η:(m ,n +2);···(m ,n )

(z12λ 1

···(1−α,λ),(1−α,λ)···

l ( r \sP (x) (1.12)

···(−α−s,λ),(1+s−α,λ);··· cs

Here all the condition of (1.1) are satisfied. Case-II: Solution for W (r, x) when r c (exterior to the spherical surface ) We consider the following solution of the problem similar to Churchill [1](p-219)

0,η:(m ,n +2);···(m ,n )

(z12λ 1

···(1−α,λ),(1−α,λ)···

l ( r \s−1

P (x) (1.13)

pi,qi:R:[pi+2,qi+2:R];··· ;[pi(r),qi(r):Rr]

. zr s=0

s=0

. zr s=0

···(−α−l,λ),(1+l−α,λ);···

(1.11)

. zr


···(−α−l,λ),(1+l−α,λ);···

(1.11)

. zr

R = (2l+1)2α−1I l

V (r, x) = 2α−1


W (r, x) = 2α−1

···(−α−s,λ),(1+s−α,λ);··· c s

Here all the condition (1.1) are satisfied. The solutions given here are formal in nature, yet the results will prove to be quite useful for application oriented mathematicians and may generate a new awakening among them.



References

[1] Churchill, R.V., Fourier series and boundary value problems, Mc-graw

hill, New Yark.

[2] Erdelyi, et al.,Tables of integraltransforms,Vol [2],Mc-graw hill, New Yark.

[3] Mathai, A.M. and Saxena, R.K.; Generalized hyper geometric functions

with application in statistics and physical sciences; springer-varlag,

Berlin, Hei- deberg and New York 1973.

[4] Mithal, P.K. and Gupta, K.C.; An integral involving generalized function

of two variables, Proc. India Acad. Sci;75 A,PP.117-123

[5] Rainville, E.D.; Special function, macwillan, New York, 1960

[6] Ronghe, A.K.:Application of −I function in a Problem on electrostatic

potential for spherical regions, Ganita, Vol.46,No.182,1995,pp 25-30 -

[7] Singh, Kaushlendra; some problem of I − functions and its application, Ph.D. Thesis submitted to APS university Rewa (M.P.)2008.

[8] Vishwakarma, S.N.and Prasad, Y.N.; Application of the general class of polynomials and multi variables I− function in heat conduction in non-homogeneous moving bar; Visnana Prasad Anusandhan Patrika Vol.41,No.1.jan 1998,pp 47-55



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