Cmjv05i01p0085

J. Comp. & Math. Sci. Vol.5 (1), 85-88 (2014)

Expansion Formula Involving I-Function Sandeep Khare1 and B. M. L. Srivastava2 1

Department of Mathematics, Govt. P.G. College, Shahdol, M.P., INDIA. 2 Retired Principal, Govt. Model Science College, Rewa, M.P., INDIA. (Received on: February 14, 2014) ABSTRACT In this paper, we present and solve a two dimensional Exponential Bessel differential equation, and obtain a particular solution of it involving I-function.

Keywords: Bessel differential equation, Particular solution, I- function.

1. INTRODUCTION The object of this paper is to formulate a two dimensional ExponentialBessel partial differential equation and obtain its double series solution. We further present a particular solution of our Exponential-Bessel equation involving

I-function. It is interesting to note that particular solution also yields a new two dimensional series expansion for I-function involving exponential functions and Bessel functions.

The I-function of one variable is defined by Saxena6 and we will represent here in the following manner:

m, n

I p , q : r [x| [(aj, j)1, n], [(aji, ji)n + 1, pi] ] = (1/2) (s) xs ds i i [(bj, j)1, m], [(bji, ji)m + 1, qi]     L where  = (– 1), m  (bj – js) j=1

(s) = r  i=1

n  (1 – aj + js) j=1

pi qi (1 – b +  s) ji ji  (aji – jis)  j=n+1

,

j=m+1

integral is convergent, when (B>0, A 0), where Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)

(1.1)

86

Sandeep Khare, et al., J. Comp. & Math. Sci. Vol.5 (1), 85-88 (2014) n



pi

m

qi

j – ji + j – ji ,

A=

j=1

j=n+1

pi

qi

j=1

j=m+1

ji – ji ,

j=1

j=1

|arg x| < ½ Bi (1, 2, …, r).

The following formulae are required in the proof:     m, n cos 2ux (sin x/2)– 2w I pi, qi:[zr (sin x/2)2h | 0

[(aj, j)1, n], [(aji, ji)n + 1, pi] ] dx [(bj, j)1, m], [(bji, ji)m + 1, qi]

  m + 1,   n+1 (1 – w – 2u, h), [(aj, j)1, n], [(aji, ji)n + 1, pi], (1 – w + 2u, h) I pi + 2, qi + 2: [z | ] r

(1.2)

(1/2 – w, h), [(bj, j)1, m], [(bji, ji)m + 1, qi], (1 – w, h)

where h > 0, |arg z| < ½ B    m, n 2k yw´  1 siny J(y) I pi, qi: [z r y | 0

  m + 1, n +1 2w´  1Ipi + 4, qi + 1: r[22kz|

[(aj, j)1, n], [(aji, ji)n + 1, pi] ] dy [(bj, j)1, m], [(bji, ji)m + 1, qi]



((1– w´)/2, k), …., (1 + (– w´)/2, k), (1 ( + w´)/2, k), ((1– w´ +)/2, k)

(1/2  w´, 2k), ……….

] (1.3)

where k > 0, |arg z| < ½ B The proof of above integrals (1.2) and (1.3) are very simple, which can be easily done with the help of4, and no need to given here.

The orthogonal property of the Bessel functions6:    x  1 Ja + 2n + 1(x) J a + 2m + 1(x) dx 0

0, if m n; =

(1.4)

(4n + 2a + 2) – 1, if m = n, Re (a + m + n) >  1.

The following orthogonal property:      0, m n e 2 i m x cos 2nx dx = /2, m = n 0 0 , m = n = 0.

(1.5)

2. TWO DIMENSIONAL EXPONENTIAL – BESSEL PARTIAL DIFFERENTIAL EQUATION Let us consider Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)

Sandeep Khare, et al., J. Comp. & Math. Sci. Vol.5 (1), 85-88 (2014)

u/t = c u/t2 + y2 u/y2 + y u/y + y2u,

87 (2.1)

In (2.5), putting t = 0, we get where u u(x, y, t) and u(x, y, 0) = f(x, y).



f(x, y) = Ar, se2 r i x J + 2s + 1(y). r = – s = 0

To solve (2.1), we assume that (2.1) has a solution of the form: u(x, y, t) = e 4cr2t + ( + 2s + 1)2t X(ix) Y(y).

(2.2) The substitution of (2.2) into (2.1) yields: c [X´´ + 4r2X] Y + X [y2Y´´ + yY´ + {y2 – ( + 2s + 1)2}Y] = 0

(2.3)

(2.6)

Multiplying both sides of (2.6) by y – 1 cos2ux J + 2w + 1(y), integrating with respect to y from 0 to and with respect to x from 0 to , then using (1.4) and (1.5), the Fourier Exponential – Bessel coefficients are given by 

Ar, s = (4/) ( + 2s + 1)  0

We see that X´´ + 4r2X = 0 has a solution X = e2 r i x and y2Y´´ + yY´ + {y2 – ( + 2s + 1)2}Y = 0 is Bessel equation1, with solution Y = J + 2s + 1(y). Therefore the solution of (2.1) is of the form: 2 2 u(x, y, t) = e 4cr t + ( + 2s + 1) t e2 r i x J + 2s + 1(y).

(2.4)

In view of the principal of superposition, the general solution of (2.1) is given by 

2 2 u(x, y, t) = Ar, se 4cr t + ( + 2s + 1)

r = –  s = 0

t + 2 r i x J + 2s + 1(y).

(2.5)

f(x, y) y

–1

0

cos2rx J + 2w + 1(y) dy dx

(2.7)

In the view of the theory of double and multiple Fourier series given by Carslaw and Jaeger2, and many other references, such as Erdelyi3, etc., the double series (2.6) is convergent, provided the function f(x, y) is defined in the region 0 < x < p, 0 < y < . In brief, the double series (2.6) converges, if the double integral on the right hand side of (2.7) exists.

In the subsequent section, we take f(x, y) as I-function and present another method to obtain Fourier exponential – Bessel coefficients Ar, s.

3. PARTICULAR SOLUTION INVOLVING I-FUNCTION The particular solution to be obtained is 



u(x, y, t) = 2w´ + 1e

4cr2t + ( + 2s + 1)2t + 2 r i x( + 2s + 1) J + 2s + 1(y).

r = – s = 0

 m  + 2, n + 2  I p + 6, q + 3: r[ 22kz| i

i

(1 w – 2r, h), ((w´ +  + 2s)/2, k), ….. , (1 – w + 2r, h), (1 + ( + 2s + 1 – w´)/2, k), (1 + (1 + w´ +  + 2s)/2, k), (1 + ( + 2s – w´)/2, k) ]

(1/2  w, h), (1/2  w´, k), ….., (1 – w, h)

Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)

(3.1)

88

Sandeep Khare, et al., J. Comp. & Math. Sci. Vol.5 (1), 85-88 (2014)

valid under the conditions of (1.2), (1.3) and (1.4).

Proof: Let     f(x, y) = (sin x/2)– 2w yw´ siny I pm,, qn:[zr (sin x/2)2h y2k| i

i

[(aj, j)1, n], [(aji, ji)n + 1, pi]

]

dx

[(bj, j)1, m], [(bji, ji)m + 1, qi]



= Ar, se2 r i x J + 2s + 1(y).

(3.2)

r = – s = 0

Equation (3.2) is valid, since f(x, y) is defined in the region 0 < x < p, 0 < y < .

Multiplying both sides of (3.2) by y – 1 J + 2w + 1(y) and integrating with respect to y from 0 to , then using (1.3) and (1.4).

Now multiplying both sides of the resulting expression by cos 2ux and integrating with respect to x from 0 to , then using (1.2) and (1.5), we obtain the value of Ar, s.

Substituting the value of Ar, s in (2.5), the expansion (3.1) is obtained.

Note 1: The value of A0, value of Ar, s.

s

is one-half the

Note 2: If we put t = 0 in (3.1), it reduces to a new two dimensional series expansion for I-function involving exponential functions and Bessel functions.

Since on specializing the parameters I-function yields almost all special functions appearing in applied mathematics and physical sciences. Therefore, the result (3.1) presented in this paper is of a general

character and hence may encompass several cases of interest.

REFERENCES 1. Andrews, L. C.: Special Functions for Engineers and applied mathematics, Macmillan Publishing Co., New York (1985).

2. Carslaw, H. S. and Jaeger, J. C.: Conduction of Heat in Solids (2nd Edt.), Clarendon Press, Oxford, 1986.

3. Erdelyi, A.: Higher Transcendental Functions, Vol. 2, Mc Graw-Hill, New York (1953).

4. Erdelyi, A.: Table of Integral Transform, Vol. 1 & 2, Mc Graw-Hill, New York.

5. Luke, Y. L.: Integrals of Bessel Functions, Mc Graw-Hill, New York (1962).

6. Saxena, V. P.: Formal Solution of Certain New Pair of Dual Integral Equations Involving H-function, Proc.

Nat. Acad. Sci. India, 52(A), III (1982).

Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)