Cmjv03i01p0079

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J. Comp. & Math. Sci. Vol.3 (1), 79-82 (2012)

Application of Orthogonal Polynomials to Non-Linear Differential Equation Involving I-Function S. S. SHRIVASTAVA and ARTI UPADHYAY Department of Mathematics, Govt. P.G. College, Shahdol, M.P. India. (Received on: 16th January, 2012) ABSTRACT In different branches of engineering and physics, many vibration problems occur, which require solving the non-linear differential equation by method of approximations. In the present paper, we present the analysis of a resistanceless circuit containing a nonlinear capacitor under the effect of external periodic force of general nature involving I-function. Keywords: Orthogonal Polynomials, non-linear, differential equation, I-function.

1. INTRODUCTION

n

The I-function introduced Saxena1, will be represented as follows:

by

[(bj, βj)1, m], [(bji, βji)m + 1, qi]

i

= (1/2πi) ∫ θ(s) xs ds

(1)

L

where i = √(– 1), m n Π Γ(bj – β js) ΠΓ(1 – aj + αjs) j=1

θ (s) =

r Σ

i=1

j=1

pi qi Π Γ(1 – bji + β jis) Π Γ(aji – αjis) j=m+1

j=1

A=

Ipm,, qn: r [x| [(aj, αj)1, n], [(aji, αji)n + 1, pi] ] i

pi

j=n+1

integral is convergent, when (B>0, A≤ 0), where

m

qi

Β = Σ αj – Σ αji + Σ β j – Σ βji , j=n+1

j=1

(2)

j=m+1

pi

qi

j=1

j=1

Σ αji – Σ βji

arg x| < ½ Bπ, ∀ i ∈ (1, 2, …, r). pi (i = 1, 2, …, r); qi(i = 1, 2, …, r); m, n are integers satisfying 0 ≤ n ≤ pi, 0 ≤ m ≤ qi, (i = 1, 2, …, r); r is finite αj, βj, αji, β ji are real and positive and aj, bj, aji, bji are complex numbers such that αj (bh + v) ≠ Bh (aj – 1 – k), for v, k = 0, 1, 2, ….. h = 1, 2, …, m; j = 1, 2, …, r; L is a contour runs from σ – i∞ to σ + i∞ (σ is real), in the complex s-plane such that the poles of

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

80

S. S. Shrivastava, et al., J. Comp. & Math. Sci. Vol.3 (1), 79-82 (2012)

s = (aj – 1 – v) | ιj

j = 1, 2, ‌, n; v = 0,1,2, ‌.

and

lie to the left hand side s = (bj + v) | β j

j = 1, 2, ‌, m; v = 0, 1, 2, ‌.

and right of L. Saxena and Kushwaha2, Khan and 3 Verma , Garde4, Shrivastava and Singh5, Singh and Shrivastava6, Nigam7, Shrivastava8 and several other authors have studied application of orthogonal polynomials to obtain the linear amplitude dependent approximate solution of the nonlinear differential equation of general type. Following Saxena and Kushwaha2, Khan and Verma3, Garde4, Shrivastava and Singh5, Singh and Shrivastava6, Nigam7, Shrivastava8 and several other authors, in this paper we will study the application of Jacobi polynomials to non-linear differential equation associated with I-function. In this chapter, we have applied the Jacobi polynomials to obtain the linear amplitude dependent approximate solution of the non-linear differential equation of general type I , , : 1 / = NF(t)

2 ,

2 ,

(5) (6)

,

where in (6), −ve sign is the velocity in the direction of x-axis. In our investigation we shall need the following results: From Erdelyi [10, p.284, (2)]:

1 x Îą 1 x P

,

x dx

, !

=

(7)

where Re Îą " 1, Re Ďƒ " 1. 2. INTEGRAL The following integral will be required in our present investigation:

,

1 x 1 x P

x I , ŕą&#x;, ŕą&#x;: Âľ 1 x/A dx

భజಉజಊ , ‌‌..,‌‌, , , , I ŕą&#x; , ŕą&#x; : Âľ2 | , , , ,‌‌...,‌‌ !

(8)

(3)

where I-function, defined in (1). The general initial conditions in which (3) can be solved may be taken as x = A(A − 1) and x = 0 at t = 0, where A(A − 1) is the amplitude of the motion. Results obtained in Pipes9 have been improved respectively by using Jacobi polynomial approximation for sinx, instead of assuming sinx equal to x, in the differential equation

(4)

provided that m is positive integer, R(1 + Îą) > 0, R(1 + β + mbj/βj) > 0 (j = 1, ‌, k), ∑ ' ∑

' ∑ ∑

( ) " 0, ∑ ∑ ' + , 0 and |arg

Âľ2−m| < (1/2)Ď€M.

3. JACOBI POLYNOMIALS AND LINEAR APPROXIMATION Jacobi polynomials can be defined as the sets of polynomials orthogonal in the

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

81

S. S. Shrivastava, et al., J. Comp. & Math. Sci. Vol.3 (1), 79-82 (2012)

interval (− A, A) with respect to the weight function (1 − x/A)Îą (1 + x/A)β. This gives , rise to the polynomials P x/A . For a function f(x), which can be expanded in terms of the Jacobi polynomials in the interval (− A, A), we obtained , , P x/A

f x ∑ a

,

where the coefficients a ,

a

,

(9)

are given by

,

, , P

, , P

a

, , x P

a

,

= a

a

,

a

,

a

,

, : Âľ

, : Âľ

,

,

(11)

!

!

,

,

!

, !

,

a

ω

, ‌‌..,‌‌‌, , I

Âľ2 | , ,‌‌...,‌‌ ŕą&#x; , ŕą&#x; :

(15) and

non-linear

a

ω

Îą ) β ) 3 Γ 2 ) Îą ) β , I ŕą&#x; , ŕą&#x; : Γ 2 ) β

‌..,‌‌‌, , , ,

¾2 | , , , ,...,‌‌

(12)

where

(16)

,

Replacing f(x) by its approximate f*(x), the equation (12) transform into x a

,

or

a

.

,

"

/ NF t (17)

x Îł x Îł Îł NF t

(18)

Îł

(19)

"

f x ωI ",# , :! ¾ 1 x/A $

(13)

by approximating f(x) in the interval (− A, A), with the help of linear Jacobi polynomials. From (10) we have

(14)

Using the result (8) and of Rainville [11, p.260] one finds that

,

x f x NF t

A

where

,

4. APPLICATION OF JACOBI POLYNOMIALS TO NON-LINEAR DIFFERENTIAL EQUATION the

A

and

where star indicates approximation.

Here we solve differential equation

, , x P

a

(10)

If the series (9) is truncated after the second term, we obtain a linear approximation f x a

f% x ωI ",# , :! ¾ 1 x/A $ %

where

, a

"

and Îł

, a "

,

,

The value of a and a given by (15) and (16) respectively.

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

are

82

S. S. Shrivastava, et al., J. Comp. & Math. Sci. Vol.3 (1), 79-82 (2012)

The approximate general solution of (18) subject to the initial conditions x = A(A − 1), â€ŤÝ”â€Źáˆś = 0 at t = 0 is given by x# A A 1

A ι β

γ 61 7 cos γt 2 ι β

Îł

;1 "

$

<

$ & = F u sin Îł t $

%

u du

(20)

which is most general approximate solution and on giving different values to F(t) in (12), one can find approximate general solution corresponding to F(t). The electric analogue of (12) can be given by a circuit consisting of a linear inductor L in series with a non-linear capacitor and a harmonic potential NF(t). If x is the charge separation of the plates of the capacitor, the differential equation of the circuit as introduced by Garde4 is given by Lx F x NF t

(21)

where f(x) is potential drop across the non-linear capacitor. Generally f(x) is in the form of a curve between voltage and charge known as saturation curve.

f(x)

N F (t)

(Resistanceless circuit containing a non-linear capacitor)

REFERENCES 1. 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). 2. Saxena, R. K. & Kushwaha, R. S. Proc. Nat. Acad. Sci. India 40(A), 65 (1970). 3. Khan, I. A. & Verma, R. C. Proc. Nat. Acad. Sci India 41(A), 191 (1971). 4. Garde, R. M. Proc. Nat. Acad. Sci India 37 (A), 109 (1967). 5. Shrivastava, B. M. and Singh, F.: Application of Jacobi Polynomials to Non-Linear Differential Equation Associated with H-function, Vijnana Parishad Anusandhan Patrika, Vol. 19, No.-1, p.25-34. January (1976). 6. Singh, M. B. & Shrivastava, B. M. L.: Proc. Nat. Acad. Sci. India 57A, IV, 7482. 7. Nigam, S. K.: A study of Special functions and polynomials with their applications, Thesis approved from A. P. S. University, Rewa. 8. Shrivastava, Sweta: Application of Jacobi Polynomials to Non-Linear Differential Equation Associated with the product of Kampe De Feriet Function and H-function of two variables, Vikram Mathematical Journal 25, 141-152 (2005). 9. Pipes, L. A.: Applied Mathematics for engineers and physicists, 2nd Ed. McGraw-Hill, New York, (1958). 10. Erdelyi, A. Tables of Integral Transform, Vol.II, McGraw-Hill, New York (1953). 11. Horn, J.: Hypergeometrische Funkionen zweir varanderliche Math. Ann. 105, p. 381-407 (1931).

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

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