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)