A Note on Differential Equation with a Large Parameter

We present here asymptotic solutions of equations of the type ( ) x f t x + = , 0 λ , where λ is a large parameter. The Bessel differential equation             x t x  − + − − = 2 2 2 1 0 4 λ λ is considered as a typical example of the above and the solutions are provided as →∞ λ . Furthermore, the behaviour of the solutions as well as the stability of the Bessel ode is investigated numerically as the parameter grows indefinitely.


Introduction
The theory of ordinary homogeneous linear differential equations of the second order, containing a large parameter, is well established [1]- [4].The aim of this paper is to investigate detailed analytical solutions of equations of the form; ( ) where 2 : f →   is 0 C and λ is a real parameter.We shall investigate the behaviour of solutions of this differential equation, and the stability of the origin as * λ λ → .Without loss of generality, we take * .λ = ∞ First, we make the following remarks: a) Any second order linear ODE of the form; ( ) ( ) 0 x a t x t x β + + =   can be reduced to ( ) 0 x q t x + =  by a suitable transformation.b) Furthermore, any equation of the form ( ) 0 x p x + =  is conservative.We shall demonstrate this shortly.This will help us in our asymptotic stability analysis.c) In Equation ( ( ) ( ) If we suppress the variable t for the moment, it then follows that; , and after transforming the interval a x b ≤ ≤ into α µ β ≤ ≤ , with further algebraic manipulations, the ode (1.2) becomes; ( ) where ( ) ( )


In the case of a finite interval ( ) is also an asymptotic expansion of ( ) , µ θ λ as λ → ∞ .Unfortunately, the ( ) , n µ θ λ is very difficult to compute.Other approximations for large λ may be obtained from formal solutions, and these are usually di- vergent.

Formal Solutions
Let us now consider the general ode; ( ) If ( ) , f t λ is a formal power series in 1 λ − with coefficients which depend on x, then two linearly indepen- dent solutions of (2.1) may also be represented by a formal power series in 1 λ − .However if the formal expan- sion of ( ) , f t λ in powers of λ contains positive powers of λ , then the formal expansion of x will be a Laurent series.We shall discover that in the case that ( ) , f t λ , as a function of λ , has a pole at λ = ∞ , we can still construct formal solutions.
In (2.1), we shall assume that ( ) where the ( ) n f t are independent of λ , and k + ∈  Furthermore, we assume that ( ) 0 f t does not vanish in the interval over which t varies.We shall adopt a first formal solution of the form; 3) into (2.1), with the convention that ( ) ( ) All summations may then be assumed over all the integers, and we obtain ( ) Picking out the coefficients of 2k n λ − we obtain; This first condition arises when ( ) Consequent upon these relations, we may restrict our summation to m k ≥ in the first sum in Equation (2.4).Now for n k = in (2.4) we get; and when we replace n by k n + in (2.4) we obtain 3) satisfies (2.1), provided that n q and v β satisfy (2.5) to (2.8).In these equations, empty sums (i.e.those with upper limit<lower limit) are interpreted as zero.Since 0 0 f ≠ we may choose a branch of ( ) q up to a constant factor, and (2.8) determines 1 2 , , q q  recurrently, up to an additive constant multiple of 0 q in each.Corresponding to the two branches of ( ) , we obtain two formal solutions of the form (2.3).

Another Formal Solution
A second type of formal solution is given by ( ) ( ) Substituting (2.9) into (2.1)we get; ( ) Equating coefficients of 2k n λ − , ( ) There are two linearly independent formal solutions of this type.The obvious connection between these two types of formal solutions can be seen from the fact that equations (2.10) and (2.11) are identical with (2.5) and (2.6), and

Remark
In the foregoing, we have assumed that ( ) , f t λ as a function of λ , has a pole of even order at λ = ∞ .If the pole is of odd order, then no solution of the form (2.3) or (2.9) exists, and instead of powers of λ , we must ex- pand in powers of 1 2 λ .

Asymptotic Solutions
We shall now demonstrate that under certain assumptions, the differential Equation (2.1) possesses a fundamental system of solutions which are represented asymptotically by the formal solutions obtained in preceding section.It actually does not matter whether we compare solution of (2.1) with ( ) ( ) ( ) where the n q and n β satisfy (2.5) to (2.8), or with ( ) ( ) where the n β satisfy (2.10) to (2.12), for the q's and β 's can be so chosen that the ratio of these two expres- sions is ( ) We now fix a positive integer N, and set; ( ) ( ) We have the following theorem.

Theorem
Let S and I be as defined above then for each fixed ( ) Uniformly in t and arg λ , as λ → ∞ in S, where the ( ) n f t are sufficiently often differentiable in I, and possesses a fundamental system of solutions, ( ) x t and ( ) x t , such that ( ) ( ) ( ) ( )

Proof
Top establish the existence and asymptotic property of ( ) x t , we substitute ( ) ( ) ( ) where ( ) ( ) uniformly in t and arg , λ λ → ∞ in S, by (3.2) and (2.10) to (2.12).Equation (3.7) may be written as By two successive integrations, and a suitable choice of the constants of integration, we obtain; where ( ) ( ) ( ) The existence of ( ) v t follows immediately from the theory of Volterra integral equations, or can be estab- lished by successive approximations.From (3.8) and (3.9), we have ( ) ( ) , uniformly in and arg , ( ) This proves (3.5) for 1 j = .The proof for 2 j = is very much similar, except that b rather than a, must be chosen as fixed limit of integration in the integral equation.

Application
The methods of the last two sections can be applied to prove the asymptotic formulae for the Bessel functions [1], viz; sech 2π tanh exp tan , as .
We observe that the functions; ( ) ( ) This equation is of the form (3.4) with ( ) ( ) ( ) ∞ are singular points of (3.10) and 1 t = is a so called transition point at which the condition (3.3) is violated for any value λ .

Stability Analysis
In Section 1.0, we claimed that any equation of the form ( ) The Bessel differential equation Clearly the origin (0, 0) is the only critical point and the corresponding Hamiltonian is; We use the above Hamiltonian to construct a Lyapunov function given by; and ( ) 0 0 p = .We note that ( ) 0, 0 0 V = , furthermore; ( ) , it follows that 0 V ≤  and hence the origin is asymptotically stable for all λ and 0 t > .

Numerical Investigation of Asymptotic Solutions
In what follows, we employ the Runge-Kutta algorithm provided by MathCAD [5] software to obtain a numeri-

Observations
For 550 λ ≥ solutions no longer exist as they become unbounded.From the graphs shown, it is clear that the given Bessel differential equation is very sensitive to the parameter λ , and as λ → ∞ the effect is to increase the oscillations until the solutions become unstable and die out.Furthermore, the phase portrait depicted shows that the Bessel differential equation represents a conservative system.This is clearly evident from the closed curves.However for 500 λ = , the phase portrait no longer appears like a closed curve but more like an explosion from the centre.

Conclusion
In this work, we have studied asymptotic solutions of equations of the type ( ) , where λ is a large parameter.We have shown that equations of this form represent a conservative system, meaning that they possess a conserved quantity, namely the Hamiltonian which is computed.As a special example, we consider the Bessel differential equation for which the behaviour of the solutions as well as the stability of the origin is investigated numerically as the parameter grows indefinitely.
the above and the solutions are provided as → ∞

Figure 1 .
Figure 1.Section of solution matrix S.