1. Introduction

The purpose of this paper is to extend the results obtained in [8] and [10].

Consider the second order linear differential equation of Hill’s type of the form

(1)

where is the second order derivative with respect to time, are independent arbitrary parameters and dots represent differentiation with respect to time. For a given the points is said to be stable if all solutions of (1) above are bounded for all and unbounded if any unbounded solution exists. Equation (1) is a linear differential equation with periodic coefficients.

Various researchers have worked on Hill’s equation with very great results. See for instance [11, 12, 13]. [8] investigated the stability of Hill’s equation with four independent parameters using Floquent theory and perturbation.

In [9] the stability of Hill’s equation with three arbitrary parameters was also investigated using Fourier analysis. The Fourier analysis method employed did not include explicit algebraic expressions for the regions of stability. [6] also investigated the stability of Hill’s equation with independent small parameters by the method of perturbations. However, the method of investigating stability and boundedness of equation (1) using a combination of numerical integration and Lyapunov functions has been rare in literature to the best of our knowledge.

Equation (1) has a lot of physical applications especially in the areas of Genetic regulatory circuit and has been widely used in Physics, Chemistry and Biology [3] and other physical phenomenon. Furthermore, the significance are found in amplitude distortion in moving coil of loud speakers, frequency modulation, dynamical systems and vibration of stretched strings, also for scattery theory and wave mechanics and for relativistic oscillators. Due to the importance of Hill’s equation in real world problems, the study of boundedness and stability of the equation has continued to attract the attention of many researchers see for instance [1, 4, 9]. Other researchers like [2], [5], [14] and [7] have investigated the stability and boundedness of linear and nonlinear differential equations.

2. Main Body

2.1. Preliminaries

Definition 2.1: Simpson’s rule is a method of numerical integration that provides an approximation of a definite integral over the interval [a, b] using parabola. Furthermore, the interval of a function over the interval [a, b] with subintervals

and subintervals length can be approximated as

(2)

as long as n is even. Let be a regular partition of [a, b] into an even number of subintervals (so n must be even) and assume that is a continuous function on [a, b], then where

(3)

Definition 2.2: Assume that otherwise the point is a singular point of

(4)

and that and are analytic at then they will have Maclaurin series expansion

(5)

with radius of convergence and respectively. That is

which converges for Then the point is called a regular singular point of (4).

Definition 2.3: The functions are analytic at if they have Taylor series expansion

(6)

with radius of convergence and respectively. That is

which converges for and which converges for .

Definition 2.4: Frobenius method of solving differential equation is a method that assumes that is a regular singular point of the differential equation.

(7)

The Frobenius series of the form can be used to solve the differential equation (7). The parameter r must be chosen so that when the series is substituted into the differential equation the coefficient of the smallest power of x is zero. This is called the indicial equation. Also, a recursive equation for the coefficient is obtained by setting the coefficient of equal to zero.

Definition 2.5: Let be the error in Simpson’s rule for a particular n i.e, . Then if on (for some constant k), then .

Definition 2.6: Let

(8)

A solution of (8) is Lyapunov stable if for each and such that if is a solution of (2.4) and then for all .

Definition 2.7: A solution of (8) is asymptotically stable if it is Lyapunov stable and if for every as .

Definition 2.8: Consider a real valued function which is continuously differentiable with V is said to be

Definition 2.9: Let the origin be an equilibrium point for . Let be a continuously differentiable function such that, and , then is stable. Moreover if then is asymptotically stable.

Corollary 2.10: Let be an equilibrium point of . Let be a positive definite function containing the origin such that in D. Let and suppose that no solution can stay identically in S, other than the trivial solution then the origin is asymptotically stable.

2.2. Results and Discussion

2.2.1. Numerical Integration Approach

We consider a differential equation of the form

(9)

where

Then equation (9) becomes

(10)

Assume that equation (6) has a solution of the form

(11)

Finding the derivative of term by term gives

(12)

(13)

Substituting for and in equation (10) we have,

(14)

When respectively we have

For a power series to vanish identically over the interval, the coefficient must be zero.

For

(15)

Hence = integer.

For we have

(16)

When , then

in (15) implies that is intermediate

For we have

(17)

(18)

For the general term , we have

(19)

which gives

(20)

From the indicial equation implies that is intermediate.

From equation (20)

Hence, one solution is

(21)

(22)

Since and are arbitrary constants, we have

(23)

Similarly when

Then

(24)

Since is an arbitrary constant, we have that

(25)

Hence, the general solution is

(26)

Since then

(27)

Let

Then,

(28)

(29)

Applying Simpson’s Integration formula on with step size

(30)

Let be the interval then,

(31)

2.2.2. Stability Analysis

Consider

(32)

Let

The first equivalent systems of (32) is

(33)

(34)

(33) and (34) can be written as

where In matrix form we have

(35)

(35) can be written as where A is the matrix.

For the characteristics polynomial

(36)

Hence the general solution is

(37)

which can be written as

(38)

Equation (38) shows that solution of Hill’s equation is periodic with respect to the independent parameters.

2.2.3. Lyapunov Direct Method

Consider the equation

(39)

At fixed point

Using we have

Hence at fixed point we have as the only equilibrium point of the system.

For the Lyapunov function, we multiply equation (39) by which gives

(40)

Integrating equation (40) we have

(41)

which gives

(42)

The energy function

But

Hence the Lyapunov function is given by

(43)

Applying definition (2.9) to equation (43) we have

Differentiating equation (43) we have

(44)

(45)

Hence the equilibrium point is unstable.

2.2.4. Numerical Solution of Hill’s Equation

Define a function that determines a vector of derivative values at any solution point (t,Y):

Define additional arguments for the ODE solver:

Solution matrix:

Table 1. Table of values for the independent variables

Figure 1. The relation between first solution function values and independent variable values

Figure 2. The relation between second solution function values and independent variable values

Figure 3. Phase portrait of Hill’s equation showing instability of the solution as a spiral source

Figure 4. The relation between the first solution function values and the independent variable values

Figure 5. The relation between the second solution function values and the independent variable values

Figure 6. The relation between first solution function values and second solution function values showing instability of Hill’s equation

Figure 7. Trajectory profile of first solution function values and independent variable values

Figure 8. Trajectory profile of second solution function values and independent variable values

Figure 9. Phase portrait of Hill’s equation of first solution function values and second solution function values showing instability around the origin

3. Conclusions

From our result, it was observed that regions of instability were basically around the equilibrium point and stable otherwise.

We also observed that this region of instability remained radially unbounded. Although, the total derivative existed, the region is unbounded, which implies that the existence of total derivative does not necessarily implied boundedness using Lyapunov methods.

ACKNOWLEDGEMENTS

The authors wishes to express their profound gratitude to the editor of American Journal of Computational and Applied Mathematics for providing a platform in which this paper was acknowledged and accepted.