ON NUMEROV METHOD FOR SOLVING FOURTH ORDER ORDINARY DIFFERENTIAL EQUATIONS

INTRODUCTION A most preferred method in industry and other applications for numerically solving ordinary differential equations (ODE) is the classical Runge-Kutta method (RK4). However, when the differential equation does not include a first order term, the Numerov method comes to mind, as it is more accurate than the RK4 by an order. More so, a great many general second order ODEs can be transformed into one without a first order term, the solution of which can be obtained via the Numerov method, which requires less computational complexity, thereby being easier to program. The Nemerov’s method, a fourth-order implicit linear multistep method (LMM), is a numerical method for solving second order ordinary differential equations wherein the first-order term is missing; that is,


INTRODUCTION
A most preferred method in industry and other applications for numerically solving ordinary differential equations (ODE) is the classical Runge-Kutta method (RK4).
However, when the differential equation does not include a first order term, the Numerov method comes to mind, as it is more accurate than the RK4 by an order. More so, a great many general second order ODEs can be transformed into one without a first order term, the solution of which can be obtained via the Numerov method, which requires less computational complexity, thereby being easier to program.
The Nemerov's method, a fourth-order implicit linear multistep method (LMM), is a numerical method for solving second order ordinary differential equations wherein the first-order term is missing; that is, The method takes the form Suppose is linear in , then by letting , the method (2) reduces to the explicit form In terms of efficiency, the Numerov method is the most preferred method when compared to the Runge-Kutta method, in that with just one evaluation of and per step a local error of is obtained, as against the Runge-Kutta method that requires six function evaluations per step to attain a local error of . More so, it is computationally easier for both This is a typical example of an equation of type (1). Another example is the equation of motion of an undamped forced harmonic oscillator, A comparison of the shooting and the matrix diagonalization forms of the finite difference method for the Schrodinger equation leads to an order doubling principle which produces an eigenvalue estimate of 8th order from the traditional Numerov method.
Several studies have made substantial contributions to the improvement or modfication of Numerov method; for example, Killingbeck and Jolicard (1999)

Definition 2
A linear multistep method is of order if and , where the term is called the error constant, is the truncation error at the point .

Definition 3
A linear multistep method is said to be consistent if it is at least first-order.

Definition 4
A linear multistep method is said to be zero-stable if as , the roots of the first characteristic polynomial satisfy , and for every the multiplicity must be simple.

Definition 5
A linear multistep method is convergent if and only if it is stable and consistent.

Derivation of the Method
Given the differential equation (1) (6) is now in form of the Numerov method (2) with the leading term of the local truncation error in the step from to expressed as Thus, the global error is of order 4.

Absolute Stability of the Numerov Method
Following Lambert (1973), the locus of the boundary of the region of absolute stability is, where and defined by (6) and (7) are explicitly expressed by and respectively. Consequently, which makes the interval of the real axis to be the boundary of the region; and the extreme values (maximum and minimum) of the function are the end points of the interval. Consequently, the interval of absolute stability is computed as .
From the foregoing sections, it is evident that the Numerov method is shown to be consistent and stable, hence its convergence.

Application of the Numerov Method to Solution of Fourth Order ODEs
Two fourth order ordinary differential equations will be considered. The exact solutions of the differential equations will be obtained analytically and the absolute value difference between the exact and approximate solutions compared.

Example 1
We consider the initial value problem: Suppose ; then ; . Then by implication, Equations (10) and (11)  From the known boundary conditions, the following values are computed.
The above are the starting values necessary to implement the first iteration step (16) and (18), where the following results are obtained and Subsequent iterations of (14) and (15) Table 1.

Example 2
We consider the boundary value problem: The exact solution of (21) is obtained analytically as where, Similar to problem 1, equations (14), (15) and (18) are employed to solve this problem with the following starting values: .

RESULTS AND DISCUSSION
In order to verify numerically whether the proposed schemes are effective, the computations of the approximate numerical solutions of the two fourth order initial and boundary value problems of ordinary differential equations presented in Examples 1 and 2 are implemented using Maple 2019 software package and the results are presented in Tables 1 and 2. In the tables, denotes the step number, is the integration points, represents the solutions of (15), stands for exact solutions of (11), is the approximate solution obtained from (14) and is the exact solution of (9) or (21), as the case may be.

CONCLUSION
The Numerov method, a two-step implicit linear multistep method for solving second order ordinary differential equations wherein the first order term is missing, has been employed to solve fourth order initial and boundary value problems in ordinary differential equations involving the second derivative. This has been achieved through a transformation of the original fourth order ordinary differential equation into a system of two coupled second order ordinary differential equations without a first derivative term, which is suitable for solution with the Numerov method. The solutions