STABLE LINEAR MULTISTEP METHODS WITH OFF-STEP POINTS FOR THE SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS

STABLE LINEAR MULTISTEP METHODS WITH OFF-STEP POINTS FOR THE SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS I. M. ESUABANA, S. E. EKORO, U. A. ABASIEKWERE, E. O. EKPENYONG, T. O. OGUMBE Department of Mathematics, University of Calabar, P.M.B. 1115, Calabar, Cross River State, Nigeria Department of Mathematics, University of Uyo, P.M.B. 1017, Uyo, Akwa Ibom State, Nigeria Copyright © 2022 the author(s). This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract: Of recent, stability has become an important concept and a qualitative property in any numerical integration scheme. In this work, we propose two stable linear multistep methods with off-step points for the numerical integration


INTRODUCTION
Differential equations are equations resulting from modeling physical phenomena in sciences, social sciences, management, etc. In particular, ordinary differential equation (ODE) models have been playing a prominent role in physics, engineering, econometrics, biomedical sciences among 2 ESUABANA, EKORO, ABASIEKWERE, EKPENYONG, OGUMBE other scientific fields. In fact, ODEs are the most widespread formalism to model dynamical systems in science and engineering. When the models appear in one or more derivatives, they are referred to as first or higher order differential equations, respectively. Systems of first order differential equation and can be expressed as: It is generally known that the solutions of models are not generally written in closed form. In order to understand these solutions, it is often necessary to construct an approximation through computational methods which this work targets to achieve. This research work is concerned with the development and analysis of two new methods for solving first order initial value problems in ordinary differential equations with the aim of achieving high computational accuracy and whose solution can compete favourably with the exact solution in some selected problems without incurring high computational cost in implementation.

PRELIMINARIES
Lately, there are several numerical methods that have been developed by researchers for approximate solutions to models [7], [8], [9], etc. This is ranging from the one-step method such as the Euler method, Runge-Kutta methods, etc to the multistep methods such as the Adam Bashforth method (AB), Adam-Moulton (AM), backward different formula (BDF), trapezoidal rule, General linear methods (GLM), etc. Each of these methods has its computational advantages and disadvantages based on the type of ODEs to be solved. The process of using numerical methods to provide approximate solutions to ODEs models is known as "numerical integration" [2]. Differential equation (1) can be further classified into initial value problems and boundary value problems (BVPS). The equation (1) can be called an initial value problem if it has specified values assigned to it called the initial conditions of the unknown function at a given point in the domain of the solutions. This is written as: 3 STABLE LINEAR MULTISTEP METHODS WITH OFF-STEP POINTS A solution to (2) is the function ( ) and satisfies the initial condition. The differential equation (1) is a boundary value problem, if the conditions can be specified in more than one point in the domain of the solution (Lambert, 1991). i.e. ′ = ( , ), ( 0 ) = y 0 , y( 1 ) = 1 , ∀ 0 , 1 Numerical methods for solving (2)  stability for the numerical integration of (1) is derived. Numerical methods with these properties are often used for special classes of ODEs especially for stiff differential equations [4], [5], [6].
The concept of stiffness shall be explained later in this research work. Most of the existing methods cannot approximate stiff differential equation due to small regions of absolute stability. The two methods are obtained by incorporating off-step points to the conventional second derivative linear multistep methods so as to overcome the constraints imposed by [1] on the stability of linear multistep methods. On the other hand, we shall examine their error constants, region of absolute stability and test their efficiency.

STATEMENT OF THE PROBLEM
Many methods have underperformed in some classes of problems in ordinary differential equations, especially stiff differential equations. This is due to the small region of absolute stability. The aim of this study is to develop, by means of interpolation and collocation, two high order hybrid methods for solving systems of first order stiff initial value problems in ordinary differential equations.

MAIN RESULTS
Derivation of the proposed hybrid methods: The first method considered in this work is expressed as Method 1:

Order of the method:
4 =+ pk Hybrid Predictors:  x + , (Gear, 1965). Derivation of proposed hybrid method 1: In order to obtain (4), we proceed by seeking the approximate solutions of the exact solution of (1) by assuming a continuous solution  We now obtain the method for k=1 3 With error constant 7 c = 67 1209600 − and order 6 p = .
We therefore generalized the nth step number to a matrix of system of difference equation to Collocating at point x + to obtain a system of equation for each value of k Now, let us consider for 1 k = , we have the system of equations Solving with the Mathematica 10.0 to obtain the values as With this we obtain the hybrid formula as Solving to obtain Through these parameters, we now obtain the method of order *4 p = and error constant 5       x + for stability and for each value of k. This method differs from the methods (4) since it has only one fixed hybrid predictor unlike the latter has two off-step points with variable hybrid parameters. Interpolation and collocation approach is adopted in its derivation as in methods (4) The first characteristic polynomial can be obtain by applying the shift operator to obtain Hence, the method is Zero-stable since a root lie inside the unit disc and a unit root on the disc.
Given the hybrid method 2 for 2 k = Adopting the boundary locus techniques, the stability plots of method 2 are shown below. From the plots above, it is seen that the method is A-stable for step number k=1 to 3 and ( ) A  − stable for step number 4 to 6. The method becomes unstable for step-number 7 and above. These methods are strong enough to be used for stiff differential equations.

Error constants of the proposed hybrid methods
Consider the operator, The Jacobian matrix is given as We used the explicit Trapezoidal rule to generate the starting method. Result is tabulated below;

Problem 3
Consider the stiff system:

CONCLUSION
This study is an extension and modification of linear multistep methods. The modification is done by introducing hybrid points and adding second derivative points to the conventional linear multistep methods. Two methods are derived each with hybrid collocation points. We employed the interpolation and collocation approach in the derivation using Mathematica software. The implementation is carried out in fixed and variable step-size. Mathematica and MATLAB programming software are used in the derivations and implementation of the two new classes of hybrid linear multistep methods with predictors. These methods have small error constants and are zero-stable and A-stable at higher orders which are important properties for numerical integrations.
The experimental results in table [5][6][7][8][9] show that the methods are competitive with other existing methods in literature.