Derivation and Application of Multistep Methods to a Class of First-order Ordinary Differential Equations

Of concern in this work is the derivation and implementation of the multistep methods through Taylor’s expansion and numerical integration. For the Taylor’s expansion me thod, the series is truncated after some terms to give the needed approximations which allows for the necessary substitutions for the derivatives to be evaluated on the differential equations . For the numerical integration technique, an interpolating polyn omial that is determined by some data points replaces the differential equation function and it is integrated over a specified interval. The methods show that they are only convergent if and only if they are consistent and stable. In our numerical examples, the methods are applied on non-stiff initial value problems of first-order ordinary differential equations, where it is established that the multistep methods show superiority over the single-step methods in terms of robustness, efficiency, stability and accuracy, the only setback being that the multistep methods require more computational effort than the single-step methods. Keywords— linear multi-step method; numerical solution; ordinary differential equation; initial value problem; stability; convergence.


INTRODUCTION
Linear multistep methods (LMMs) are very popular for solving initial value problems (IVPs) of ordinary differential equations (ODEs). They are also applied to solve higher order ODEs. LMMs are not self-starting hence, need starting values from single-step methods like Euler's method and Runge-Kutta family of methods.
The general -step LMM is as given by Lambert [1] (2) where and jj  are expressed as continuous functions of and are at least differentiable once [2].
According to [3], the existing methods of deriving the LMMs in discrete form include the interpolation approach, numerical integration, Taylor series expansion and through the determination of the order of LMM. Continuous collocation and interpolation technique are also used for the derivation of LMMs, block methods and hybrid methods.
In this study, we present the general multistep method, some of its different types and examine their characteristics. In lig ht of this, we investigate the stability and convergence of these methods, compare the multistep methods with the single -step methods in operational time, accuracy and user-friendliness via some numerical examples.
In practice, only a few of the initial value differential equations that originate from the study of physical phenomena have exact solutions. The introduction however, of the multistep methods as numerical techniques is used in finding solutions to problems that have known exact solutions and in extension handle those problems whose exact solutions are not known. We shall limit this study to only non-stiff initial value problems of first-order ordinary differential equations.

The Linear Multistep Methods
The general linear multistep method is given by implicit [4].

The Adams methods
These are the most important linear multistep methods for non -stiff initial value problems. It is the class of multistep methods (3) with then we have the Adams-Bashforth methods. And, if it is given by k n k n k 1 j n j j0 then we have the Adams -Moulton methods [5].

Predictor-Corrector (P-C) method
The multistep methods are often implemented in a 'predictor-corrector' form. In this way, a preliminary calculation is done using the explicit form of the multistep method then corrected using the implicit form of the multistep method. This is done by two calculations of the function  at each step of this computation.

Order of linear multistep methods
We can associate the linear multistep method (3) where the q C constants [1].

Definition 1
The difference operator (6) and the linear multistep method (3) associated with it are of order  if, in (7),    . The following formulae for the constants q C in terms of the coefficients j  and j  are given as: for q 2,3,

II. CHARACTERISTICS OF THE METHODS
With the number of approximations involved during computations using the multistep methods, the problem of consistency, stability and convergence call for discussion. The approximation in a one -step method depends directly on previous approximations alone, while the multistep method uses at least two of the previous approximations.

Consistency
The linear multistep method (3) is consistent if it has order 1   [4]. From (8), (9) and (10), it follows that the method is consistent if and only if the following two conditions hold. Let n x tend to x( r ) in the limit, that is Hence, we have or replacing the first term on the right hand side of the equation by the term on the right hand side of (3) we have In the limit, both terms on the right hand side vanish. Therefore, the left hand side becomes zero.  converges to the solution of the initial value problem (4) then the conditions (11) and (12) must hold [1].
It follows from conditions (11) and (12)    [1]. The stability of the multistep technique with respect to round -off error is clearly dictated by the magnitude of the zeros of the first polynomial above. However, the methods we have discussed in this work are zero -stable by virtue of their characteristics. The following are motivated by the types of zeros of the characteristic polynomial. are simple roots, then the difference method satisfies the root condition [7].

Theorem 3.2 i)
The methods that satisfy the root condition with 1   as the only root of the characteristic equation with magnitude equal to 1 are said to be strongly stable. That is, the roots lie on the unit disc. ii) If a method satisfies the root condition and has more than one distinct root with magnitude equal to 1, it is said to be weakly stable. iii) If a method does not satisfy the root condition, it is said to be unstable. A multistep method is said to be stable if and only if it satisfies the root condition [7].

Convergence
One basic property that is demanded of an acceptable linear multistep technique is the convergence of the solution   n x that is generated by the method, in some sense, to the theoretical solution   xr as the step-size h goes to zero. A linear multistep method is convergent if and only if it is consistent and stable, otherwise it is not convergent [8]. If a method is consisten t but not stable, then it is not convergent. Also, if a method is stable but n ot consistent then it is not convergent.

Obtaining Starting Values
A multistep method is not self-starting, that is, a k-step multistep scheme requires some k previous values These k values that are needed to start the application of the multistep method are gotten by a single step method such as Taylor series method, Euler method or Runge-Kutta method. The starting method should be of the same or even lower order than the order of the multistep method itself.

Taylor series method
Let us consider the initial value problem Let us consider a numerical solution to (13) above using a k-step multistep method of order  . We require that the starting values i x , i 1,2, ,k 1  should be calculated to an accuracy that is at least as high as the accuracy of the multistep method itself. That is, we require that If enough partial derivatives of ( r,x )  with respect to r and x exist, then we will use a truncated Taylor series to estimate i x to any required degree of accuracy [9]. Thus, we have This approach is theoretically flawless. Nevertheless, the evaluations of the total derivatives can be excessively tedious an d may not be adopted for an efficient computation.

Euler method
This is another method that can be employed to generate all the needed starting values for a linear multistep method. Consider the equation Let us suppose that the solution to the initial value problem (15) above has two continuous derivatives on the interval   a,b , so that for each i 1,2, ,N 1  , we have and, since x(r) satisfies our differential equation, we have By deleting the remainder term, the Euler method becomes  . The Euler method is gotten when the Taylor series method above is of order 1   . The simplicity of this method may be used to illustrate the techniques we intend to adopt in starting the multistep methods.

Runge-Kutta method
This method can also be applied to generate starting values for any multistep method. We consider the equation We note that the Runge-Kutta methods are not unique due to the manner in which they are derived. However, any Runge -Kutta methods of the same order are equivalent.

Runge-Kutta method of order two
This method uses two evaluations and it is given by

Runge-Kutta method of order four
This method uses four evaluations. It is given by We note however, that this is not unique. The Runge-Kutta method of order four s hall be used in obtaining the starting values for the implementation of the multistep methods adopted in this work.

Derivations
Any specific linear multistep may be derived in a number of different ways. We shall consider a selection of different approaches which cast some light on the nature of the approximation involved.

Derivation through Taylor expansions Euler method
Let us consider the Taylor series expansion for If we truncate this expansion after two terms and substitute for x ( r )  from the differential equation (4) x  , we get which is an explicit linear one-step method [11]. This shall be used in solving the numerical examples in this work.

Mid-point rule
and choose 0  , 0  and 1  so as to make the approximation as accurate as possible.
The following expansions are used: Substituting these two equations into (20) and collecting the terms on the left -hand side gives

Derivation through numerical integration
This technique can be used to derive only a subclass of linear multistep methods consisting of those methods for which Then, the truncation error becomes This is the Simpson's rule and it is the most accurate implicit linear two -step method [1].

Adams-Bashforth methods
Though any form of the interpolating polynomials could be used for the derivations, the Newton backward -difference formula will be used for the purpose of convenience. For us to derive an explicit k-step Adams-Bashforth method, we now form the backward-difference polynomial The coefficient s( s 1)( s 2 ) ( s k 1) does not change sign on the interval   0,1 [12].  [7].
Adams-Bashforth two-step explicit method: The local truncation error is The local truncation error is  [7].

Numerical Examples
We will now solve the following problems using some of the methods discussed in this work and the results displayed in tables along with the results of the corresponding exact solutions.