In this article, we propose a new computational method for second order initial value problems in ordinary differential equations. The algorithm developed is based on a local representation of theoretical solution of the second order initial value problem by a non-linear interpolating function. Numerical examples are solved to ensure the computational performance of the algorithm for both linear and non-linear initial value problems. From the results we obtained, the algorithm can be said computationally efficient and effective.
Many phenomena that occur in chemical, biological, engineering, physical and social sciences can be modelled mathematically in the form of either ordinary or partial differential equations. However it is difficult to obtain exact solution for these differential equations especially if it is nonlinear, by analytical means.So we consider an approximate solution to these problems.There are numerous ways by which an approximate solution can be constructed.In numerical analysis a concept of approximation play very important role. Thus solving these practical problems which modelled as differential equation approximately, is one of the main preoccupations in numerical analysis.
Consider second order initial value problems in ordinary differential equations of the form
subject to initial conditions
In the literature, problems of the form(1.1) are conventionally solved by reducing the differential system to first order equations. Some eminent authors have contributed in this specific area of research [1,2,3,4,11]. Another approach to investigate the solution of such problems were and referred to as shooting method either simple or multiple .In recent years researchers[5,6] applied a nonstandard method and obtained competitive results to those obtained with other method. So, much research have reported on the numerical integration of initial value problems in literature, many of them are excellent work.But a concept to develop a new algorithm to solve equation (1.1) can not be over emphasized.
In this article, we develop a new single step algorithm capable of solving equations of the form (1.1).The similar algorithm was first reported  in study of first order initial value problems.Having seen the performance of the algorithm for solution of first order initial value problems, we are motivated and challenged to investigate what will happen if a similar idea is used to derive an algorithm for solution of second order initial value problems.
The existence and uniqueness of the solution to initial value problem(1.1) is assumed.Further we assume that problems (1.1) is well posed with continuous derivatives and that the solution depends differentially on the initial conditions.The specific assumption on f(x, y, y’) to ensure existence and uniqueness will not be considered[8,9,10].
This paper is divided into five sections.Section 2 deals with the derivation and development of the algorithm while truncation error and convergence of the algorithm are developed in Section 3.The stability of the algorithm is discussed in section 4 while numerical experiments on four model problems are presented in section 5.
2 Development of Algorithm
We define N, the finite number of the nodal points of the interval [a, b], in which the solution of the problem (1.1) is desired as
where the terms in right side of expression (2.2) are defined as, the step length
Suppose we have to determine a number yj, which is numerical approximation to the value of the theoretical solution y(x) of problem(1.1) at the nodal point xj, j = 1,2…, N and other similarly notations like fj defined as f(xj,yj,
where a0,a1,a2,and a3 are undetermined coefficients.
To determine these undetermined coefficients, let impose these following conditions.
The interpolating function and its first derivative w.r.t. x must coincide with y(x) and y′(x) the theoretical solution and derivative of solution w.r.t. xof the problem (1.1) at x = xj and x = xj+1 i.e.
The second and third derivatives w.r.t x, of the interpolating function respectively coincide with f(x,y,y′) and derivative of f(x,y,y′) w.r.t. x at x = xj i.e.
Solving the system of equation (2.6) for a0,a1,…., we will obtain
From equation (1.4) we have
So we will obtain our single step implicit algorithm.
Thus we have developed single step implicit algorithm of the form
where ϕ and φ are increment functions.These increment functions depend on h, fj and
3 The Local truncation error and Convergence
Substituting the value of yn+1 from (2.9) in (3.11), and expanding y(xn+h) in Taylor series about point xn, so we have
where b = max (xn) in [a,b]. Thus local truncation error Tn+1 is bounded.We know x0 and y(x0) exactly then using algorithm (2.9), we can compute yn+1, n = 0,1,2,3,…..N with maximum error
4 Stability property
subject to initial conditions y(x0) = y0, y′(x0) = λ y0. Apply the method (2.9) to this test equation, we obtained a finite difference equation, assuming the negligible contribution of the terms with O(h2) and O(
where the stability function E(hλ) is an approximation to ehλ. For the alogorithm (2.9) to be converge
5 Numerical experiment
In this section, four numerical examples linear and nonlinear were considered, to illustrate our algorithm (2.9) and to demonstrate computationally its efficiency and accuracy.In tables, we have shown maximum absolute error computed on the nodal points in the interval of integration for these examples in their solution and derivative of solution. Let yi and
All computations in the examples consider were performed in the GNU FORTRAN environment version -99 compiler(2.95 of gcc) running on a MS Window 2000 professional operating system.
Consider the initial value problem,
Maximum absolute error in y(x) =
Consider nonlinear initial value problem
The exact solution in [0,1] is y(x) = (1 + x)–2. The maximum absolute error in y(x) and y′(x) are given Table 2.
Maximum absolute error in y(x) = (1 + x)–2 and y′(x) for Example 5.2.
Consider nonlinear initial value problem
Maximum absolute error in y(x) = (1 + x)–1 and y′(x) for Example 5.3.
Consider nonlinear initial value problem
The exact solution in [0,1] is y(x) = sin2 (
Maximum absolute error in y(x) = sin2(
In this paper, we have described a new method that is efficient, stable and convergent for solving second order initial value problems in ordinary differential equations.The implementation of the method is simple.The results we obtained for examples shows that method is computationally efficient and accurate. Our future works will deal with extension of the present method to solve higher order boundary value problems and improving its order of accuracy.
Communicated by Juan L.G. Guirao
Lambert J. D., Numerical Methods for Ordinary Differential systems, John Wiley, England (1991).
Henrici P., Discrete Variables Methods in Ordinary Differential Equation, John Wiley and Sons New York (1962).
Collatz L., Numerical Treatment of Differential Equations (3/e), Springer Verlag, Berlin (1966).
Mickens R.E., Nonstandard Finite Difference Models of Diffferential Equations, World Scientific, Singapore (1994).
Sunday J. and Odekunle M.R., A New Numerical Integrator for the Solution of Initial Value Problems in Ordinary Differential Equations, The Pacfic Journal of Science and Technology, Vol.-13, no. 1, pp. 221-227 (2012).
Fatunla S.O., A New Algorithm for Numerical Solution of Ordinary Differential Equations, Computer and Mathematics with Applications, no. 2, 247-253 (1973).
Keller H. B., Numerical Methods for Two Point Boundary Value Problems, Blaisdell Waltham Mass. (1968).
Stoer J. and Bulirsch R., Introduction to Numerical Analysis (2/e), Springer-Verlag, Berlin Heidelberg (1991).
Baxley J.V., Nonlinear Two Point Boundary Value Problems in Ordinary and Partial Differential Equations (Everrit, W.N. and Sleeman, B.D. Eds.), 46-54, Springer-Verlag, New York (1981).
Gear C.W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall (1971).
Jain M. K., Iyenger S. R. K. and Jain R.K., Numerical Methods for Scientific and Engineering Computations (2/e), Wiley Eastern Ltd. New Delhi (1987).