Implicit One-Step Legendre Polynomial Hybrid Block Method for the Solution of First-Order Stiff Differential Equations

DOI: 10.9734/BJMCS/2015/16252 Editor(s): (1) Andrej V. Plotnikov, Department of Applied and Calculus Mathematics and CAD, Odessa State Academy of Civil Engineering and Architecture, Ukraine. (2) Zuomao Yan, Department of Mathematics, Hexi University, China. (3) Paul Bracken, Department of Mathematics, The University of Texas-Pan American Edinburg, TX 78539, USA. Reviewers: (1) Anonymous, Nigeria. (2) Anonymous, Turkey. (3) Mfon Okon Udo, Department of Mathematics/Statistics, CRUTECH, Calabar, Nigeria. Complete Peer review History: http://www.sciencedomain.org/review-history.php?iid=1035&id=6&aid=9104


Introduction
In this paper, we are concerned with the solution of stiff first-order differential equations of the form, where f is assumed to be Lipchitz continuous in y and  is a given initial value.
Different methods have been proposed for the solution of [1] ranging from predictor-corrector methods to hybrid methods. Despite the success recorded by the predictor-corrector methods, its major setback is that the predictors are in reducing order of accuracy especially when the value of the step-length is high and moreover the results are at overlapping interval [2]. Hybrid methods have advantage of incorporating function evaluation at off-step points which affords the opportunity of circumventing the 'the Dahlquist zero stability barrier' and it is actually possible to obtain convergent k-step methods with order 2 1 k  up to 7 k  , [3]. The method is also useful in reducing the step number of a method and still remains zero-stable, [1].
Stiff differential equations were first encountered in the study of the motion of springs varying stiffness, from which the problem derives its name. Stiffness occurs when some components of the solution decay much more rapidly than others. These problems are of frequent occurrence in the mathematical formulation of physical situations in control theory and mass action kinetics, where processes with widely varying time constants are usually encountered. Historically, two chemical engineers, Curtis and Hirschfelder in 1952 proposed the first set of numerical integration formulas (both implicit backward differentiation formulas) that are well suited for stiff differential equations. Stiff equations pose stability problems for most numerical integrators, [4].
In particular, Many scholars have proposed various forms of methods for the solution of (1) by adopting power series, Chebyshev and Lagrange polynomials as basis functions. In this paper, we develop an implicit hybrid block method that gives better stability condition by using Legendre polynomial as our basis function.

Formulation of the Implicit One-Step Hybrid Block Method
We consider the first six terms of the Legendre polynomial as our basis function. This is given by, Interpolating ( where the coefficients of n y and n j f  are given by, gives a discrete hybrid block method of the form, , T T m n n n n n n n n n y y y y y y y y 323  11  53  19  1440 120 1440 2880  31  1  1  1  90  15  90  360  51  90  21  3  160  40  160  320  16  2  16  7  45 15

Order of the Implicit One-step Hybrid Block Method
Let the linear operator   ( ); L y x h associated with the block (7) be defined as, Expanding (8) For implicit one-step hybrid block method,   Expanding (11) in Taylor series gives,

Convergence and Region of Absolute Stability of the Implicit One-Step Hybrid Block Method
Definition 5 [9]: Region of absolute stability is a region in the complex z plane, where z h   . It is defined as those values of z such that the numerical solutions of ' y y    satisfy 0 j y as j    for any initial condition.
We shall adopt the boundary locus method to determine the region of absolute stability of the implicit one-step hybrid block method. This gives the stability polynomial below, The stability region is shown in the figure below.

Fig. 1. Stability Region of the Implicit One-Step Hybrid Block Method
The stability region obtained in Fig. 1 above is A-Stable, since it contains the whole of the left-half complex plane, [6].

Numerical Experiments
The implicit one-step hybrid block method formulated shall be tested on some set of stiff differential equations and compare the results with solutions from some methods of similar derivation. The numerical results were obtained using MATLAB.
The following notations shall be used in the tables below; ERR-|Exact Solution-Computed Solution| EJA-Error in [10] EOK-Error in [11] Problem 1 A certain radioactive substance is known to decay at the rate proportional to the amount present. A block of this substance having a mass of g 100 originally is observed. After 40hours, its mass reduced to g 90 . Test for the consistency of the implicit one-step hybrid block method on this problem for This stiff problem is modeled by the differential equation, where N represents the mass of the substance at any time t and  is a constant which specifies the rate at which this particular substance decays. The theoretical solution to (15) is given by; Source: [12] This problem was solved using Half-step hybrid method of order six. We present the result for problem 1 using the newly derived method in Table 1 below.

Problem 2
Consider the highly stiff ODE with the exact solution, Source: [11].
This problem was earlier discussed by [9], he showed that many predictor-corrector and block methods become unstable with this problem, including the hybrid methods. However, the newly derived block method is used for the integration of this problem within the interval 0 0.1 t   . The authors in [10] solved this stiff problem by adopting a new 3-point block method with step size ratio at 1 r  and of order five. We present the result for problem 2 using the newly derived method in Table 2 below.

Conclusion
We formulated an A-stable implicit one-step hybrid block method for the solution of stiff first-order ordinary differential equations where we adopted Legendre polynomial as our basis function. The method developed was also found to be zero-stable, consistent and convergent. The hybrid block method was also found to perform better than some existing methods in view of the numerical results obtained.