A STUDY ON APPROXIMATE SOLUTIONS AND ACCURACY OF PARALLEL ALGORITHMS FOR SOLVING SYSTEM OF ODES

. The purpose of this paper is, to study the numerical computation of system of ordinary differential equations of first order with initial value problems and all steps of parallel algorithms of R.K. methods are tested in MATLAB (2009a) [2]. In this part, we use four classical RK2 order methods to introduce the basic ideas associated with initial value problems for solving system of ODEs of two or more equations. At the end we make a comparison between these numerical methods for approximate solutions and numerical errors. Numerical examples are given to illustrate the computational accuracy and robustness of these parallel algorithms


INTRODUCTION
Numerical Analysis is the area of mathematics and computer science that creates algorithm and implements numerical methods for solving the problems of continuous mathematics.In the field of Engineering and Science, we come across physical and natural phenomena which, when represented by mathematical models happen to be differential PARALLEL ALGORITHMS FOR SOLVING SYSTEM OF ODES equations.Solving system of linear simultaneous differential equations is one of the most important and challenging problems in science and engineering applications.It arises in a wide variety of practical applications in Physics, Chemistry, Biosciences, Engineering, etc. System of linear differential equations arises in various theoretical research fields as well as applications in science and engineering.After the availability of computers, we go to numerical methods which are suited for computer operations.For example, system of Lorentz equations and system of coupled equations, deflection of a beam, etc. are represented by differential equations.Hence solution of differential equation is a necessity in such studied.There are number of differential equations which we studied in calculus to get closed form solutions.But all differential equations do not possess closed form solutions or finite form solutions.Even they possess closed form solutions; we do not know the method of getting it.In such situations depending upon the need of the hour, we go in for numerical solutions of differential equations.In researches, especially after the advent of computer, the numerical solutions of the differential equations have becomes easy for manipulation.The dynamic behavior of systems is an important subject.A mechanical system involves displacements, velocities, and accelerations.An electric or electronic system involves voltages, currents, and time derivatives of these quantities.An equation that involves one or more derivatives of the unknown function is called an ordinary differential equation, abbreviated as Ode.The problems of solving an ode are classified into initial-value problems and boundary value problems, depending on how the conditions at the endpoints of the domain are specified.All the conditions of an initial-value problem are specified at the initial point.On the other hand, the problem becomes a boundary-value problem if the conditions are needed for both initial and final points.The ode in the time domain are initial-value problems, so all the conditions are specified at the initial time, such as x = 0.For notations, we use x as an independent variable.It is important to note that our focus here is on the practical use of numerical methods in order to solve some typical problems, not to present any consistent theoretical background.Today there are numerous methods that produce numerical approximations to the solution of differential equations.There are many excellent and exhaustive texts on these subjects that may be consulted.[3][4][5][6][7].Our purpose is here, to compare the accuracy of various algorithms and how we can solve the systems of ODEs numerically and also checks the stability analysis using Matlab [2].Methods to solve systems of first-order differential equations are generalizations of the methods for a single first-order equation presented.[3]

MATERIALS AND METHODS
where k1 and k2 are mention above.Thus we have four parameters c2, a21, w1 and w2 are chosen to make y j+1 closer to y (x j+1 ).and to be determined.The values of c2, a21, w1 and w2 are evaluated by setting the second order equation to Taylor series expansion to the second order term.Now Taylor series expansion about xj gives We also have Comparing the coefficients of h and h 2 in (3) and ( 4) we obtain Thus we get three equations of four unknowns and their solution is Where c2 is not equal to zero.It is not possible the compare of h 3 as there are five terms in (3) and only three terms in (4).Therefore Runge-Kutta method using two evaluations of f is A. HASAN The free parameter c2 is usually taken between 0 and 1.Sometimes c2 is chosen such that one of the wi's in the formula (2) is zero.For example, the choice c2 =1/2 makes w1=0 Case (i) If c2=1/2, we get which is called Mid-point method.It reduces to the mid-point quadrature rule when f(x,y) is independent of y.
Case (iii).If c2=2/3 we get k1 = h f (xj , yj) k2 = h f (xj+2/3 h , yj+2/3 k1) Which is called nearly Optimal method.It may be noted that the explicit Runge Kutta methods using two evaluations of f have one arbitrary parameter and have produced second order methods.
Case (iv).If c2=3/4, we get k1 = h f (xj, yj) k2 = h f (xj+3/4 h , yj+3/4 k1) Which is called Raltson's Method and have two functions of evaluations.Ralston (1962) and Ralston and Rabinowitiz (1978) determined that choosing ω2 = 2/3 provides a minimum bound on the truncation error for the second order RK algorithms.[7] STABILITY ANALYSIS: [1] Consider the initial value problem for the test equation as Where λ may be real or complex number.The exact solution of the initial value problem is PARALLEL ALGORITHMS FOR SOLVING SYSTEM OF ODES () = ( 0 ) λ(− 0 ) =  0  λ(− 0 ) ( Substituting  =  +1 and  =   in (11) Hence the propagation factor (λℎ) is independent of the parameter c2 .Therefore the stability intervals or regions of all second order Runge kutta methods is same.Now for λ real and λ < 0, the condition is satisfied when λℎ ∈ (−2,0) Hence the region of stability of second order is (−2, 0).The plots of the stability regions for the second -order Runge-Kutta algorithms is shown in below Figure.
This stability region is larger than those of multi-step methods.In particular, the stability regions of the multi-stage schemes grow with increasing accuracy while the stability regions of multistep methods decrease with increasing accuracy.When analyzing multi-step methods, the next step would be to determine the locations in the λh-plane of the stability boundary (i.e.where |w| = 1).This however is not easy for Runge-Kutta methods and would require the solution of a higher-order polynomial for the roots.Instead, the most common approach is to simply rely on a contour plotter in which the λh-plane is discretized into a finite set of points and |w| is evaluated at these points.[5] A. HASAN Then, the |w| = 1 contour can be plotted.Above Figure is Stability boundaries for second-order Runge-Kutta algorithms (stable within the boundaries).

MATLAB CODES:
According to [10][11].Write all programs in M-file and save it as

CONVERGENCE ANALYSIS OF METHODS
The numerical solutions yi will contain errors.We shall be concerned with the effect of these errors on the solutions, To finding the numerical solution of ordinary differential equations, two types of errors occurs.Round-off errors and Truncation errors.Rounding errors originate from the fact that computer can only represent numbers using a fixed and limited number of significant figures.Thus, such numbers cannot be represented exactly in computer memory.The discrepancy introduced by this limitation is called Round-off error.Truncation error in numerical analysis arises when approximations are used to estimate some quantity.The accuracy of the solution will depend on how small we take the step size ℎ.A numerical method is said to be convergent if the numerical solution approaches the exact solution as the step size ℎ goes to 0. A method is convergent if, as more grid points are taken or step size is decreased.[8]

NUMERICAL RESULTS AND COMPARATIVE DISCUSSION
In this section, we employ the different techniques, obtained in this paper to solve system of simultaneous the ordinary differential equations with initial value problems and compare them.We use the stopping criteria up to fifteen decimal places.We have  +1 = ( +1 ) −  +1 ,  = 0,1, … … … ,  − 1 for computer programs (2010) [9].All programs are written in Matlab 2009a.Let us consider the initial value Problem1.
The exact solution for this problem is  1 () = cos() + sin() −     From the Table6 and Table7.The calculated results are displayed for average percentage errors at different step size.Reasonably good results are obtained even for a large step size and the approximation can be improved by decreasing the step size.According to the results Midpoint method give good result when compare with exact solution.Since midpoint method has minimum error as compared to other methods.From the Table8 we see that when we reduce the step size h/2 the ratios of % errors are equivalent to 2 2 .This verifies that each method is second order RK method.Thus the accuracy of Midpoint method is good among them.Hence we conclude that Midpoint method is most effective scheme overall.

CONCLUSIONS
The RK2 methods are simple methods of solving first-order system of ODE, particularly suitable for quick programming because of their great simplicity, although their accuracy is not high.First, the midpoint method is more accurate than other RK2 methods.Second, it is more stable.The above plots shows the results obtained from different algorithms at different step size.
One can easily adopt this paper as needed for a different type of problems for solving system ordinary differential equations for more variables first order linear equations.Inner working of the numerical methods will be clear, especially for the students.In using numerical procedure, such as Midpoint method, one must always keep in mind the question of whether the results are accurate enough to be useful.In the preceding examples, the accuracy of the numerical results could be ascertained directly by a comparison with the solution obtained analytically.Thus from