A Study on Linear and Nonlinear Stiff Problems. Using Single-Term Haar Wavelet Series Technique

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1 Int. Journal of Math. Analysis, Vol. 7, 3, no. 53, HIKARI Ltd, A Study on Linear and Nonlinear Stiff Problems Using Single-Term Haar Wavelet Series Technique S. Sekar Department of Mathematics Government Arts College (Autonomous) Salem 636 7, India sekar_nitt@rediffmail.com E. Paramanathan Research and Development Centre Bharathiar University Coimbatore 64 46, India paramulect@yahoo.co.in Copyright 3 S. Sekar and E. Paramanathan. 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 In this article presents a numerical study of linear and nonlinear stiff problems using the single-term Haar wavelet series (STHW) technique. The obtained discrete solutions using the STHW are found to be very accurate and are compared with the exact solutions of the linear and nonlinear stiff problems and also with the classical fourth order Runge-Kutta method (RK). The results obtained show that STHW technique is more useful for solving linear and nonlinear stiff problems and the solution can be obtained for any length of time. Mathematics Subject Classification: 4A45, 4A46, 4A58 Keywords: linear stiff differential equation, non linear stiff differential equations, Runge-Kutta methods, Haar wavelets, single-term Haar wavelet series

2 66 S. Sekar and E. Paramanathan Introduction In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal function basis. Single-term Haar wavelet series (STHW) plays an important role in both the analysis and numerical solution of stiff differential equations. STHW can have a significant impact on what is considered a practical approach and on the types of problems that can be solved. However, working with stiff differential equations places special demands on STHW codes. In science and engineering, stiff differential equations often have to be solved. Although some cases can be solved analytically, the majority of stiff differential equations are too complicated to have analytical solutions. Even when analytical solutions can be found, they are not always useful in practice since the computational cost involved is very high. In recent years, there has been an increased interest in several methods were arisen to solve the stiff differential equations. STHW can have a significant impact on what is considered a practical approach and on the types of problems that can be solved. S. Sekar and team of his researchers [3-9] introduced the STHW to study the time-varying nonlinear singular systems, analysis of the differential equations of the sphere, to study on CNN based hole-filter template design, analysis of the singular and stiff delay systems and nonlinear singular systems from fluid dynamics, numerical investigation of nonlinear volterra-hammerstein integral equations, to study on periodic and oscillatory problems, and numerical solution of nonlinear problems in the calculus of variations. Among the models using differential equations (DE), ordinary differential equations are frequently used to describe various physical problems, for example, motions of the planet in a gravity field like the Kepler problem, the simple pendulum, electrical circuits and chemical kinetics problems. An ordinary differential equation (ODE) has the form y ( x) = f ( x, y( x) ) () where x is the independent variable which often to time in a physical problem and the dependent variable, y(x), is the solution. Moreover, since y(x) could be an N dimensional vector valued function, the domain and range of the differential equation, f and the solution, y are given by N N f : R R R, N y : R R. The above equation () where f is a function of both x and y are called non-autonomous. However, by simply introducing an extra variable which is always exactly equal to x, it can be easily rewritten in an equivalent autonomous form below where f is a function of y only.

3 Study on linear and nonlinear stiff problems 67 ( x) f ( y( x) ) y = () Even though many problems are naturally expressed in the non-autonomous form, the autonomous form of differential equation () is preferred for most of the theoretical investigations. Furthermore, the autonomous form has some advantages in numerical analysis since it gives a greater possibility that numerical methods can solve the differential equation exactly. The differential equation by itself is not enough to find a unique solution. Hence, some other additional information is needed. However, if all components of y are given at a certain value of x i.e. initial conditions then the differential equation is called as an initial value problem (IVP) which is closely and naturally involved with physical modeling. An initial value problem with the given initial condition ( x ) y structure in non-autonomous form and ( x) = f ( x y( x) ), y ( x ) = y y, ( x) f ( y( x) ), y ( x ) = y y = y = has the in autonomous form. In this paper, we introduce the STHW technique for finding the numerical solution of linear and nonlinear stiff problems with more accuracy. Single-term Haar wavelet series technique The orthogonal set of Haar wavelets h i ( t) is a group of square waves with magnitude of ± in some intervals and zeros elsewhere [3]. In general, j j h () ( ) n t = h t k, n = + k, j j, k <, n, j, k Z, t < h () = t,, t < Namely, each Haar wavelet contains one and just one square wave, and is zero elsewhere. Just these zeros make Haar wavelets to be local and very useful in solving stiff systems. Any function y(t), which is square integrable in the interval [,). Can be expanded in a Haar series with an infinite number of terms () = j y t cihi ( t), i = + k, i= (3) j j, k <, n, j, t, where the Haar coefficients [ ]

4 68 S. Sekar and E. Paramanathan () t j c = y h ( t) dt i are determined such that the following integral square error ε is minimized: m () () = y t cihi t dt, ε i= j m =, j {} N usually, the series expansion Equation (3) contains an infinite number of terms for a smooth y(t). If y(t) is a piecewise constant or may be approximated as a piecewise constant, then the sum in Eq. () will be terminated after m terms, that is m T y t c h ( t) = c h t, t, () ( ) ( )() [ ] h i= i i m i c( )( ) [... m t cc c m ], T ( m)( t) = [ h ( t) h ( t)... hm ( t) ], m T = (4) where T indicates transposition, the subscript m in the parantheses denotes their dimensions. The integration of Haar wavelets can be expandable into Haar series with Haar coefficient matrix P[3]. h ( m)( τ ) dτ P( m m) h( m)() t,t [, ] where the m-square matrix P is called the operational matrix of integration and single-term P ( ) =. Let us define [3] T h t h t M t, (5) and M ( )() t = h (). t ( m )( ) ( m ) ( ) ( m m)( ) Equation (5) satisfies M ( m m)( t) c ( m ) = C ( m m ) h ( m ) ( t), where c ( m ) is defined in Equation (4) and C ( ) = c. 3 Stiff problems Even if there exists the numerical solution to a differential equation, certain types of differential equations are difficult to solve, in fact, they need certain types of numerical methods. This phenomenon known as stiffness was first recognized by Curtiss and Hirschfelder [] in 95. Stiffness occurs when some components of the solution decay much more rapidly than others. These problems have highly stable exact solutions but have highly unstable numerical solutions. There are several ways of characterizing stiffness and one way of understanding is looking at the Lipschitz constant. Stiff problems typically have a large Lipschitz constant; however, many of them have a more moderate size one-sided Lipschitz constant.

5 Study on linear and nonlinear stiff problems Definition. N N Butcher [] The function f : [ a, b] R R is said to satisfy a one-sided Lipschitz condition if there exists a one-sided Lipschitz constant l, such that for N x a, b and all y, z R, all [ ] f ( x y) f ( x, z),, y z l y z where the norm is defined by y = y, y assuming that there exists an inner-product on R N. Therefore, the Lipschitz constant could be large while the one-sided Lipschitz constant could be small, or even negative. This theorem leads me to deduce the following result. 3.. Theorem. Butcher [] If f satisfies a one-sided Lipschitz condition with one-sided y x = f ( x, y x ), then for all x x, Lipschitz constant l, and y and z are solutions of ( ) ( ) y ( x) z( x) exp( l( x x )) y( x ) z( x ). Notice from this result that the distance between any two solutions will not increase rapidly or may even decrease if the equation has an adequate one-sided Lipschitz constant. Since stiffness is closely related to the behaviour of perturbations to a given solution, it is important to find out the effect of small perturbations with a one-sided Lipschitz condition. Consider y ( x) = f ( x, y( x) ) (6) with y(x), a solution, and ε Y ( x), a small perturbation to the given solution. Replace y(x) in the equation (6) by y( x) + εy ( x) and expand the solution in a series in powers of ε up to the second order, then get f y ( x) + ε Y ( x) = f ( x, y( x) ) + ε Y ( x). (7) y Subtract the equation (6) from (7) and simplify it, then finally obtain the equation which controls the behaviour of the perturbation, f Y ( x) = Y ( x) = J ( x) Y ( x) y where J(x) is the Jacobian matrix of f ( x, y( x) ). I can use the spectrum of eigen values of J(x) to characterise stiffness. The eigen values of J(x) determine the growth rate of the perturbation with a moderate change in the value of the solution and a very small change in J(x) in a time interval Δ x. The existence of one or more large and negative values of λ where λ σ ( J ( x) ) where x Δx indicates that stiffness is present.

6 63 S. Sekar and E. Paramanathan 4. Examples of stiff problem Stiffness can be understood by the practical difficulty found in numerical calculation as well. The stiff problems are impossible or very difficult to solve by explicit methods, mainly because the small bounded stability region of explicit methods forces the numerical method to take very small step sizes for the smooth solution. Two examples of stiff problems are given here to observe how explicit and implicit methods work for these problems. 4.. Example Stiff linear problem. Consider the stiff system of three linear ordinary differential equations with corresponding initial conditions is of the autonomous form. y ( x) y( x) y( ) ( ) ( ) ( ) = y x = y x, y ( ) ( ) ( ) y 3 x L L y3 x y3 ε where L= - 5 and ε =. The analytic solution is y( x) sin( x) which is drawn in ( ) y x = cos( x) ( ) ( ) ( ) y3 x sin x + ε exp Lx Figure. Figure Analytical Solution of Example 4., Stiff linear problem The results of using the RK and STHW methods for solving this stiff problem on the interval of [,] are presented in Figures -7. The Figures -4 show that the RK method definitely seems to have difficulty approximating y 3 while y and y are computed without difficulties. Especially the approximations with n= and n=5 are hopeless. However, the STHW method performs perfectly well even for n as low as 4 as shown in Figures 5-7.

7 Study on linear and nonlinear stiff problems 63 Figure RK solution of Example 4. with n = 4 and 8 Figure 3 RK solution of Example 4. with n = and 6 Figure 4 RK solution of Example 4. with n = and 3

8 63 S. Sekar and E. Paramanathan Figure 5 STHW solution of Example 4. with n = 4 and 8 Figure 6 STHW solution of Example 4. with n = and 6 Figure 7 STHW solution of Example 4. with n = and Example Stiff nonlinear problem (The Kaps problem). Consider the stiff system of two dimensional Kaps problem with corresponding initial conditions is of the non autonomous form.

9 Study on linear and nonlinear stiff problems 633 y y Analytic solution is ( x) ( x) = y y( x) + y ( x) ( x) y ( x) ( + y ( x) ) ( x) ( x) ( x) ( ) x y, y ( ) = ( ) y exp = y exp which is drawn in Figure 8. In Figures 9-4, the computed solutions of this problem using the RK and STHW method on the interval of [,] are displayed. Even using a large number of steps, the RK method performs poorly. However the STHW method easily gives a good approximation. From these two examples, it is clearly confirmed that the RK method is not suitable but the STHW method is appropriate for stiff problems. Figure 8 Analytical Solution of Example 4., Stiff nonlinear problem. Figure 9 RK solution of Example 4. with n = 4 and 8

10 634 S. Sekar and E. Paramanathan Figure RK solution of Example 4. with n = and 6 Figure RK solution of Example 4. with n = and 3 Figure STHW solution of Example 4. with n = 4 and 8

11 Study on linear and nonlinear stiff problems 635 Figure 3 STHW solution of Example 4. with n = and 6 Figure 4 STHW solution of Example 4. with n = and 3 5. Conclusions The RK method is simple. It uses only four pieces of information from the past and evaluates the driving function only four per step. However, the RK method is not very practical for computational purpose since considerable computational effort is required to improve accuracy. From the Figures -7 and 9-4, one can predict that the error is very less in STHW when compared to the RK method. This STHW provided a momentum for advancing numerical methods for solving linear and nonlinear stiff problems.

12 636 S. Sekar and E. Paramanathan References [] J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, U.K. 3. [] C. F. Curtiss and J. O. Hirschfelder, Integration of Stiff Equations, Proc. Nat. Acad. Sci., 38(95), [3] S. Sekar and A. Manonmani, A study on time-varying singular nonlinear systems via single-term Haar wavelet series, International Review of Pure and Applied Mathematics 5(9), [4] S. Sekar and G. Balaji, Analysis of the differential equations on the sphere using single-term Haar wavelet series, International Journal of Computer, Mathematical Sciences and Applications 4(), [5] S. Sekar and M. Duraisamy, A study on cnn based hole-filler template design using single-term Haar wavelet series, International Journal of Logic Based Intelligent Systems 4(), 7-6. [6] S. Sekar and K. Jaganathan, Analysis of the singular and stiff delay systems using single-term Haar wavelet series, International Review of Pure and Applied Mathematics 6(), [7] S. Sekar and R. Kumar, Numerical investigation of nonlinear volterra-hammerstein integral equations via single-term Haar wavelet series, International Journal of Current Research 3(), [8] S. Sekar and E. Paramanathan, A study on periodic and oscillatory problems using single-term Haar wavelet series, International Journal of Current Research (), [9] S. Sekar and M. Vijayarakavan, Analysis of the non-linear singular systems from fluid dynamics using single-term Haar wavelet series, International Journal of Computer, Mathematical Sciences and Applications 4(), 5-9. Received: August 9, 3