NUMERICAL SOLUTION OF DELAY DIFFERENTIAL EQUATIONS VIA HAAR WAVELETS

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1 TWMS J Pure Appl Math V5, N2, 24, pp NUMERICAL SOLUTION OF DELAY DIFFERENTIAL EQUATIONS VIA HAAR WAVELETS S ASADI, AH BORZABADI Abstract In this paper, Haar wavelet benefits are applied to the delay differential equations (DDEs) A discretized form of DDEs at collocation points based on some useful properties of Haar wavelets transforms original problem into a nonlinear algebraic equations Finally, the numerical experiments are given to demonstrate the conclusions Keywords: Delay differential equation, haar wavelet, collocation method, numerical solution AMS Subject Classification: 34K28, 42C5, 65L3 Introduction Delay differential equations (DDEs) arise in many areas of mathematical modelling, as physiological and pharmaceutical kinetics, chemical kinetics, population dynamics, the navigational control of ships and aircraft, infectious diseases, and more general control problems Given the importance of applications of DDEs, many researchers have addresssed this issue [, 4, 2, 7] Since analytical solutions of the DDEs may be obtained only in very restricted cases, many methods have been proposed for numerical approximation of them Some of these techniques include Hermite interpolation [3], θ-methods [], Runge-Kutta methods [, 9], Parallel continuous Runge-Kutta methods [2], Spline collocation methods [5], Collocation methods [6], linear multistep methods [9] Recently, Haar wavelets as a useful mathematical tool, have been applied extensively for signal processing in communications, physics researches and optimal control problems [3] Haar wavelets have the simplest orthogonal series with compact support and this characteristic introduces Haar wavelets as a good candidate for application to differential equations In this paper, using the benefits of Haar wavelets, a novel collocation approach is introduced to find approximate solutions of the following delay differential equation ẋ(t) = f ( t, x(t), x(t τ ), x(t τ 2 ),, x(t τ s ) ) < t t f, x(t) = ξ(t) t, () where x(t) R n, f : R (s+)n+ R n is a nonlinear smooth function and τ i >, i =,, s are constant delays In the next sections, first Haar wavelets and their properties are introduced Then the approximation of a function by Haar wavelets is discussed By introducing operational integration matrix and delay operational matrix, a discretization method is established Finally, by some numerical examples the proficiency of the given approach is examined School of Mathematics and Computer Science, Damghan University, Damghan, Iran soasmath84@gmailcom, borzabadi@duacir Manuscript received January 23 22

2 222 TWMS J PURE APPL MATH, V5, N2, 24 2 Haar wavelets and its properties The orthogonal set of Haar wavelets φ i (t) is a group of square waves with magnitude + or - in some intervals and zeros elsewhere, φ (t) =, t <, (2) { if t < φ (t) = 2, if 2 t <, (3) if k t < k +, φ i (t) = φ (2 j 2 j 2 j 2 j+ t k) = if k + t < k +, (4) 2 j 2 j+ 2 j 2 j otherwise i = 2 j + k, j =,, M, k =,, 2 j, integer m = 2 j, (j =,,, J), indicates the level of the wavelet, k =,,, m is the translation parameter, maximal level of resolution is J and the maximal value of i is M = 2 J+ A simple calculation shows that { φ i (t)φ l (t)dt = M if i = l = 2j + k, if i l Consequently, the functions φ i (t) are orthogonal This allows us to transform any function square integrable on the interval time [,] into Haar wavelets series 2 Function approximation We just pointed out that a square integrable function can be expressed in terms of Haar orthogonal basis on interval l [, ] However, before the procession to this transfer, it is necessary to unify the time interval Using a linear transformation, the actual time t can be expressed as a function of l via t = [(t f t )l + t ], where t is the initial time and t f is the final time in a square integrable function f(t) Any function f(t) which is square integrable in the interval [, ] can be expanded in a Haar series with an infinite number of terms f(t) = a i φ i (t), i = 2 j + k, j, k < 2 j, t [, ), (6) i= where the Haar coefficients a i = 2 j are determined in such a way that the integral square error ε = (5) f(t)φ i (t)dt, (7) M (f(t) a i φ i (t)) 2 dt (8) is minimum Here ε is vanished when M tends to infinity Usually, the series expansion of (5) contains an infinite number of terms for smooth f(t) If f(t) is a piece wise constant or may be approximated as a piecewise constant, then the summation (6) will be terminated after M terms, that is, f(t) M i= i= a i φ i (t) = A T Ψ M (t), (9) where the coefficient vector A = [a, a,, a M ] T and Ψ M (t) = [φ, φ,, φ M ] T Let us define the collocation points t k = (k 5)/M, (k =,, M) With these chosen

3 S ASADI, AH BORZABADI: NUMERICAL SOLUTION OF DELAY 223 collocation points, the function is discretized into a series of nodes with equivalent distances Let the Haar matrix H M M be the combination of Ψ M (t) at all the collocation points Thus, φ (t ) φ (t ) φ (t M ) φ (t ) φ (t ) φ (t M ) H M M = [Ψ M (t ),, Ψ M (t M )] = φ M (t ) φ M (t ) φ M (t M ) M M () For example, H 2 2 = [ Ψ 2 (t ) Ψ 2 (t ) ] [ ] = Therefore, the function f(t) may be approximated as 2 2 f(t k ) c T MH M M, () where c T M = [c, c 2,, c M ] are the wavelet coefficients The integration of the vector Ψ M (t) defined in (9) can be approximated by t Ψ M (t )dt = P M M Ψ M (t), (2) where P M M is M M operational integration matrix which satisfies the following recursive formula [8], [ ] P M M = P M/2 M/2 2M H 2M H M/2 M/2 M/2 M/2 where (M/2) (M/2) is a null matrix of order(m/2) (M/2), P = [ ], (3) 2 22 The delay operational matrix The delay function Ψ M (t τ) is the shift of Ψ M (t) in (9) along the time axis by τ and to estimate them respect to Ψ M (t) the delay operational matrix D(τ) is defined as Ψ M (t τ) D(τ)Ψ M (t), t > τ, t <, (4) where τ [, ] is the delay constant Here is shown how to generate the delay operational matrix D(τ) = [d ij ] for τ 2 with four basis functions These basis functions are given by φ, φ, φ 2, φ 3 By (4) and (4) we have φ (t τ) φ (t) φ (t τ) φ 3 (t τ) = [d ij(τ)] φ (t) φ 3 (t), i, j =, 2, 3, 4, φ 4 (t τ) φ 4 (t) where φ (t τ) =, τ t <, and φ (t τ) = { ifτ t < 2 + τ, if 2 + τ t <, and φ 2 (t τ) = φ 3 (t τ) = { ifτ t < 4 + τ, if 4 + τ t < 2 + τ, { if 2 + τ t < τ, if τ t <,

4 224 TWMS J PURE APPL MATH, V5, N2, 24 To find the entries d ij (τ), i, j =, 2, 3, 4, we use the inner product For example if τ =, we have d =< φ (t τ), φ (t) >= φ (t τ)φ (t)dt = 9, d 3 =< φ 3 (t τ), φ (t) >= φ 3 (t τ)φ (t)dt = If we calculate all d ij (τ) as d and d 3, the 4 4 operational matrix D(τ) will be obtained as D 4 4 () = In a similar manner if we use the vector function φ(t) with dimension 2 n+, then 2 n+ 2 n+ delay matrix D(τ) with τ 2 can be obtained Note that for any dimension if τ = then n matrix is diagonal and we have d il = 4 4 { 2 j if i = l = 2 j + k, φ i (t)φ l (t)dt = ifi l, 3 The collocation approach We discretize the functions φ i (t) in (4) by dividing the interval l [, ], to M equidistance intervals with distance parameter t = /M and introduce the collocation points l k = (k 5)/M, k =,, M, where M is the number of nodes used in the discretization and also is the maximum wavelet index number Also we approximate state variables ẋ(l) by Haar wavelets with M collocation points, ie, ẋ(l) c T x Ψ M (l), (5) where Haar coefficient vector c T x are defined as c T x = [c x,, c xm ] Using the operational integration matrix P M M defined in (2), x(l) = l ẋ(l )dl + x() = l c T x Ψ M (l )dl + x() = c T x P M M Ψ M (l) + x() (6) As stated in (3), the expansion of the matrix Ψ M (l) at the M collocation points will yield the M M Haar matrix H M M and by (6) it can be concluded that ẋ(l k ) = c T x Ψ M (l k ), x(l k ) = c T x P M M Ψ M (l k ) + x(), k =,, M, (7) Now we focus on the analysis of time-delayed systems Choose N i as following manner, N i = Mτ i + 5, (8) i =, 2,, s and let N N 2 N s, (where denotes the bracket function) Using (2), (4) and (), { ξ(l τi ) if l < τ x(l τ i ) = i Cx T i =, 2,, s (9) P M M D(τ i )Ψ M (l) + x() if τ i l < t f By substituting x(l k ), ẋ(l k ) and x(l k τ i ), (i =, 2,, s) in () and using (4)-(9), we have c T x Ψ M (τ k ) = (t f t ) { f(l k, c T x P M M Ψ M (l k ) + x(), ξ(l k τ ), ξ(l k τ 2 ),, ξ(l k τ s )) }, k =,, N, (2),

5 S ASADI, AH BORZABADI: NUMERICAL SOLUTION OF DELAY 225 c T x Ψ M (l k ) = (t f t ) { f ( l k, c T x P M M Ψ M (l k ) + x(), c T x P M M D(τ )Ψ M (l k ) + x(), ξ(l k τ 2 ),, ξ(l k τ s ) )}, k = N +,, N 2, (2) c T x Ψ M (l k ) = (t f t ) { f(l k, c T x P M M Ψ M (l k ) + x(), c T x P M M D(τ )Ψ M (l k ) + x(), c T x P M M D(τ 2 )Ψ M (l k ) + x(),, c T x P M M D(τ s )Ψ M (l k ) + x()) }, k = N s +,, M, (22) In this way, the DDE problem is transformed into nonlinear algebraic equations for the coefficients c T x From the equations (2)-(22), the vector unknown c T x using an iterative approach, eg Newton iterative method, may be evaluated Since the first and last collocation points are not set as the initial and final time, the initial and final variables are calculated according to x() = ξ(), (23) x f = x(m) + ẋ(m)/2m (24) 4 Numerical results In this section, two DDE problems and the results of applying the discussed collocation approach for them is considered Also the obtained approximate solutions have been compared with exact ones Example We consider the equation ẋ(t) = x(t /2) + sin(t /2) + cos(t), < t, (25) ξ(t) = sin(t), t (26) Using the Haar wavelets collocation method with M = 6 collocation points we have τ = 2, N = 8, (27) c T x h k = sin(l k /2) + sin(l k /2) + cos(l k ), k =, 2,, N, (28) c T x h k = c T x P M M D(/2)Ψ M (l k ) x() + sin(l k /2) + cos(l k ), k = N +,, M, (29) and boundary constraints x() = ξ() =, (3) x(t f ) = c T x P M M Ψ M (l M ) + x() + c T x Ψ M (l M )/2M, (3) where c x are all variables which must be obtained at the collocation points The approximate solution with respect to the number of nodes with exact solution of equation are shown in Fig Example 2 In the second example the two-dimensional DDE equation ẋ (t) = x 2 (t), ẋ 2 (t) = x 2 (t) + exp () x 2(t /) + x (t), (32) with delay functions ξ (t) = exp ( t), ξ 2 (t) = exp ( t), for t is considered The approximate solution with using 6 collocation points by applying the collocation approach, in comparison with the exact solution are shown in Figs 2 and 3

6 226 TWMS J PURE APPL MATH, V5, N2, 24 5 Conclusions In this paper an collocation method using the profits of Haar wavelets is presented Applying the given approach is simple and expected that along with the increase of collocation points the approximate solution converges to exact solution of DDE Of course the approximate solution depends on the applied method of solving the nonlinear equation, which will be generated in the process of yielding the mentioned approach and so, like any iterative scheme the nonlinearity of the functional f in () has direct impact on the precision of the approximate solution References [] Baker CTH, Paul CAH, (994), Computing stability regions Runge-Kutta methods for delay differential equations, IMA J Numer Anal, V(4), pp [2] Baker CTH, Paul CAH, (993), Parallel continuous Runge-Kutta methods and vanishing lag delay differential equations, Adv Comp Math, V(), pp [3] Dai R, JECochran, Jr, (29), Wavelet Collocation Method for Optimal control Problems, J Optim Theory Appl, V(43), pp [4] Driver RD, (977), Ordinary and Delay Differential Equations, Applied Mathematical Sciences, Springer- Verlag, New York-Heidelberg, 2p [5] El-Hawary HM, Mahmoud SM, (23), Spline collocation methods for solving delay-differential equations, Appl Math Comput, V(46), pp [6] Engelborghs K, Luzyanina T, Int l Hout KJ, Roose, D, (2), Collocation methods for the computation of periodic solutions of delay differential equations, SIAM J Sci Comput, V(22), pp [7] Gopalsamy K, (992), Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 74p [8] Gu JS, Jian WS, (996), The Haar Wavelet operation matrix of integration, Int J Syst Sci, V(27)(7), pp [9] Houwen PJ, Sommelier BP, (984), Stability in linear multistep methods for pure delay equations, J Comp Appl Math, V(), pp55-63 [] Kolmanovskii VB, Nosov VR, (986), Stability of Functional-Differential Equations, Mathematics in Science and Engineering, Academic Press, Inc [Harcourt Brace Jovanovich, Publishers], London, 8 p [] Koto T, (23), Stability of θ-methods for delay integro-differential equations, J Comput Appl Math, V(6), pp [2] Kuang K, (993), Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, Academic Press, Inc, Boston, MA, 9p [3] Oberle HJ, Pesch HJ, (98), Numerical treatment of delay differential equations by Hermite interpolation, Numer Math, V(37), pp [4] Oppelstrup, J, (978), The RKFHB4 method for delay differential equations, Lecture Notes in Mathematics, V(63), pp33-46

7 S ASADI, AH BORZABADI: NUMERICAL SOLUTION OF DELAY Haar wavelet with 6 nodes Haar wavelet with 32 nodes 7 Numeric solution for x time Figure Approximate and exact solution for Example 4 2 Haar wavelet with 6 nodes exact solution 8 6 x Time Figure 2 Approximate and exact solution x for Example x Haar wavelet with 6 nodes exact solution Time Figure 3 Approximate and exact solution x 2 for Example 4

8 228 TWMS J PURE APPL MATH, V5, N2, 24 Solayman Asadi was born in Sari, Iran, 987 He received his BSc in pure mathematics in Ferdowsi University of Mashhad in 2, MSc in applied mathematics in 22 from Damghan University in IRAN His research areas include optimal control, optimization and analysis Akbar Hashemi Borzabadi received his BSc from Birjand University, Birjand, Iran and MSc and PhD from Ferdowsi University of Mashhad, Mashhad, Iran Now he is an Associate Professor at the School of Mathematics and Computer Science, Damghan University, Iran and his research areas are optimal control, optimization, approximation theory and numerical analysis