Numerical solution of delay differential equations via operational matrices of hybrid of block-pulse functions and Bernstein polynomials

Transcription

1 Computational Methods for Differential Equations Vol, No, 3, pp Numerical solution of delay differential equations via operational matrices of hybrid of bloc-pulse functions and Bernstein polynomials M Behroozifar Faculty of Basic Sciences, Babol University of Technology, Babol, Mazandaran, Iran m behroozifar@nitacir S A Yousefi Department of Mathematics, Shahid Beheshti University, Tehran, Iran s-yousefi@sbuacir Corresponding Author Abstract In this paper, we introduce hybrid of bloc-pulse functions and Bernstein polynomials and derive operational matrices of integration, dual, differentiation, product and delay of these hybrid functions by a general procedure that can be used for other polynomials or orthogonal functions Then, we utilize them to solve delay differential equations and time-delay system The method is based upon expanding various time-varying functions as their truncated hybrid functions Illustrative examples are included to demonstrate the validity, efficiency and applicability of the method Keywords Delay differential equation, Bernstein polynomial, Hybrid of bloc-pulse function, Operational matrix Mathematics Subject Classification 65L5, 34K6, 34K8 Introduction Delays occur frequently in biological, chemical, electronic and transportation systems [] Time-delay systems are therefore a very important class of systems whose control and optimization have been of interest to many investigators Orthogonal functions and polynomial series have received considerable attention in dealing with various problems of dynamic systems The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus greatly simplifying the problems [, 3, 7, 9,, 9] The approach is based on converting the underlying differential equations into integral equations through integration, approximating various signals involved in the equation by truncated polynomials series and using the operational matrices to eliminate the integral, derivation and delay operations Special attention has recently been given to applications 78

2 CMDE Vol, No, 3, pp of Walsh function [3], Haar wavelets [6], Chebyshev polynomials [7], Bessel functions [6], Legendre polynomials [], Legendre wavelets [7], Sine-cosine wavelets [8] and Bernstein polynomials [3] In recent years the different inds of hybrid functions [4, 5, 8, 9,,, 3, 4, 9, ] were applied to solve the delay problem Bernstein polynomials have some properties that distinguish it from other polynomials for example continuity and partition of unity property All of the Bernstein polynomials vanish at the initial and end points of the interval [a, b] except for the first polynomial and the last polynomial are equal to at x = a and x = b, respectively, which provides greater flexibility to impose boundary conditions The Bernstein polynomials, although not based on orthogonal polynomials, can also be applied to analyze various problems Specifically, in [4, 5] Bernstein polynomials have been used for solving the partial differential equation In this paper, hybrid of bloc-pulse functions and Bernstein polynomials are introduced and derived operational matrices of integration P, dual Q, differentiation D, product Ĉ and delay Del by a general procedure These matrices are applied to evaluate the solution of delay differential equation or delay differential system by expanding the candidate function as hybrid functions with unnown coefficients and reducing the delay problem to a set of algebraic equations This paper is organized as follows: In section we describe the basic formulation of Bernstein polynomials which is required for our subsequent development Section 3 is devoted to the definition of hybrid of bloc-pulse functions and Bernstein polynomials and function approximation using these functions In section 4 we calculate the operational matrices of integration, dual, differentiation, product and delay In section 5, we report our numerical findings which demonstrate the validity, accuracy, efficiency and applicability of the operational matrices by considering some test problems Section 6 consists of a brief summary Properties of Bernstein polynomials The Bernstein polynomials of mth-degree are defined on the interval [a, b] as [] where B i,m (x) = ( ) m (x a) i (b x) m i i (b a) m, i =,,, m, ( ) m = i m! i!(m i)!

3 8 M BEHROOZIFAR AND S A YOUSEFI These Bernstein polynomials form a basis on L [a, b] There are m + polynomials of mth-degree For convenience, we set B i,m (x) =, if i < or i > m A recursive definition can also be used to generate the Bernstein polynomials over [a, b] so that the ith Bernstein polynomial of mth-degree can be written (b x) B i,m (x) = b a B (x a) i,m (x) + b a B i,m (x) It can readily be shown that each of the Bernstein polynomials is positive and the sum of all the Bernstein polynomials is unity for all real x [a, b], ie, m i= B i,m(x) = (unity partition property) It is easy to show that any given polynomial of mth-degree can be expanded in terms of these basis functions 3 Properties of hybrid functions 3 Hybrid of bloc-pulse functions and Bernstein polynomials Since interval [a, b) can be shifted to [, ), therefor we concentrate on [, ) Hybrid of bloc-pulse functions and Bernstein polynomials are defined on [, ) for i =,,, m and n =,,, N, { n Bi,m (t n) ψ i,n (t) = t < n+ otherwise, (3) where m is the degree of Bernstein polynomial on [, ], n is transmission parameter, N denotes the number of subinterval of [, ] and the parameters, N will be specified 3 Function Approximation Suppose that H = L [, ] and {ψ i,n (t)} m,n i=,n= H be the set of hybrid functions of Bernstein polynomials of mth-degree and Y = Span{ψ i,n (t) i =,, m, n =,,, N } and f be an arbitrary element in H Since Y is a finite dimensional vector space, f has the unique best approximation out of Y such as y Y, that is y Y ; y Y f y f y, where f =< f, f >= f (x)dx Since y Y, there exist the unique coefficients c i,n such that where f y = N n= i= m c i,n ψ i,n = c T ϕ, ϕ T = [ψ,, ψ,,, ψ m,, ψ m,,, ψ,n, ψ,n,, ψ m,n, ψ m,n ], c T = [c,, c,,, c m,, c m,,, c,n, c,n,, c m,n, c m,n ],

4 CMDE Vol, No, 3, pp and c T can be obtained by where c T < ϕ, ϕ >=< f, ϕ >, < f, ϕ >= f(x)ϕ(x) T dx and < ϕ, ϕ > is a N(m+) N(m+) matrix which is said dual operational matrix of ϕ and denoted by Q then ( c T = Q =< ϕ, ϕ >= ϕ(x)ϕ(x) T dx, ) f(x)ϕ(x) T dx ( Q ) (3) In the following lemma we present an upper bound for the error approximation Lemma 3 Suppose that the function g : [, ) R is m + times continuously differentiable, g C m+ [t, t f ] and Y = Span{ψ i,n (t) i =,, m, n =,,, N } If c T ϕ is the best approximation g out of Y then the mean error bounded is presented as follows: g c T M ϕ (m + )! m+ m + 3, where M = max x [t,t f ] g (m+) (x) Proof We consider the Taylor polynomial of order m for function g on [ n, n+ ) y n (x) = g( n ) + g ( n )(x n ) + + g(m) ( n )(x n )m m! for n =,,, N which we now g(x) y n (x) g (m+) (η) (x n )m+ (m + )! (33) where η ( n, n+ ) Since ct ϕ is the best approximation g out of Y, y n Y and using (33) we have g c T ϕ = N n= n+ n g(x) c T ϕ(x) dx = g(x) y n (x) dx N n= n+ n N n= n+ n g(x) c T ϕ(x) dx [g (m+) (η) (x n ] )m+ dx (m + )!

5 8 M BEHROOZIFAR AND S A YOUSEFI M (m + )! N n= n+ n (x n )m+ dx = and by taing square root we have the above bound M [(m + )! ] m+ (m + 3), The presented upper bound of the error depends on (m+)! m+ m+3 which shows that the error reduces to zero very fast as m increase This is one of the advantages of hybrid of bloc-pulse functions and Bernstein polynomials 4 Operational matrices of hybrid of bloc-pulse functions and Bernstein polynomials 4 Operational matrix of integration The operational matrix of integration P is given by x ϕ(t)dt P ϕ(x), x < For obtaining an explicit formula for P we denote the vector B,m (x n) ψ,n (x) B,m (x n) B(x n) = = ψ,n (x) B m,m (x n) ψ m,n (x) for n =,,, N B(x) x < B(x ) x < ϕ(x) = B(x ) x < 3 It is easy to see that: therefore B(x (N )) N x <, (4) ( x) r x i dx = (r + i + ) ( ), i, r N {} r+i i B i,m (x)dx = m +, i =,,, m

6 CMDE Vol, No, 3, pp so then B(x n)dx = On the other hand we now x B(x)dx = (m+) (m+) (m+) m+ m+ m+, n =,,, N (4) B(t)dt P B(x), x (43) which P is the operational matrix of integration of B(x) and the details of obtaining this matrix is given in [3] Now, we want to obtain the operational matrix of integration P using (4), (43) and the property of partition of unity of Bernstein polynomial x B(t)dt x < x B(t)dt = B(t)dt + x B(t)dt + B(t)dt x < B(t)dt + + x N B(t)dt N P B(x), x < (m+) (m+) + = (m+) B(x ), x < = (m+) (m+) (m+) (m+) = (m+) B(x (N )), N x < N x < N

7 84 M BEHROOZIFAR AND S A YOUSEFI which is a matrix (m + ) (m + ) that all of its elements is, therefore B(x) x [ ] P B(t)dt =, (m + ),, B(x ) (m + ) B(x (N )) [ ] P =, (m + ),, ϕ(x) (m + ) Similarly, we have B(t )dt x < x x B(t )dt = B(t )dt + x B(t )dt B(t )dt + B(t )dt + + x N B(t )dt = B(x) x < x < N x < N = P B(x ) x < (m+) (m+) (m+) = (m+) B(x ) x < 3 then x (m+) (m+) (m+) = B(t )dt = (m+) B(x (N )) N x < N [, P ], (m + ),, ϕ(x) (m + ) which is the zero matrix (m + ) (m + ) The same process can be done for the rest of the vectors, therefore the operational matrix of integration P is

8 CMDE Vol, No, 3, pp obtained as follows P = (m+) P P (m+) (m+) (m+) (m+) P (m+) P 4 Dual operational matrix We define the dual operational matrix of ϕ in the preceding section that Q = ϕ(x)ϕ T (x)dx In [3] the dual operational matrix of B(x) is presented We have Q = B(x n)b(x n) T dx = B(x)B T (x)dx n+ n B(x n)b(x n) T dx = Q, (44) B(x n)b(x j) T dx =, n j (45) for j, n =,,, N By using (44) and (45) Q = ϕ(x)ϕ T (x)dx B(x)B(x)T dx B(x)B(x (N ))T dx = B(x )B(x)T dx B(x )B(x (N ))T dx B(x (N ))B(x)T dx B(x (N ))B(x (N ))T dx Q = Q Q

9 86 M BEHROOZIFAR AND S A YOUSEFI 43 Operational matrix of differentiation The operational matrix of differentiation D is given by dϕ(x) dx = Dϕ(x) We have db(x) = D B(x) (46) dx which D is the operational matrix of differentiation of B(x) and the details of obtaining this matrix is given in [3] The operational matrix of differentiation D is obtained using (46) as follows d dx ϕ(x) = d dx B(x) x < d dx B(x ) x < d dx B(x ) x < 3 d N dxb(x (N )) D B(x) x < x < N so = D B(x ) x < D B(x ) x < 3 D B(x (N )) N x < N D D D = D 44 Operational matrix of product Suppose that C T = [C T, CT,, CT N ] is an arbitrary N(m + ) matrix which Ci T is (m + ) matrix for i =,,, N, then Ĉ is N(m + ) N(m + ) operational matrix of product whenever C T ϕ(x)ϕ(x) T ϕ(x) T Ĉ We now Ci T B(x)B(x) T B(x) T Ĉ i, i =,,, N

10 CMDE Vol, No, 3, pp which Ĉi is operational matrix of product of Bernstein polynomials presented in [3], then C T ϕ(x)ϕ T (x) = B(x)B(x) T C T B(x )B(x ) T = B(x (N ))B(x (N )) T C T B(x)B(x) T C T B(x )B(x ) T CN B(x T (N ))B(x (N )) T B(x) T Ĉ B(x ) T Ĉ = ϕ(x)t Ĉ B(x (N )) T Ĉ N which Ĉ Ĉ Ĉ =, Ĉ N which is (m + ) (m + ) zero matrix 45 Operational matrix of delay The operational matrix of delay Del is given by ϕ(x τ) (Del)ϕ(x), τ > Suppose that τ be a positive real number, then we let [5] = { τ, N = τ τ Z + [ τ ] otherwise, (47) which [] denotes greatest integer smaller than or equal to a number From (4) B((x τ)) x τ < ϕ(x τ) = B((x τ) ) x τ < B((x τ) ) x τ < 3 B((x τ) (N )) N x τ < N

11 88 M BEHROOZIFAR AND S A YOUSEFI B(x ) x < thus B(x ) x < 3 = B(x (N )) B(x 3) 3 x < 4 B(x N) N x < N N x < N+ I I I Del = I which I and are (m + ) (m + ) identity matrix and (m + ) (m + ) zero matrix, respectively 5 Illustrative example In this section some examples are given to demonstrate the applicability and accuracy of our method In all examples the pacage of Mathematica (7) has been used to solve the test problems considered in this paper Example Consider the delay differential equation ẋ(t) = + x(t x(t) =, x() = ), t < t <, (5) with the exact solution + t, t < x(t) = ( )t + t, t < Since τ = and using (47), we let = and N = in (4) We approximate the unnown function ẋ(t) as ẋ(t) = c T ϕ(t) (5)

12 CMDE Vol, No, 3, pp which c is the unnown vector, so x(t) = c T P ϕ(t) + = (c T P + d T )ϕ(t) where = d T ϕ(t) Using the delay operational matrix x(t ) = (ct P + d T )ϕ(t ) = (ct P + d T )(Del)ϕ(t) (53) Substituting (5) and (53) in (5) we have then c T = d T + (c T P + d T )(Del) c T = (d T + d T Del) (I P (Del)), (54) which I is identity matrix with appropriate dimension The equation set (4) is solved by m = and m = 3 and the error values in some points is presented in Table and the the error function is plotted in Figure for m = 3 Numerical findings show the high accuracy of aproximate solution despite τ = t m = m = Table rror values of x(t) for Example

13 9 M BEHROOZIFAR AND S A YOUSEFI Figure Error function of x(t) for m = 3 in example Example Consider the delay differential equation ẋ(t) = t x(t 3 ), t <, x(t) =, 3 t <, x() = We approximate the function ẋ(t) as (55) ẋ(t) = c T ϕ(t), (56) so x(t) = c T P ϕ(t) + = (c T P + d T )ϕ(t) where = d T ϕ(t) Using the delay operational matrix x(t 3 ) = (ct P + d T )ϕ(t 3 ) = (ct P + d T )(Del)ϕ(t) (57) If we approximate the function t as t ϕ(t) T e (58) and substitute (56), (57) and (58) in (55), we will have then c T ϕ(t) = (c T P + d T )(Del)ϕ(t)ϕ(t) T e c T = (c T P + d T )(Del)ê (59) which ê is operational matrix of product that ϕ(t)ϕ(t) T e ê ϕ(t) If we solve the set (59) with m = 6 and = 3, we will get c = c = c = c 3 = c 4 = c 5 = c 6 =, c 7 = 9, c 8 = 8 7 c 9 = 5 35, c = 45, c = 8 35, c = 7, c 3 = 8 9, c 4 = 8 9, c 5 = c 6 = , c 7 = 79 79, c 8 = , c 9 = 37 79, c = 76 8 and c T = [c, c,, c ] x(t) = c T P ϕ(t) + t < 3 = 3 t t < 3 t t5 + t t t <

14 CMDE Vol, No, 3, pp which is the exact solution Example 3 Consider the following delay system with delay in both control and state [3, 5] ẋ(t) + x(t) + x(t 4 ) = u(t 4 ) t <, x(t) = u(t) =, 4 t, u(t) =, t > (5) The exact solution is [5], t < 4, x(t) = e (t 4 ), 4 t <, e (t 4 ) + ( + 4t)e (t ), t < 3 4, 6 e (t 4 ) + ( + 4t)e (t ) ( t + 4t )e (t 3 4 ), 3 4 t < Using (5) we now then, 4 t, u(t) =, t >, u(t, t 4 ) = 4, = u T ϕ(t), t > 4, which the vector u can be calculated by (3) Suppose that so ẋ(t) = c T ϕ(t) x(t) = c T P ϕ(t) where c is unnown vector Using the delay operational matrix and (5) c T = u T (I + P + P (Del)) ( ) (5) where I is identity matrix with appropriated dimension We solve (5) with m = 3, = 4 and m = 6, = 4 and present the error of x(t) for some points in Table and plot the error function of x(t) for m = 6, = 4 in Figure From numerical results, it can be guessed that the method is convergent of order two

15 9 M BEHROOZIFAR AND S A YOUSEFI t m = 3, = 4 m = 6, = Table Error of x(t) for Example 3 Figure Error function of x(t) in Example 3 for m = 6, = 4 Example 4 Consider the delay differential equation [3] ( ) ( ) ( x (t) x (t = 4 ) ) ( ) x (t) 5 5t x (t 4 ) + with ( x (t) x (t) ) ( = ), [ 4, ] The exact solutions are [8], t < 4 x (t) = 3 t 4 + t, 4 t < t t t3 5 t4, t < t t t t t t6, 3 4 t <

16 CMDE Vol, No, 3, pp and t, t < 4 x (t) = t t 5 3 t3, t t 5 4 t t t5, t t t t t t6 5 t7, We approximate the function x (t) and x (t) as x (t) = c T ϕ(t), by initial conditions 4 t < t < t < x (t) = u T ϕ(t) (5) x (t) = c T P ϕ(t), Using the delay operational matrix x (t) = u T P ϕ(t) x (t 4 ) = ct P ϕ(t 4 ) = ct P (Del)ϕ(t), x (t 4 ) = ut P ϕ(t 4 ) = ut P (Del)ϕ(t), If we approximate the functions (53) = d T ϕ(t), t = ϕ(t) T e, (54) and use the operational matrix of product ê and substitute (5), (53) and (54) in main problem, we will have { c T = u T P (Del), u T = 5 c T P (Del) 5 u T P (Del)ê + d T (55), which ê is the operational matrix of product ϕ(t)ϕ(t) T e = ê ϕ(t) Solving (55) by N = = 4 and m = 7 gives the exact values for x (t) and x (t) 6 Conclusions In this paper the operational matrices of integration, dual, differentiation, product and delay for hybrid of bloc-pulse functions and Bernstein polynomials are obtained An upper bound for the error of approximation is given The presented upper bound of error suggests rapidly convergent to the exact solution when m The hybrid of bloc-pulse functions and Bernstein polynomials are used to solve delay differential systems and delay differential equations The problem has been reduced to solve a set of algebraic equations It is also shown that the hybrid of bloc-pulse functions and Bernstein polynomials provide an exact solution when the exact solutions are piecewise

17 94 M BEHROOZIFAR AND S A YOUSEFI polynomial The illustrative examples demonstrate that the proposed method is valid References [] M I Bhatti and P Bracen, Solutions of differential equations in a Bernstein polynomial basis, Journal of Computational and Applied Mathematics, 5, (7), 7 8 [] RY Chang, ML Wang, Shifted Legendre direct method for variational problems, J Optim Theory Appl 3, (983), [3] CF Chen, CH Hsiao, A Walsh series direct method for solving variational problems, J Franlin Inst 3, (975), 65-8 [4] HY Chung, YY Sun, Analysis of time-delay systems using an alternative method, Int J Control 46, (987), 6-63 [5] KB Datta, BM Mohan, Orthogonal Functions in Systems and Control, World Scientific, Singapore, 995 [6] JS Gu, WS Jiang, The Haar wavelets operational matrix of integration, Int J Syst Sci 7, (996), [7] IR Horng, JH Chou, Shifted Chebyshev direct method for solving variational problems, Int J Syst Sci 6, (985), [8] C Hwang, MY Chen, Analysis of time-delay systems using the Galerin method, Int J Control 44, (986), [9] C Hwang, YP Shih, Laguerre series direct method for variational problems, J Optim Theory Appl 39, (983), [] M Jamshidi, CMWang, A computational algorithm for large-scale nonlinear timedelays systems, IEEE Trans Syst Man Cybernetics SMC-4, (984), -9 [] K Malenejad, Y Mahmoudi, Numerical solution of linear Fredholm integral equation by using hybrid Taylor and bloc-pulse functions, Appl Math Comput 49, (4), [] HR Marzban, HR Tabrizidooz, M Razzaghi, Solution of variational problems via hybrid of bloc-pulse and Lagrange interpolating, IET Control Theory Appl 3,(9), [3] HR Marzban, M Razzaghi, Solution of time-varying delay systems by hybrid functions, Math and Com in Sim 64, (4), [4] HR Marzban, M Razzaghi, Optimal control of linear delay systems via hybrid of bloc-pulse and Legendre polynomials, J Franlin Inst 34, (4), [5] HR Marzban, M Razzaghi, Analysis of Time-delay Systems via Hybrid of Bloc-pulse Functions and Taylor Series, J Vibration and Con, (5), [6] PN Parasevopoulos, P Slavounos, and GCH Georgiou The operation matrix of integration for Bessel functions, Journal of the Franlin Institute, 37, (99), [7] M Razzaghi, S Yousefi, The Legendre wavelets operational matrix of integration, Int J Syst Sci 3 (4),(), [8] M Razzaghi, S Yousefi, Sine-cosine wavelets operational matrix of integration and its applications in the calculus of variations, Int J Syst Sci vol 33, no,(), 85-8 [9] M Razzaghi, M Razzaghi, Fourier series direct method for variational problems, Int J Control 48, (988), [] M Razzaghi, HR Marzban, Direct method for variational problems via hybrid of bloc-pulse and Chebyshev functions, Math Probl Eng 6, (), [] XT Wang, Numerical solution of delay systems containing inverse time by hybrid functions, Appl Math Comput 73, (6),

18 CMDE Vol, No, 3, pp [] XT Wang, Numerical solutions of optimal control for time delay systems by hybrid of bloc-pulse functions and Legendre polynomials, Appl Math Comput 84, (7), [3] S A Yousefi, M Behroozifar, Operational matrices of Bernstein polynomials and their applications, Int J Syst Sci 4, (), [4] S A Yousefi, M Behroozifar, Mehdi Dehghan, The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of mass, Journal of Computational and Applied Mathematics 35, (), [5] S A Yousefi, M Behroozifar, Mehdi Dehghan, Numerical solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomials, Applied Mathematical Modelling 36, (),