A Taylor polynomial approach for solving differential-difference equations

Transcription

1 Journal of Computational and Applied Mathematics 86 (006) wwwelseviercom/locate/cam A Taylor polynomial approach for solving differential-difference equations Mustafa Gülsu, Mehmet Sezer Department of Mathematics, Faculty of Science, Mugla University, Mugla, Turey Received 9 June 004; received in revised form 7 February 005 Abstract The purpose of this study is to give a Taylor polynomial approximation for the solution of mth-order linear differential-difference equations with variable coefficients under the mixed conditions about any point For this purpose, Taylor matrix method is introduced This method is based on first taing the truncated Taylor expansions of the functions in the differential-difference equations and then substituting their matrix forms into the equation Hence, the result matrix equation can be solved and the unnown Taylor coefficients can be found approximately In addition, examples that illustrate the pertinent features of the method are presented, and the results of study are discussed Also we have discussed the accuracy of the method We use the symbolic algebra program, Maple, to prove our results 005 Elsevier BV All rights reserved MSC: 39A0; 4A0; 65Q05 Keywords: Taylor polynomials and series; Taylor polynomial solutions; Taylor matrix method; Differential-difference equations Introduction In recent years, the studies of differential-difference equations, ie equations containing shifts of the unnown function and its derivatives, are developed very rapidly and intensively [ 4,8,] Problems involving these equations arise in studies of control theory [4] in determining the expected time for Corresponding author Tel: ; fax: address: mgulsu@muedutr (M Gülsu) /$ - see front matter 005 Elsevier BV All rights reserved doi:006/jcam

2 350 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) the generation of action potentials in nerve cells by random synaptic inputs in the dendrites [3], in the modelling of the activation of a neuron [3], in the wors on epidemics and population [8], in the twobody problems in classical electrodynamics in the physical systems whose acceleration depends upon its velocity and its position at earlier instants, and in the formulation of the biological reaction phenomena to X-rays [8] Also, the differential-difference equations occur frequently as a model in mathematical biology and the physical sciences [] A Taylor method for solving Fredholm integral equations has been presented in [5] and then this method has been extended by Sezer to Fredholm integro-differential equations [7] and second-order linear differential [9,0] In this study, the basic ideas of the above studies are developed and applied to the mth-order linear differential-difference equation (which contains only negative shift in the differentiated term) with variable coefficients [8, pp 8, 9] m P (x)y () (x) + R P r (x)y(r) (x τ) = f(x), =0 r=0 R m, τ > 0, τ x 0, () with the mixed conditions m =0 [a i y () (a) + b i y () (b) + c i y () ]=μ i () i = 0()(m ), a c b and the solution is expressed in the form y(x) = y (n) n! (x c) n, a c b, N m (3) which is a Taylor polynomial of degree N at x = c, where y (n), n = 0()N are the coefficients to be determined Here P (x), Pr (x) and f(x)are functions defined on a x b; the real coefficients a i, c i,b i and μ i are appropriate constants The rest of this paper is organized as follows Higher-order linear differential-difference equation with variable coefficients and fundamental relations are presented in Section The newscheme are based on Taylor matrix method The method of finding approximate solution is described in Section 3 To support our findings, we present result of numerical experiments in Section 4 Section 5 concludes this article with a brief summary Fundamental relations Let us consider the linear differential-difference equation with variable coefficients () and find the truncated Taylor series expansions of each term in expression () at x =c and their matrix representations We first consider the desired solution y(x) of Eq () defined by a truncated Taylor series (3) Then we

3 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) can put series (3) in the matrix form where [y(x)]=xm 0 Y, (4) X =[ (x c) (x c) (x c) N ], ! y (0) 0 0 0! y () y () M 0 =!, Y = y (N) N! Nowwe consider the differential part P (x)y () (x) of Eq () and can write it as the truncated Taylor series expansion of degree N at x = c in the form P (x)y () (x) = By the Leibnitz s rule we evaluate [P (x)y () (x)] (n) x=c = n n! [P (x)y () (x)] (n) x=c (x c)n (5) i=0 ( ) n i P (n i) y (i+) and substitute in expression (5) Thus expression (5) becomes P (x)y () (x) = n i=0 (n i)!i! P (n i) y (i+) (x c) n (6) and its matrix form [P (x)y () (x)]=xp Y, (7)

4 35 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) where P (0) !0! P () 0 0 P (0) 0 0 0!0! 0!! P () 0 0 P () P (0) 0 0!0!!! 0!! P = P (N ) 0 0 P (N ) P (N ) P () P (0) (N )!0! (N )!! (N )!!!(N )! 0!(N )! P (N +) 0 0 P (N ) P (N ) P () P () (N + )!0! (N )!! (N )!!!(N )! 0!(N )! P (N ) 0 0 P (N ) P (N 3) P () P ( ) (N )!0! (N )!! (N 3)!!!(N )! ( )!(N )! P (N) 0 0 P (N ) P (N ) P (+) P () N!0! (N )!! (N )!! ( + )!(N )!!(N )! (N+)x(N+) Nowin a similar way we consider the difference part P r (x)y(r) (x τ) of Eq () and can write it as the truncated series expansion of degree N at x = c in the form P r (x)y(r) (x τ) = By the Leibnitz s rule we evaluate n! [P r (x)y(r) (x τ)] (n) x=c (x c)n (8) n ( ) [Pr (x)y(r) (x τ)] (n) n x=c = P (n i) i r y (i+r) (c τ) i=0 and substitute in expression (8) Thus expression (8) becomes P r (x)y(r) (x τ) = and its matrix form n i=0 (n i)!i! P r (n i) y (i+r) (c τ)(x c) n [P r (x)y(r) (x τ)]=xp r Y τ, Y τ =[y (0) (c τ) y () (c τ) y (N) (c τ)] T, (9) where P r (r) can be obtained by substituting the quantities Pr instead of P () in relation (7)

5 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) Nowsubstituting quantities (x τ) instead of x in (3) and differentiating both side with respect to x we obtain y (0) y (n) (x τ) = (x τ c) n, n! y () (x τ) = y () (x τ) = y (N) (x τ) = n= n= n=n or the matrix form for x = c y (n) (n )! (x τ c)n, y (n) (n )! (x τ c)n, y (n) (n N)! (x τ c)n N (0) Y τ = X τ Y, () where ( τ) ( τ) ( τ) N 0!!! N! ( τ) ( τ) N 0 0!! (N )! ( τ) N X τ = 0 0 0! (N )! ! (N+)x(N+) Putting relation () in (9), the matrix representation becomes [Pr (x)y(r) (x τ)]=xp r X τy () Let the function f(x)be approximated by a truncated Taylor series f(x)= f (n) n! (x c) n Then we can put this series in the matrix form [f(x)]=xm 0 F, (3)

6 354 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) where F =[f (0) f () f (N) ] T Substituting the matrix forms (7), () and (3) corresponding to the functions P (x)y () (x), P r (x)y(r) (x τ) and f(x), into Eq (), and then simplifying the resulting equation, we have the matrix equation ( m P + =0 ) R Pr X τ Y = M 0 F (4) r=0 The matrix equation (4) is a fundamental relation for mth-order linear differential-difference equation with variable coefficients () On the other hand, if we tae (+τ) instead of ( τ) in Eq () we can obtain the fundamental relation, as (4), of the equation m P (x)y () (x) + =0 R r=0 P r (x)y(r) (x + τ) = f(x), τ > 0 (5) Next, we can obtain the corresponding matrix forms for conditions () as follows Using relation (0), we find the matrix representations of the functions in (), for the points a, b and c, in the forms [y () (a)]=pm Y, (6) [y () (b)]=qm Y, (7) [y () ]=RM Y, (8) where P =[ (a c) (a c) (a c) 3 (a c) N ], Q =[ (b c) (b c) (b c) 3 (b c) N ], R =[ ],

7 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) ! ! M = (N )! (N+)x(N+) Substituting the matrix representations (6) (8) into Eq (), we obtain the matrices system m =0 {a i P + b i Q + c i R}M Y =[μ i ] (9) Let us define U i as m U i = {a i P + b i Q + c i R}M [u i0 u i u in ], i = 0()m (0) =0 Thus, the matrix forms of conditions () become U i Y =[μ i ], i = 0,,,m () 3 Method of solution Let us consider the fundamental matrix equation (4) corresponding to the mth-order linear differentialdifference equation with variable coefficients () We can write Eq (4) in the form WY = M 0 F, () where ( m W =[w ij ]= P + =0 ) R Pr X τ, i = 0()N, j = 0()N (3) r=0

8 356 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) The augmented matrix of Eq () becomes w 00 w 0 w 0N ; w 0 w w N ; [W;M 0 F]= w N0 w N w NN ; f (0) 0! f ()! f (N) N! (4) We nowconsider the matrix equations () corresponding to conditions () Then the augmented matrices of Eqs () become [U i ;μ i ]=[u i0 u i u in ; μ i ], i = 0()(m ), (5) where the elements u i0,u i,,u in are defined in relation (0) Consequently, to find the unnown Taylor coefficients y (n), ()N, related with the approximate solution of the problem consisting of Eq () and conditions (), by replacing the m rowmatrices (5) by the last m rows of augmented matrix (4), we have new augmented matrix f (0) w 00 w 0 w 0N ; 0! f () w 0 w w N ;! ; [W ;F ]= f (N m) w N m,0 w N m, w N m,n ; (N m)! u 00 u 0 u 0N ; μ 0 u 0 u u N ; μ ; u m,0 u m, u m,n ; μ m or the corresponding matrix equation W Y = F (6)

9 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) so that w 00 w 0 w 0N w 0 w w y (0) N y () W w N m,0 w N m, w N m,n y () =, Y = u 00 u 0 u 0N, u 0 u u N y (N) u m,0 u m, u m,n [ F f = (0) 0! f ()! If det W = 0, we can write Eq (6) as Y = (W ) F f (N m) (N m)! μ 0 μ μ m ] T and the matrix Y is uniquely determined Thus the mth-order linear differential-difference equation with variable coefficients () with conditions () has a unique solutionthis solution is given by the truncated Taylor series y(x) = y (n) n! (x c) n (7) In the augmented matrix [W ;F ],ifwetaeu ij = 0 and μ i = 0, we may obtain the general solution of Eq () In the augmented matrix [W;M 0 F], ifdetw = 0, we may obtain the particular solution of Eq () We can easily chec the accuracy of this solution as follows: Since the Taylor polynomial (3) is an approximate solution of Eq (), when the solution y(x) and its derivatives are substituted in Eq (), the resulting equation must be satisfied approximately; that is, for x = x i [a,b] m R E(x i ) = P (x i )y () (x i ) + Pr (x i)y (r) (x i τ) f(x i ) 0 or =0 r=0 E(x i ) 0 i ( i is any positive integer) If max (0 i ) = 0 ( is any positive integer) is prescribed, then the truncation limit N is increased until the difference E(x i ) at each of the points becomes smaller than the prescribed 0

10 358 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) Numerical experiment In this section, we report on numerical results of some examples, selected differential-difference equations, solved by matrix method described in this paper For Examples 4, we have reported in Tables 5, the values of exact solution y(x), polynomial approximate solution y N (x), absolute error y(x) y N (x) and estimation error (denoted by exact, Present, e N,E N, respectively) at selected points of the given interval Example Consider the following third-order linear differential-difference equation with constant coefficients: y (x) y (x) y(x) + (e )y (x ) + y (x ) + y(x ) = e 7 with conditions y(0) =, y (0) = 0, y (0) =, x 0 and approximate the solution y(x) by the Taylor polynomial y(x) = 5 n! y(n) x(x c) n, (8) Table Numerical results of Example N x Exact Present method e N E N N = E E-3 0E E E E- 033E E- 0786E E N = E E E E-4 000E E E E- 04E E- 009E- N = E E-6 058E E E E-4 076E E-3 058E E-3 03E-3

11 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) Table Numerical results of Example N x Exact Present method e N E N N = E E-4 075E E E E E E N = E E-4 004E E E E-3 00E E- 0794E E- 04E- N = E E E E-4 044E E-3 035E E-3 066E E E-3 Table 3 Numerical results of Example 3 N x Exact Present method e N E N N = E E E-3 05E E- 0856E E- 088E E- 054E- N = E E-3 000E E E- 003E E- 004E E- 07E- N = E E E E-3 00E E- 087E E- 0783E-

12 360 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) Table 4 The maximum error for ε = and for different δ values δ e N = 5 e N = 6 e N = Table 5 The maximum error for ε = and for different δ values δ e N = 5 e N = 6 e N = E where N = 5, c = 0, τ =, P 0 =, P (x) =, P 3 (x) =, P0 (x) =, P (x) =, P (x) = (e ), f(x)= e 7 Then, for N = 5, the matrix equation (4) becomes where [P 0 + P + P 3 + (P + P + P 3 )X ]Y = M 0 F, P 0 =, P = , P 3 = , P0 = ,

13 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) e e 0 0 P = , P = e e, X = , M 0 = , F =[ ] T, Y =[y (0) (0) y () (0) y () (0) y (3) (0) y (4) (0) y (5) (0)] T For the conditions y(0) =, y (0) = 0 and y (0) =, the augmented matrices become [U 0 ;μ 0 ]=[ ; ], [U ;μ ]=[ ; 0 ], [U ;μ ]=[ ; ] from (5) Using the matrices P 0, P, P 3, P0, P, P, X, M 0, and F, we find matrices W and F in (6) as e 3 e 8 + e 30 6 e e 3 e 8 + e W 7 = e 5 3 e, F =[ ] and thus the solution Y =[ ] T (9) 0

14 36 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) Exact N = 5 N = 7 N = 9 6 y(x) x Fig Numerical and exact solution of Example for various N Substituting the elements of column matrix (9) into (8), we obtain the approximate solution in terms of the Taylor polynomial of degree five about x = 0as y(x) = + 05x x x x 5 We use the absolute error to measure the difference between the numerical and exact solutions In Table the solutions obtained for N = 5, 7, 9 are compared with the exact solution y(x)= x + x + e x [6] (see Fig ) Example Secondly we can tae the problem y (x) + y (x) 4y(x) + y (x ) + y (x ) y(x ) = 6x + 0x + 8, y(0) =, y (0) = so that N = 4, c= 0, τ =, P 0 (x) = 4, P (x) =, P (x) =, P0 (x) =, P (x) =, P (x) =, f(x)= 6x + 0x + 8 Then for N = 4, the matrix equation is obtained as [P 0 + P + P + (P 0 + P + P )X ]Y = M 0 F Following the previous procedures, we find matrices W and F in (6) as W = , F =[8 0 6 ] T

15 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) and its solution Y =[ ] (30) By using the elements of column matrix (30), we obtain the solution y(x) = + x x x x4 The solutions obtained for N = 4, 6, 8 are compared with the exact solution y(x) = e x + x, which are given in Table Example 3 We nowconsider the equation with variable coefficients y (x) + xy (x) + xy(x) + y (x ) + y(x ) = e x, y(0) =, y (0) = The exact solution is y(x) = e x For numerical results see Table 3 Example 4 (Kadalbajoo and Sharma [4, Example ]) εy (x) + y (x δ) y(x) = 0, δ x 0, y(0) =, y() = The exact solution is y(x) = ( em )e mx + (e m )e m x e m e m, where m = + 4(ε δ) (ε δ) For numerical results, see Tables 4 and 5, m = + + 4(ε δ) (ε δ) 5 Conclusions High-order linear differential-difference equations with variable coefficients are usually difficult to solve analytically In many cases, it is required approximate solutions The present method is based on computing the coefficients in the Taylor expansion of solution of a linear differential-difference equation To get the best approximating solution of the equation, we tae more terms from the Taylor expansion of functions; that is, the truncation limit N must be chosen large enough From the tabular points shown in Table, it may be observed that the solution found for N = 7 shows close agreement for various values of x i In particular, the solution of Example for N = 8 shows a very close approximation to the exact solution at the points in interval x 0 In Example 4, we compare the results for different ε and δ values and it is seen that when ε, our results are in a good agreement with the exact solutionalso, if 0 < ε <, some modifications are required

16 364 M Gülsu, M Sezer / Journal of Computational and Applied Mathematics 86 (006) If the Taylor polynomial solutions are looed for about the points in the given conditions, we see that there exists a solution which is closer to the exact solution In the matrix [W;M 0 F], ifdetw = 0, we can obtain a particular solution of Eq () If the conditions given in () are not used and if ran[w] =ran[w;m 0 F]=N + in (), then by replacing the last m rows of the matrix [W;M 0 F] with zero, the general solution may be obtained A considerable advantage of the method is that Taylor coefficients of the solution are found very easily by using the computer programs We use the symbolic algebra program, Maple, to find the Taylor coefficients of the solution The method can be developed and applied to system of linear difference equations Also, the method may be used to solve integrodifferential-difference equations in the form m n P (x)y () (x) + Ph (x)y(h) (x h) =0 = f(x)+ λ b a h= p K i (x, t)y (i) (t) dt + μ i=0 x a q K j (x, t)y (j) (t) dt but some modifications are required Note that the presented method can be used for solving the differentialdifference equations with positive shift, too j=0 Acnowledgements The author thans the anonymous referees and the Editor, L Wuytac, for their very valuable discussions and suggestions, which led to a great improvement of the paper References [] DD Bainov, MB Dimitrova, AB Dishliev, Oscillation of the bounded solutions of impulsive differential-difference equations of second order, Appl Math Comput 4 (000) 6 68 [] J Cao, J Wang, Delay-dependent robust stability of uncertain nonlinear systems with time delay, Appl Math Comput 54 (004) [3] MK Kadalbajoo, KK Sharma, Numerical analysis of boundary-value problems for singularly-perturbed differentialdifference equations with small shifts of mixed type, J Optim Theory Appl 5 (00) [4] MK Kadalbajoo, KK Sharma, Numerical analysis of singularly perturbed delay differential equations with layer behavior, Appl Math Comput 57 (004) 8 [5] KP Kanwal, KC Liu, A Taylor expansion approach for solving integral equation, Internat J Math Ed Sci Tech 0 (3) (989) 4 44 [6] H Levy, F Lesman, Finite Difference Equations, The Macmillan Company, NewYor, 96 [7] S Nas, S Yalçinbas, M Sezer, A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations, Internat J Math Ed Sci Technol 3 () (000) 3 5 [8] TL Saaty, Modern Nonlinear Equations, Dover Publications, Inc, NewYor, 98 [9] M Sezer, Taylor polynomial solution of Volterra Integral equations, Internat J Math Ed Sci Technol 5 (5) (994) [0] M Sezer, A method for approximate solution of the second order linear differential equations in terms of Taylor polynomials, Internat J Math Ed Sci Technol 7 (6) (996) [] H Zuoshang, Boundness of solutions to functional integro-differential equations, Proc Amer Math Soc 4 () (99)