Parameter Fitted Scheme for Singularly Perturbed Delay Differential Equations

Transcription

1 International Journal of Applied Science and Engineering , 4: Parameter Fitted Sceme for Singularly Perturbed Delay Differential Equations Awoke Andargiea* and Y. N. Reddyb a b Department of Matematics; Bair Dar University, Bair Dar, Etiopia Department of Matematics, National Institute of Tecnology, Warangal, India Abstract: In tis paper, we presented a parameter fitted sceme to solve singularly perturbed delay differential equations of second order wit left and rigt boundary. In tis tecnique, approximating te term containing negative sift by Taylor series, we modified te singularly perturbed delay differential equations. We introduced a fitting parameter on te igest order derivative term of te modified problem. Te fitting parameter is to be determined from te sceme using te teory of singular Perturbation. Finally, we obtained a tree term recurrence relation tat can be solved using Tomas algoritm. Te applicability of te metod is tested by considering four linear problems (two problems on left layer and two problems on rigt layer). It is observed tat wen te delay parameter is smaller tan te perturbation parameter, te layer beavior is maintained. Keywords: Delay differential equations; singular perturbation; parameter fitted. 1. Introduction A singularly perturbed differential-difference equation is an ordinary differential equation in wic te igest derivative is multiplied by a small parameter and involving at least one delay or advance term. In recent papers [9-13] te terms negative or left sift and positive or rigt sift ave been used for delay and advance respectively. Te smootness of te solutions of suc singularly perturbed differential-difference equation deteriorates wen te parameter tends to zero. Suc problems are found trougout te literature on epidemics and population dynamics were tese small sifts play an important role in te modeling of various real life penomena [18]. Boundary value problems in differential-difference equations arise in a very natural way in studying variation problems in control teory were te problem is complicated by te effect of time delays in signal transmission [7]. Te differential-difference equation plays an important role in te matematical modeling of various practical penomena in te biosciences and control teory. Any system involving a feedback control will almost always involve time delays. Tese arise because a finite time is required to sense information and ten react to it. For a detailed discussion on differential-difference equation one may refer to te books and ig level monograps: Bellen[2], Driver[16], Bellman and Cooke [17]. Lange and Miura [5] gave an asymptotic approac in te study of a class of boundary value problems for linear second order differential-difference equations in wic te igest order * Corresponding autor; awoke248@yaoo.com 2013 Caoyang University of Tecnology, ISSN Received 1 June 2012 Revised 5 Marc 2013 Accepted 26 April

2 Awoke Andargie and Y. N. Reddy derivative is multiplied by a small parameter. It as been sown tat te layer beavior can cange its caracter and even be destroyed as te sifts increase but remain small. In [7], similar boundary value problems wit solutions tat exibit rapid oscillations are studied. Based on finite difference sceme, fitted mess and B-spline tecnique, piecewise uniform mes an extensive numerical work ad been initiated by M. K. Kadalbajoo and K. K. Sarma in teir papers [9-14] for solving singularly perturbed delay differential equations. In [8], M. Gulsu presented matrix metods for approximate solution of te second order singularly perturbed delay differential equations. It is well known tat te classical metods fail to provide reliable numerical results for suc problems (in te sense tat te parameter and te mess size cannot vary independently). Lange and Miura[3-5] gave asymptotic approaces in te study of class of boundary value problems for linear second order differential difference equations in wic te igest order derivative is multiplied by small parameter. Te aim of tis paper is to provide a parameter fitted sceme to solve singularly perturbed delay differential equations of second order wit left or rigt boundary. Te effect of small sifts on te boundary layer solution of te problem as been considered. In tis tecnique, by approximating te term containing negative sift by Taylor series expansion, we modify te singularly perturbed delay differential equations. We introduce a fitting parameter on te igest order derivative term of te new equation. Te fitting parameter is to be determined from te upwind sceme using te teory of singular Perturbation. We obtain a tree term recurrence relation tat can be solved using Tomas algoritm. Te applicability of te metod is tested by considering four problems wic ave been widely discussed in literature (two linear problems on left layer, two linear problems on rigt layer. It is observed tat wen te delay parameter is smaller tan te perturbation parameter, te layer beavior is maintained. 2. Te parameter fitted sceme To describe te metod, we first consider a linear singularly perturbed delay differential two-point boundary value problem of te form: y ( x ) a( x) y ( x ) b( x) y ( x) f ( x), 0 x 1 (1) wit y (0) ( x); x 0 (1a) and y (1) (1b) were is a small positive parameter (0< <<1), b( x ), f ( x ) are bounded functions in (0,1) and, are known constants. Furtermore, we assume tat a( x ) M 0 trougout te interval [0,1], were M is a positive constant. Under tese assumptions, (1) as a unique solution y(x) wic in general, displays a boundary layer of widt O( ) at x 0. Assuming (0) ; x 0, we can rewrite (1) in te form: y ( x ) a( x) y ( x ) b( x) y ( x) f ( x), 0 x 1 (2) wit y (0) ; x 0 (2a) and y (1) (2b) Approximating y ( x ) by te linear interpolation, we ave y ( x ) y ( x ) y ( x ) 362 (3)

3 Parameter Fitted Sceme for Singularly Perturbed Delay Differential Equations Substituting equation (3) in to equation (2), we get ( a ( x) y ( x ) a ( x ) y ( x ) b( x) y ( x) f ( x) (4) For Possible coices of suc tat 0 a( x) 1 (ere varies as, and ( x ) vary), from te teory of singular perturbations it is known tat te solution of (4) and (3) is of te form [O Malley [15]; pp 22-26]. x a ( x ) b( x ) dx a( x) a (0) y( x) y0 ( x) ( y0 (0)) e 0 a ( x) (5) O( ) were y0 (x ) is te solution of a ( x ) y 0 ( x ) b( x ) y 0 ( x ) f ( x ), y0 (1). By taking te Taylor s series expansion for a(x) and b(x) about te point 0 and restricting to teir first terms, (5) becomes, y ( x) y0 ( x ) ( y0 (0)) e a (0) b (0) x a (0) O( ) (6) Now we divide te interval [0, 1] into N equal parts wit constant mes lengt. 0 x0, x1,..., x N 1 be te mes points. Ten we ave xi i; i 0,1,..., N. From (6) we ave y (i) y0 ( x ) ( y0 (0)) e a 2 (0) b (0)) i a (0) and lim y (i) y0 (0) ( y0 (0)) e (7) a 2 (0) b (0) i a (0) (8) 0 were Let Te upwind sceme corresponding to equation (4) is a ( yi 1 2 yi yi 1 ) i ( y i 1 y i ) bi y i = f i ; i 1, 2,...N 1 2 (9) were a( xi ) ai ; b( xi ) bi ; f ( xi ) f i ; y( xi ) yi. Now, we introduce a fitting factor ( ) in te above sceme (9) as follows a ( ) ( yi 1 2 yi yi 1 ) i ( y i 1 y i ) bi y i = f i ; i 1, 2,...N 1 2 (10) wit y (0)= and y (1)=. Te fitting factor ( ) is to be determined in suc a way tat te solution of (10) converges uniformly to te solution of (1)-(2). Multiplying (10) by and taking te limit as 0; we get lim ( ) y (i ) 2 y ( i ) y ( i ) a ( i ) y ( i ) y (i ) 0 0 (11) 363

4 Awoke Andargie and Y. N. Reddy By substituting (5) in (11) and simplifying, we get te constant fitting factor ( a (0) 4 a 2 (0) b (0) ) a (0) [1 e ] a 2 (0) b(0) [sin(( ) / 2)]2 a (0) (12) Te equivalent tree term recurrence relation of equation (10) is given by: Ei yi 1 Fi yi G i yi 1 Hi ; i=1,2,3,.,n-1 a a were; Ei 2 ; Fi 2 i bi ; Gi 2 i ; H i f i 2 (13) Tis gives us te tri diagonal system wic can be solved easily by Tomas Algoritm. Tomas Algoritm A brief discussion on solving te tree term recurrence relation using Tomas algoritm wic also called Discrete Invariant Imbedding (Angel & Bellman [6])is presented as follows: Consider te sceme given in (13): Ei yi 1 Fi yi G i yi 1 Hi ; i=1,2,3,.,n-1 subject to te boundary conditions y0 y(0) ; and y N y(1) (13a) We set yi Wi yi 1 Ti for i = N-1, N-2,. 2, 1. were Wi W (x i ) and Ti T( x i ) wic are to be determined. From (13b), we ave (13b) yi 1 Wi 1yi Ti 1 (13c) By substituting (13c) in (13), we get Ei Wi 1yi Ti 1 Fi yi Gi yi 1 Hi. E T Hi Gi yi 1 i i 1 yi Fi E i Wi 1 Fi E i Wi 1 (13d) By comparing (13d) and (13b), we get te recurrence relations Gi Wi Fi Ei Wi 1 (13e) E T Hi. Ti i i 1 Fi Ei Wi 1 (13f) To solve tese recurrence relations for i=0,1,2,3,,n-1, we need te initial conditions for W0 and T0. For tis we take y0 W0 y1 T0. We coose W0 0 so tat te value of T0. Wit tese initial values, we compute Wi and Ti for i=1,2,3,.,n-1 from (13e-13f) in forward process, and ten obtain yi in te backward process from (13b)and (13a). Te conditions for te discrete invariant embedding algoritm to be stable are (see [1-2]): Ei 0, Fi 0, Fi Ei Gi and Ei Gi (13g) In tis metod, if te assumptions a( x) 0, b( x) 0 and ( a( x)) 0 old, one can easily sow tat te conditions given in (13g) old and tus te invariant imbedding algoritm is stable. 364

5 Parameter Fitted Sceme for Singularly Perturbed Delay Differential Equations 3. Rigt-end boundary layer We now assume tat a( x ) M 0 trougout te interval [0, 1], were M is some negative constant. Tis assumption merely implies tat te boundary layer for equation (1)-(2) will be in te neigborood of x=1. From te teory of singular perturbations it is known tat te solution of (4) and (2) is of te form [cf. O Malley [15]; pp 22-26]. 1 b( x) a( x) a ( x ) a ( x ) dx a(1) y ( x) y0 ( x) ( y0 (1)) e x a ( x) (14) O( ) were y0 (x ) is te solution of a ( x ) y 0 ( x ) b( x ) y 0 ( x ) f ( x ), y0 (0). Similar to te left layer problems, for possible coices of suc tat 0 a( x) 1, we ave y ( x) y0 ( x) ( y0 (1)) e a (1) b (1) (1 x ) a (1) O ( ) (15) From (15) we ave lim y (i) y0 (0) ( y0 (1)) e a 2 (1) b (1) 1 ( i ) a (1) (16) 0 were,, We introduce a fitting factor ( ) in te sceme corresponding to equation (4) a ( ) ( yi 1 2 yi yi 1 ) i ( yi yi 1 ) bi y i = f i ; 1 i N-1 2 (17) wit y (0)= and y (1)=. Multiplying (17) by and taking te limit as 0; we get te value of te fitting factor: ( a (0) 4 a 2 (1) b (1) ) a (1) [1 e ] 2 a (1) b(1) [sin(( ) / 2)]2 a (1) (18) Te equivalent tree term recurrence relation of equation (17) is given by: Ei yi 1 Fi yi G i yi 1 Hi ; i=1,2,3,.,n-1 (19) a a were; Ei 2 i ; Fi 2 i bi ; Gi 2 ; H i f i Tis gives us te tri diagonal system 2 wic can be solved easily by Tomas Algoritm. Description of te Metod To solve problems of te form given in equation (1)-(2), we followed te procedure: Step 1. Modify te Singularly Perturbed Delay Differential Equation (1) to te form (4). Step 2. Introduce te fitting parameter in (4) and determine its value using te teory of singular perturbation. 365

6 Awoke Andargie and Y. N. Reddy Step 3. Solve te tri-diagonal system (13) and (19) wit te boundary conditions (2) using Tomas Algoritm by taking different values of te perturbation and te delay parameter. 4. Numerical examples To demonstrate te applicability of te metod, we considered four numerical experiments (two problems wit left-end and two wit rigt-end boundary layer). We presented te absolute error compared to te exact solution of te problems. For te examples not aving te exact solution, te absolute error is calculated using te double mes principle. Example 1. Consider a singularly perturbed delay differential equation wit left layer: y ( x ) y ( x ) y ( x) 0; 0 [0,1] wit y (0) 1 and y (1) 1. Te exact solution is given by: y ( x ) were, m ( ) 2( ) [(1 e m2 )e m1x (e m1 1)e m2 x e m1 e m2 and m ( ). 2( ) Example 2. Now we consider an example of variable coefficient singularly perturbed delay differential equation wit left layer: y ( x ) e 0.5 x y ( x ) y ( x) 0; 0 [0,1] wit y (0) 1 and y (1) 1. Example 3. Consider a singularly perturbed delay differential equation wit left layer: y ( x) y ( x ) y ( x) 0; 0 [0,1] wit y(0) 1 and y(1) 1. Te exact solution is given by: y( x) were, m ( ) 2( ) [(1 e m2 )e m1x (e m1 1)e m2 x e m 2 e m1 and m ( ). 2( ) Example 4. Now we consider an example of variable coefficient singularly perturbed delay differential equation wit rigt layer: y ( x ) e x y ( x ) xy ( x) 0; 0 [0,1] wit y (0) 1 and y (1) Discussion and conclusions We presented a parameter fitted sceme to solve singularly perturbed delay differential equations of second order wit left and rigt boundary. Approximating te term containing negative sift by Taylor series, we modified te singularly perturbed delay differential equations. We introduced a variable fitting parameter on te igest order derivative term of te new 366

7 Parameter Fitted Sceme for Singularly Perturbed Delay Differential Equations equation and determined its value using te teory of singular Perturbation; O Malley [15]. We solved te resulting tree term recurrence relation using Tomas algoritm. To test te applicability of te new metod, four problems were considered by taking different values of te delay parameter, te perturbation parameter and te same mes size. We considered δ to increase from 0.1ε to 0.9ε. For left layer problems, it is observed tat as δ increase from 0.1ε to 0.9ε, ϒ decreases and absolute maximum error decreases for te constant coefficient problem and increases for te variable coefficient problem. For rigt layer problems, it is observed tat as δ increase from 0.1ε to 0.9ε, ϒ increases and absolute maximum error increases for te constant coefficient problem and decreases for te variable coefficients problem. Overall, it is observed tat wen te delay parameter is smaller tan te perturbation parameter, te layer beavior is maintained. Figure 1. Grap for solution of example 1. for ε=0.1 and different values of δ Table 1. Results for example 1. for ε=0.1, = δ= E-02 δ= E-03 δ= E-03 δ= E-03 δ= E

8 Awoke Andargie and Y. N. Reddy Figure 2. Grap for solution of example 1. for ε=0.01 and different values of δ Table 2. Results for example 1. for ε=0.01,= δ= E-03 δ= E-03 δ= E-03 δ= E-04 δ= E-04 Figure 3. Grap for solution of example 2. for ε=0.1 and different values of δ 368

9 Parameter Fitted Sceme for Singularly Perturbed Delay Differential Equations Table 3. Results for example 2. for (ε=0.1) δ= E-04 δ= E-03 δ= E-03 δ= E-03 δ= E-03 Figure 4. Grap for solution of example 2. for ε=0.01 and different values of δ Table 4. Results for example 2. for (ε=0.01) δ= E-04 δ= E-04 δ= E-04 δ= E-04 δ= E

10 Awoke Andargie and Y. N. Reddy Figure 5. Grap for solution of example 3. for ε=0.1 and different values of δ Table 5. Results for example 3. for (ε=0.1) δ= E-02 δ= E-02 δ= E-02 δ= E-02 δ= E-02 Figure 6. Grap for solution of example 3. for ε=0.01 and different values of δ 370

11 Parameter Fitted Sceme for Singularly Perturbed Delay Differential Equations Table 6. Results for example 3. for (ε=0.01) δ= E-03 δ= E-03 δ= E-03 δ= E-03 δ= E-03 Figure 7. Grap for solution of example 4. for ε=0.1 and different values of δ Table 7. Results for example 4. for (ε=0.1) Abs. Ma. Err. δ= E-03 δ= E-04 δ= E-04 δ= E-04 δ= E

12 Awoke Andargie and Y. N. Reddy Figure 8. Grap for solution of example 4. for ε=0.01 and different values of δ Table 8. Results for example 4. for (ε=0.01) δ= E-02 δ= E-03 δ= E-03 δ= E-03 δ= E-03 References [ 1] Andargie, A. and Reddy, Y. N An exponentially fitted special second order finite difference metod for solving singular perturbation problems. Applied Matematics and Computation, 190: [ 2] Bellen, A. and Zennaro, M Numerical Metods for Delay Differential Equations. Oxford University Press. Oxford. [ 3] Lange, C. G. and Miura, R. M Singular perturbation analysis of boundary value problems for differential difference equations, V. Small sifts wit layer beavior. SIAM J. Appl. Mat., 4: [ 4] Lange, C. G. and Miura, R. M Singular Perturbation Analysis of Boundary Value Problems for differential difference equations, (VI). Small sifts wit Rapid Oscillations. SIAM J. Appl. Mat., 54: [ 5] Lange, C. G. and Miura, R. M Singular perturbation analysis of boundary value 372

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