A Class of Multi-Scales Nonlinear Difference Equations

Transcription

1 Applied Mathematical Sciences, Vol. 12, 2018, no. 19, HIKARI Ltd, A Class of Multi-Scales Nonlinear Difference Equations Tahia Zerizer Mathematics Department, College of Sciences Jazan University, Jazan, Kingdom of Saudi Arabia Copyright c 2018 Tahia Zerizer. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. Abstract In this paper, we give an iteration algorithm to compute asymptotic solutions for a class of nonlinear difference equations containing small parameters of multiple scales. We consider two inds of perturbations. Mathematics Subject Classification: 39A10 Keywords: Perturbed difference equations, Multiple scales 1 Introduction Applied mathematicians and control scientists deal with practical systems involving nonlinear difference equations with small parameters of multiple time scales. The different scales arrange the convenience to reduce order and separate time-scale by using the singular perturbation methodology to reduce the complexity of these systems. Recently in [9], we developed an iterative method that gives asymptotic solutions for difference equations with one parameter, this procedure was initially introduced in various linear problems, see [3, 4, 5, 6, 7, 8]. In the present paper, we extend this procedure for a class of nonlinear equations containing several small multiscale parameters and we also allow variations for the boundary values. Let (E,. ) be a Banach space, E E be closed bounded, non-empty sets such that E +1 E. Let U l, : E 0 E r E, A : E 0 E r 1 E be p-differentiable

2 912 Tahia Zerizer functions, we consider the multi-scale difference equations j=1 εj U j, (x,, x +n+j ) + A (x,, x +n ) = 0, 0 N n m, (1) said of left end perturbation, satisfying the boundary conditions x = α (ε), = 0,, n 1, x N = β (ε), = 0,, m 1. (2) We assume that for ε δ < 1, α (ε) and β (ε) have the asymptotic representations α (ε) = α (0) + εα (1) + + ε p α (p), β (ε) = β (0) + εβ (1) + + ε p β (p). (3) The linear case of BVP (1) (2) is studied in [4]; in the next Section we prove under suitable assumptions, the existence and uniqueness of a solution x (ε), and we describe how to compute the coefficients of the asymptotic development x (ε) = x (0) + ε + ε 2 x (2) + + ε p + O(ε p+1 ). (4) In Section 3, the method is promptly extended to difference equations with a right-end perturbation. 2 Main Results 2.1 Reduced Problem Cancelling the parameter ε, the order of the difference equation in (1) drop to n providing the reduced problem x = α (0), = 0,, n 1, A (x,, x +n ) = 0, 0 N n m, (5) x N = β (0), = 0,, m 1, (6) with uncoupled boundary conditions, the values x 0,..., x N m can be recursively computed from the IVP (5) without needing the final conditions; pursuant to the singular perturbation theory of ODES, it is stated as singular perturbation and there is a boundary layer behavior at (6). Hypothesis 1. Suppose D n A (x (0) the ranges of the functions x A (x (0),, x(0) +n 1, x) 0, for all x E n, and,, x(0) +n 1, x), include zero. Proposition 2.1 If hypothesis 1 holds, then problem (5) (6) has a unique solution.

3 A class of nonlinear perturbed difference equations Preliminaries It is an asset that the BVP (1) (2) may be reconsidered as a system of equations depending on a parameter. Let X = (x 0,..., x N ), we introduce the function F (ε, X) = (F 0 (ε, X),..., F N (ε, X)), where F = x α (ε), = 0,, n 1, F +n = j=1 εj U j, (x,, x +n+j ) + A (x,, x +n ), = 0,, N n m, F N = x N β (ε), = 0,, m 1, BVP (1) (2) is equivalent to F (ε, x 0, x 1,..., x N ) = 0. By the Implicit Function Theorem [2], and under some assumptions, there exists a formal function of class C p, we denote g(ε) = (g 0 (ε),..., g N (ε)), such that F (ε, g(ε)) = 0, or j=1 εj U j, (g (ε),, g +n+j (ε)) + A (g (ε)..., g +n (ε)) = 0, = 0,..., N n m g (ε) = α (ε), = 0,, n 1, g N (ε) = β (ε), = 0,, m 1. (7) To find the coefficients of the Taylor polynomial expansion g (ε) = g (0) + εġ(0) 1! + ε 2 g (0) 2! + + ε p g(p) (0) + O(ε p+1 ), (8) p! we compute the sequential derivatives of (7) and we use the Faà Di Bruno Formula [1]. To give concise formulas, we drop the arguments and we denote the partial derivative f(x 0,x 1,...,x n) by D 0 0 D Dn n f. x 0 0 x xn n Lemma 2.2 Assume that the functions g, U j, and A satisfy (7), and that all the necessary derivatives are defined. Then we have for p m, n l=0 D la g (p) +l = 0 p 0 p 1 0 p m ( p 1)1!(p 1)!D p(1) 0 ( m)m!(p m)!d p p(m) 0 0 D p(0) n n p!d p(0) 0 p (i!)(0) i A p (g(i) )q(0) i0 (g (i) p n j=0 q(0) ij! 0...D p (1) U p 1 1, (g(i) )q(1) i0 (g (i) + )q(1) i p 1 p 1 (i!)(1) i j=0 q(1) ij! 0...D p (m) n+m n+m U m, (g(i) )q(m) i0 (i!)(m) i n+m j=0 q(m) ij! +n )q(0) in (g (i) +n+m )q(m) in+m. (9) Agreeing that g (p) = 0 for p < 0, the coefficients (l) i, q (l) ij nonnegative integer solutions of the Diophantine equations 0 (l) (l) (p l) (l) p l l i q(l) i0 + q(l) i1 + + q(l) i n+l = (l) p (l) j = q (l) 1j + q(l) 2j + + q(l) p l j = p l, l = 0,, m, and p (l), are all i, i = 1,, p l, l = 0,, m,, j = 0,, n + l, l = 0,, m,, l = 0,, m, (l) = p (l) 0 + p (l) p (l) n+l = (l) 1 + (l) (l) p l j (10)

4 914 Tahia Zerizer in 0 p m j=1 we fix (0) p = 0; the left side of (9) corresponds to (0) p = 1. Proof 2.3 By induction, we can easily prove for p 1, j l=0 ( p ) l j!ε j l (j l)! d p l U j, (g (ε),, g +n+j (ε)) + dp A (g (ε),, g +n (ε)) = 0, dε p l dε p we cancel the parameter ε in (11), we obtain m j=1 p j! dp j U j, (g (0),, g +n+j (0)) + dp A (g (0),, g +n (0)) = 0. j (dε) p j dε p (11) (12) We find the formula (9) by expanding Faa Di Bruno Formula [1] into (12), and by arranging the equation so that on the left hand side, we have the terms corresponding to p (0) = 1 in the diophantine equations (10). 2.3 Description of the Method It is understood that the coefficients of 0 th order in (4) satisfy the reduced problem (5) (6). To determine the coefficients of higher order, we substitute = g(p) (0), = 0,, N, (13) p! into (9), then 1 st order coefficients are defined by the iteration D n A (x (0) = α (1), = 0,..., n 1, l=0 D la (x (0),, x(0) + ), = 0,..., N n m,, = 0,..., m 1.,, x(0) +n )x(1) +n = n 1 U (x (0) N = β(1),, x(0) +n )x(1) +l (14) which starts from the initial values without needing the final conditions. The coefficents of 2 nd order satisfy x (2) = α (2), = 0,..., n 1, D n A x (2) +n = n 1 l=0 D la x (2) +l 0 l<r n D ld r A 2 +l l=0 D lu 1, +l U 2,(x (0) = 0,..., N n m, 1 2! n l=0 D2 1+l A x (2) N = β(2) = 0,..., m 1, +l x(1) +r,, x(0) +n+2 ), (15)

5 A class of nonlinear perturbed difference equations 915 the arguments are removed to give shorter formulas, the coefficients are computed recursively using the initial values and the 1 st order solution determined in the preceding stage. In general, agreeing that = 0 for p < 0, we have 0 = α (p) 0, 1 = α (p) 1,, n 1 = α (p) D n A +n = 0 p 1 0 p m 0 p n 1, D p(1) D p (1) U p 1 1, (x(i) )q(1) i0 (x (i) + )q(1) i p 1 j=0 q(1) ij! (m) D p(m) D p n+m n+m U m, (x(i) )q(m) i0 n+m j=0 q(m) ij! D p(0) 0 0 D p(0) n n A p (x(i) )q(0) i0 (x (i) +n )q(0) in p n j=0 q(0) ij! (x (i) +n+m )q(m) in+m n 1 l=0 D la +l, (16) the initial values are used in the iteration process while the final value are fixed N = β(p) = 0,..., m 1. (17) Algorithm. After completing the coefficient calculations, we replace in (4) then we find the desired p th order approximate solution for the BVP (1) (2). The process is validated in the following theorem. Theorem 2.4 If hypothesis 1 holds, then there exists ɛ > 0, such that for all ε < ɛ, the BVP (1) (2) has a unique solution (x (ε)) =0,,N satisfying (4); the coefficients x (0), x(1), x(2), x(n), are the solutions of the problems (5) (6), (14), (15), (16) (17), respectively. Proof 2.5 We denote: X := (ε, X), F X = (ε, F (ε, X)) and D F its Jacobian. From H 1 we deduce that F is locally invertible since the determinant of its Jacobian at X(0) is equal to ( ) N n m i=0 D n A i x (0) i,, x (0) i+n which does not cancel. Because D F is continuous, we have ρ > 0, X B X(0), ρ : D F X D F ( 1 X(0)) < 2 D F( X (0) ) 1. ρ Let ɛ = 2 D F( X, and G (0) ) 1 Y ( X) = X D F( X (0) ) 1 F( X) Y. For Y < ɛ, we verify that G Y is a contraction from B X(0), ρ to itself, so G Y has a unique fixed point. Therefore, there is a unique X in B X(0), ρ, such that Y = F( X). Let g(ε) = (g 0 (ε),, g N (ε)), ( if ε < ) ɛ, then for (ε, 0,, 0) B (0, ɛ), there is a unique (ε, g(ε)) in B X(0), ρ, such that (ε, 0,, 0) = F(ε, g(ε)). It means that the BVP (1) (2) has a unique solution for ε < ɛ. Moreover g is C p ( ɛ, ɛ) as are F and F 1. By the chain rule

6 916 Tahia Zerizer we have ġ(ε) = F(ε,g(ε)) ε Lemma F(ε,g(ε)) X, higher derivatives of g are given in Hypothesis 2. We assume that α (i) A, β (i) B, A and B are δ δ constants, the functions U and A are differentiable at any order (smooth). Theorem 2.6 If Hypothesis 2 holds, there exists ɛ > 0, such that for all ε < ɛ, the BVP (1) (2) has a unique solution (x (ε)) =0,,N satisfying x (ε) =, where x(0) are the solutions of the problems p=0 εp, x(1), x(2), x(p) (5) (6), (14), (15), (16) (17), respectively. 3 Right End Perturbation The results obtained in Section 2 can be easily extended to equations presenting a right-end perturbation. Consider the boundary value problem j=1 εj U j, (x,, x +n+j ) + A (x +n,, x +n+m ) = 0, 0 N n m (18) x = α (ε), = 0,, n 1, x N = β (ε), = 0,, m 1. (19) The linear case of (18) (19) is studied in [3, 4]. In an analogous way to the method developed for left-end perturbation, similar results lined to (18) (19) may be obtained without difficulties. Deleting the parameter in (18), follows x = α (0), = 0,, n 1, (20) A (x +n,, x +n+m ) = 0, 0 N n m, x N = β (0), = 0,, m 1. (21) The boundary layer behavior is located at the initial values (20), to solve (21) we compute bacward using the final values. Hypothesis 3. Suppose that D 0 A ( (x, x (0) +2,, x(0) +n+m ) ) 0 x E +n, and the ranges of the functions x A x, x (0) +,, x(0) +n+m contain 0. Proposition 3.1 If hypothesis 3 holds, then problem (21) has a unique solution. By the Implicit Function Theorem, we can find under some assumptions, a function g(ε) = (g 0 (ε),, g N (ε)), such that j=1 εj U j, (g (ε),, g +n+j (ε)) + A (g +n (ε),, g +n+m (ε)) = 0, = 0,..., N n m, g = α (ε), = 0,, n 1, g N (ε) = β, = 0,, m 1. (22)

7 A class of nonlinear perturbed difference equations 917 Lemma 3.2 Assume that the functions g, U and A satisfy (22), and that all the necessary derivatives are defined. Then we have for n 2, l=0 D la g (p) +n+l = 0 p 0 p 1 0 p m p!d p(0) 0 ( p 1)1!(p 1)!D p(1) D p (1) U 1, ( m)m!(p m)!d p p(m) 0 (i!)(m) i 0 D p(0) m m A p (g(i) +n )q(0) i0 (g (i) +n+m )q(0) in p (i!)(0) i p 1 (g(i) p 1 p 1 (i!)(1) i 0...D p (m) n+m n+m U m, n+m j=0 q(m) ij! p m j=0 q(0) ij! )q(1) i0 (g (i) + )q(1) i j=0 q(1) (g(i) ij!... )q(m) i0 Agreeing that g (p) = 0 for p < 0, the coefficients (l) i, q (l) ij nonnegative integer solutions of the Diophantine equations 0 (l) l i q(l) i0 + q(l) in 0 p the left side of equation (23). (g (i) +n+m )q(m) in+m. (23) and p (l), are all j (l) (p l) (l) p l = p l, l = 0,, m, 0 i q(0) i0 + q(0) i1 + + q(0) im = (0) i, i = 1,, p, i1 + + q(l) i n+l = (l) i, i = 1,, p l, l = 1,, m, p (l) j = q (l) 1j + q(l) 2j + + q(l) p l j, l = 0,, m, (24) we fix (0) p = 0; the case p (0) = 1 is omitted and corresponds to We can already indicate the main result of this section. From (8), (13) and (23), we deduce that the coefficients of the 1 st order development, satisfy = α (1) D 0 A (x (0) +n,, x(0) +n+m )x(1) l=1 D la (x (0) +n,, x(0) +n+m )x(1) +n+l N = β(1), = 0, n 1, +n = U 1,(x (0),, x(0) + ), = 0,..., N n m,, = 0,, m 1. (25) The coefficients n,,, N m are calculated bacward from the m 1 final values regardless of the initial value which are fixed at 0; the 0 th order solution is needed. In what follows, the arguments are removed. For 2 nd order development, we have the iterative process x (2) = α (2), = 0, n 1, D 0 A (x (0) +n,, x(0) +n+m )x(2) +n = l=1 D la (x (0) +n,, x(0) 0 l<r m D ld r A +n+l x(1) +n+r 1 2! l=0 D2 l A U 2, (x (0),, x(0) +n+2 ), = 0,..., N n m, x (2) N = β(2) ( +n+l +n+m ) )x(2) +n+l 2, = 0,, m 1. (26)

8 918 Tahia Zerizer For pth order development, p 2, agreeing that = 0 for p < 0, we have l=0 D la +n+l = 0 p 0 p 1 0 p m = α (p), = 0, n 1, D p(1) D p (1) 0 D p(0) m m A p (x(i) +n )q(0) i0 (x (i) +n+m )q(0) in p m j=0 q(0) ij! U p 1 1, (x(i) )q(1) i0 (x (i) + )q(1) i p 1... j=0 q(1) ij! (m) D p(0) 0 D p(m) D p n+m n+m U m, N = β(p) (x(i) )q(m) i0 n+m j=0 q(m) ij!, = 0,, m 1, (x (i) +n+m )q(m) in+m, (27) the computations are done bacward from the final values while the initial value remains fixed. The proof of the following theorems are left to the reader. Theorem 3.3 If Hypothesis 3 holds, then ɛ > 0, ε < ɛ, the BVP (18) (19) has a unique solution (x (ε)) =0,,N, which satisfy (4), where x (0)., x(2), x(p), are the solutions of (20) (21), (25), (26), (27), respectively. Theorem 3.4 If Hypothesis 2 holds, ɛ > 0, such that ε < ɛ, the BVP has a unique solution (x (ε)) =0,,N, which satisfy x (ε) = p=0 εp where x (0), x(1), x(2), x(p), are solutions of (20) (21), (25), (26), (27), respectively. 4 Conclusion, As the results show, the theory of singular perturbation for difference equations includes the same list of ingredients as for the singular perturbation theory for ODEs, a separation of time scales, an order reduction, and boundary layer phenomena. Instead of singularly perturbed differential equations, we can find homogeneous development for singularly perturbed difference equations. Initial value problems on finite time interval could be treated by the same methods. References [1] R. L. Mishov, Generalization of the Formula of Faa Di Bruno for a Composite Function with a Vector Argument, International Journal of Mathematics and Mathematical Sciences, 24 (2000),

9 A class of nonlinear perturbed difference equations 919 [2] S. G. Krantz, H. R. Pars, The Implicit Function Theorem, History, Theory, and Applications, Springer, Birhäuser, [3] T. Sari and T. Zerizer, Perturbations for Linear Difference Equations, Journal of Mathematical Analysis and Applications, 305 (2005), [4] T. Zerizer, Perturbation Method for Linear Difference Equations with Small Parameters, Advances in Difference Equations, 2006 (2006), [5] T. Zerizer, Perturbation Method for a Class of Singularly Perturbed Systems, Advances in Dynamical Systems and Applications, 9 (2014), no. 2, [6] T. Zerizer, Problems for a Linear Two-Time-Scale Discrete Model, Advances in Dynamical Systems and Applications, 10 (2015), no. 1, [7] T. Zerizer, Boundary Value Problems for Linear Singularly Perturbed Discrete Systems, Advances in Dynamical Systems and Applications, 10 (2015), no. 2, [8] T. Zerizer, Boundary Value Problem for a Three-Time-Scale Singularly Perturbed Discrete System, Dynamics of Continuous, Discrete and Impulsive Systems (Series A), 23 (2016), [9] T. Zerizer, A Class of Nonlinear Perturbed Difference Equations, International Journal of Mathematical Analysis, 12 (2018), no. 5, Received: July 21, 2018; Published: August 8, 2018