ON DISCRETE ANALOGUES OF NONLINEAR IMPLICIT DIFFERENTIAL EQUATIONS PHAM KY ANH AND LE CONG LOI Received 16 February 2005; Revised 26 Sep tember 2005; Accepted 27 September 2005 This paper deals with some classes of nonlinear implicit difference equations obtained via discretization of nonlinear differential-algebraic or partial differential-algebraic equa- tions. The unique solvability of discretized problems is proved and the compatibility be- tween index notions for nonlinear differential-algebraic equations and nonlinear implicit difference equations is studied. Copyright © 2006 P. K. Anh and L. C. Loi. This is an open access article distributed un- der the Creative Commons Attribution License, which permits unrestricted use, distri- bution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The importance of implicit difference equations (IDEs) seems to flow from two sources. First, in real world situations it has been found that many problems are modeled by sin- gular discrete systems, such as the Leslie population growth model, the Leontief dynamic model of multisector economy, singular discrete optimal control problems and so for th. Second, implicit discrete systems appear in a natural way of using discretization tech- niques for solving differential-algebraic equations (DAEs) and partial differential-algebraic equations (PDAEs). Recently [1, 2, 6], a class of implicit difference equations, called index-1 IDEs has been investigated. The solvability of initial-value problems (IVPs) as well as boundary-value problems (BVPs) associated with index-1 IDEs has been studied. In [1] a connection be- tween linear index-1 DAEs and linear index-1 IDEs has been revealed. In particular, the compatibility between index notions for linear index-1 DAEs and linear index-1 IDEs has been established. Until now, we have not found any results on the unique solvability of nonlinear im- plicit difference systems obtained via discretization by using explicit schemes for nonlin- ear DAEs and PDAEs. This problem will be studied in the paper. The paper is organized as follows. In Section 2 we show that the explicit Euler method applied to nonlinear index-1 DAEs leads to nonlinear index-1 IDEs. Moreover, the Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 43092, Pages 1–19 DOI 10.1155/ADE/2006/43092 2 On discrete analogues of nonlinear implicit differential equations convergence of the explicit Euler method for nonlinear index-1 DAEs is established. The results of this section are a “nonlinear version” of the corresponding re sults in [1]. Section 3 deals with the unique solvability of a discretized problem for degenerated parabolic equations. In Section 4 two numerical examples are given and finally Section 5 summa- rizes the main results of this work. 2. Compatibility of index notions for nonlinear DAEs and IDEs According to Griepentrog and M ¨ arz [5], a nonlinear DAE f  x  (t),x(t),t  = 0, t ∈ J :=  t 0 ,T  , (2.1) where the function f : R m × R m × J → R m is continuous in t and continuously differen- tiable in the first two variables, is said to be of index-1 if (i) the null-space Ker(∂f/∂y)(y,x,t) ≡ ᏺ(t) does not depend on y,x ∈ R m ,and there exists a smooth projection Q ∈ C 1 (J,R m×m )suchthat Q 2 (t) = Q(t); ImQ(t) = ᏺ(t) ∀t ∈ J. (2.2) (ii) the matrix G(y,x,t): = (∂f/∂y)(y, x,t)+(∂f/∂x)(y,x,t)Q(t) is nonsingular ∀y, x ∈ R m and ∀t ∈ J. Together with (2.1) we consider a nonlinear IDE f n  x n+1 ,x n  = 0(n ≥ 0), (2.3) where the functions f n : R m × R m → R m are supposed to be continuously differentiable. We recall the following definition. Definit ion 2.1 ([2, Definition 3.2]). Equation (2.3) is called an index-1 IDE if (i) the subspaces ᏺ n := Ker(∂f n /∂y)(y,x)areindependentofy,x ∈ R m and have the same dimension, that is, dimᏺ n = m− r for some integer r between 1 and m − 1, (ii) the matrices G n (y,x):= (∂f n /∂y)(y,x)+(∂f n /∂x)(y,x)Q n−1,n are nonsingular for all y,x ∈ R m and n ≥ 0, where the so-called connecting operators Q n−1,n are de- fined as follows. Let Q n−1 and Q n be arbitrary projections onto subspaces ᏺ n−1 and ᏺ n , respectively. Then Q n−1 = V n−1  QV −1 n −1 and Q n = V n  QV −1 n ,whereV n−1 , V n are nonsingular matri- ces and  Q = diag(O r ,I m−r ). Here O r and I m−r stand for zero and identity matrices, re- spectively. We define an operator connecting two subspaces ᏺ n−1 and ᏺ n as Q n−1,n = V n−1  QV −1 n . For definiteness, we put ᏺ −1 := ᏺ 0 ; Q −1 := Q 0 and V −1 := V 0 . It can be verified (cf. [2]) that the matrices G n (y,x) are nonsingular if and only if S n (y,x) ∩ ᏺ n−1 ={0}∀y,x ∈ R m ; ∀n ≥ 0, (2.4) where, as in the DAE case, S n (y,x) denotes the set  ξ ∈ R m : ∂f n ∂x (y,x)ξ ∈ Im ∂f n ∂y (y,x)  . (2.5) P. K. Anh and L. C. Loi 3 Since condition (2.4) does not depend on the choice of connecting operators, the cor- rectness of the index-1 notion for nonlinear IDEs is guaranteed. Now we discretize (2.1) by the explicit Euler scheme, namely f  x n+1 − x n τ ,x n ,t n  = 0, n = 0, N − 1, (2.6) where t n = t 0 + nτ; τ := (T − t 0 )/N, n = 0,N. The following theorem ensures the compat- ibility of index notions for DAE (2.1)andIDE(2.6). Theorem 2.2. Suppose the DAE (2.1) is of index-1 and the matrices G −1 (y,x,t) and (∂f/ ∂x)(y,x,t) are uniformly bounded. Then for sufficiently small τ, the discretized equation (2.6) is also an index-1 IDE. Proof. Fortheproofofthetheoremwefirstreduce(2.6) to its normal form (2.3). Then we will show that (2.3) is of index-1 by verifying all the conditions of Definition 2.1.Letthe DAE (2.1) be of index-1. Then the null-space ᏺ(t) = Ker(∂f/∂y)(y,x,t) does not depend on y, x ∈ R m and is smooth in t. In particular, dimᏺ(t) ≡ m − r for some integer r be- tween 1 and m − 1. Further, the matrix G(y,x,t):= (∂f/∂y)(y,x,t)+(∂f/∂x)(y,x,t)Q(t), where Q(t) = V(t)  QV −1 (t) is a smooth projection on ᏺ(t), is nonsingular. To re duce ( 2.6)to(2.3), we put f n (y,x):= f ((y − x)/τ,x, t n )(n = 0,N − 1). First ob- serve that ∂f n ∂y (y,x) = 1 τ ∂f ∂y  (y − x)/τ,x,t n  , ∂f n ∂x (y,x) = ∂f ∂x  (y − x)/τ,x,t n  − 1 τ ∂f ∂y  (y − x)/τ,x,t n  . (2.7) Clearly, Ker(∂f n /∂y)(y,x) = Ker(∂f/∂y)((y − x)/τ,x,t n ) = ᏺ(t n ) ≡ ᏺ n .LetP(t):= I − Q(t); Q n := Q(t n ); P n := P(t n ); V n := V(t n ), n ≥ 0andᏺ −1 := ᏺ 0 ; Q −1 := Q 0 ; V −1 := V 0 . We define connecting operators Q n−1,n = V n−1  QV −1 n , n ≥ 0. To prove the index-1 prop- erty of (2.6 ) we have to verify the nonsingularity of the matrix ¯ H n (y,x):= ∂f n ∂y (y,x)+ ∂f n ∂x (y,x)Q n−1,n ≡ 1 τ H n (y,x), (2.8) where H n (y,x):= ∂f ∂y ((y − x)/τ, x,t n )+  τ ∂f ∂x  (y − x)/τ,x,t n  − ∂f ∂y  (y − x)/τ,x,t n   Q n−1,n . (2.9) Letting ¯ G(y,x,t): = (∂f/∂y)(y,x,t)+τ(∂f/∂x)(y,x,t)Q(t) and using the relation Q(t) = G −1 (y,x,t) ∂f ∂x (y,x,t)Q(t), (2.10) 4 On discrete analogues of nonlinear implicit differential equations we find ¯ G(y,x,t) = G(y,x,t) − (1 − τ) ∂f ∂x (y,x,t)Q(t) = G(y,x,t)  I − (1 − τ)G −1 (y,x,t) ∂f ∂x (y,x,t)Q(t)  = G(y,x,t)  I − (1 − τ)Q(t)  = G(y,x,t)  P(t)+τQ(t)  . (2.11) From the identity (P(t)+τQ(t)) −1 ≡ (1/τ)(τP(t)+Q(t)), it fol lows ¯ G −1 (y,x,t) = 1 τ  τP(t)+Q(t)  G −1 (y,x,t). (2.12) Now we will express H n (y,x)intermsofG((y − x)/τ,x, t n )andprojectionsP n , Q n .Ob- serving that (∂f/∂y)((y − x)/τ,x,t n )Q n = 0andQ n−1,n − Q n = (V n−1 − V n )  QV −1 n , after a short computation we find H n (y,x) = ¯ G  (y − x)/τ,x,t n  ×  I +   τP n + Q n  G −1  (y − x)/τ,x,t n  ∂f ∂x  (y − x)/τ,x,t n  − P n  ×  V n−1 − V n   QV −1 n  . (2.13) By assumption, the matrices G −1 ((y − x)/τ,x,t n )and(∂f/∂x)((y − x)/τ,x,t n )areuni- formly bounded. Further, P(t), Q(t)andV −1 (t) are continuous, hence they are uni- formly bounded on J. The smoothness of V(t) on the compact seg ment J implies that V n−1 − V n   c 1 τ,wherec 1 = max t∈J V  (t). Thus the norm of the matrix M n (y,x):=   τP n + Q n  G −1  (y − x)/τ,x,t n  ∂f ∂x  (y − x)/τ,x,t n  − P n   V n−1 − V n   QV −1 n (2.14) will be bounded by cτ, where the constant c is determined by the bounds of G −1 , ∂f/∂x, P, Q, V −1 and V  . This fact implies the nonsingularity of the matrix ¯ H n (y,x) = 1 τ ¯ G  (y − x)/τ,x,t n  I + M n (y,x)  , (2.15) provided τ<τ 0 := 1/c.TheproofofTheorem 2.2 is complete.  Now for finding a solution of (2.1), satisfying the initial condition P  t 0  x  t 0  − x 0  = 0, (2.16) we use the explicit Euler method, that is, we seek for a solution of (2.6) satisfying condi- tion P 0  x 0 − x 0  = 0. (2.17) P. K. Anh and L. C. Loi 5 Theorem 2.3. Under the assumptions of Theorem 2.2, the explicit Euler method applied to the IVP (2.1), (2.16)doesconverge. Proof. The proof is divided into three steps. First, the H’adamard theorem is used for decomposing (2.1) into a system of an inherent ODE and an algebraic constraint. The second step is de voted to the similar decomposition for the discretized equation (2.6). The last step deals w i th the convergence of the explicit Euler method. Step 1. Since Q(t) is a projection onto ᏺ(t) = Ker(∂f/∂y)(y,x,t) it implies f (y,x,t) − f  P(t)y,x, t  =  1 0 ∂f ∂y  sy+(1−s)P(t)y,x, t  Q(t)yds=0, ∀y,x ∈R m ; ∀t∈J. (2.18) Thus the IVP (2.1), (2.16) is reduced to the problem f  P(t)x  (t),P(t)x(t)+Q(t)x(t),t  = 0, t ∈ J, P  t 0  x  t 0  = P  t 0  x 0 . (2.19) Putting u( t): = P(t)x(t)wecometotheequivalentIVP f  P(t)x  (t),u(t)+Q(t)x(t), t  = 0, t ∈ J, (2.20) u  t 0  = P  t 0  x 0 . (2.21) To establish the convergence of the explicit Euler method, we will treat the DAE (2.20)in aslightlydifferent way than that of [5, 9]. Since P(t)andQ(t) are smooth projections and dim(ImP(t)) = r, dim(ImQ(t)) = m− r, ∀t ∈ J, there exist linear homeomorphisms ξ t : R r −→ Im P(t), ζ t : R m−r −→ Im Q(t), (2.22) such that ξ t and ζ t depend continuously on t ∈ J. For fixed ¯ u ∈ R m and t,t 1 ,t 2 ∈ J we consider an operator F t; ¯ u;t 1 ,t 2 : R m → R m mapping every z = (z T 1 ,z T 2 ) T ∈ R m ,wherez 1 ∈ R r and z 2 ∈ R m−r ,into f (ξ t 1 z 1 , ¯ u + ζ t 2 z 2 ,t). For the sake of simplicity we denote F t; ¯ u := F t; ¯ u;t,t .Thus F t; ¯ u (z) = f  ξ t z 1 , ¯ u + ζ t z 2 ,t  , (2.23) and the Frechet derivative of F t; ¯ u (z)isdeterminedby F  t; ¯ u (z)w = ∂f ∂y  ξ t z 1 , ¯ u + ζ t z 2 ,t  ξ t w 1 + ∂f ∂x  ξ t z 1 , ¯ u + ζ t z 2 ,t  ζ t w 2 , (2.24) where w = (w T 1 ,w T 2 ) T ∈ R m , w 1 ∈ R r , w 2 ∈ R m−r . Consider an equation F  t; ¯ u (z)w = q, (2.25) 6 On discrete analogues of nonlinear implicit differential equations where q ∈ R m , or equivalently, ∂f ∂y  ξ t z 1 , ¯ u + ζ t z 2 ,t  ξ t w 1 + ∂f ∂x  ξ t z 1 , ¯ u + ζ t z 2 ,t  ζ t w 2 = q. (2.26) Observing that ξ t w 1 ∈ ImP(t), ζ t w 2 ∈ ImQ(t), hence P(t)ξ t w 1 = ξ t w 1 , Q(t)ζ t w 2 = ζ t w 2 , from (2.26)wefind ∂f ∂y  ξ t z 1 , ¯ u + ζ t z 2 ,t  ξ t w 1 + ∂f ∂x  ξ t z 1 , ¯ u + ζ t z 2 ,t  Q(t)ζ t w 2 = q. (2.27) Taking into account the relation (2.10)and G −1 (y,x,t) ∂f ∂y (y,x,t) = P(t), (2.28) from (2.27)weget P(t)ξ t w 1 + Q(t)ζ t w 2 = G −1  ξ t z 1 , ¯ u + ζ t z 2 ,t  q. (2.29) Multiplying both sides of (2.29)byP(t)andQ(t), respectively we find ξ t w 1 = P(t)G −1  ξ t z 1 , ¯ u + ζ t z 2 ,t  q, ζ t w 2 = Q(t)G −1  ξ t z 1 , ¯ u + ζ t z 2 ,t  q. (2.30) Thus (2.25) has a unique solution w = (w T 1 ,w T 2 ) T .Moreover, w  c    w 1   +   w 2     cq (2.31) for some positive constant c since G −1 (ξ t z 1 , ¯ u + ζ t z 2 ,t), P(t), Q(t), ξ −1 t , ζ −1 t are uniformly bounded. It follows that     F  t; ¯ u (z)  −1     c. (2.32) By the H’adamard theorem on homeomorphism (see [4, 10]), F t; ¯ u is a homeomorphism between R m and R m .Forfixed ¯ u ∈ R m and t ∈ J, the equation F t; ¯ u (z) = 0 (2.33) has a unique solution z = ϕ( ¯ u,t) =  ϕ T 1 ( ¯ u,t),ϕ T 2 ( ¯ u,t)  T , (2.34) where ϕ 1 ( ¯ u,t) ∈ R r , ϕ 2 ( ¯ u,t) ∈ R m−r . Moreover, by the implicit function theorem, ϕ( ¯ u,t) is continuously differentiable in ¯ u and continuous in t and ϕ  ¯ u ( ¯ u,t) =−  F  t; ¯ u (z)  −1 ∂f ∂x  ξ t z 1 , ¯ u + ζ t z 2 ,t  . (2.35) P. K. Anh and L. C. Loi 7 The last relation shows that ϕ  ¯ u ( ¯ u,t) is uniformly bounded because [F  t; ¯ u (z)] −1 is uni- formly bounded by (2.32)and(∂f/∂x)(y,x,t) is uniformly bounded by assumption. The application of the above mentioned arguments to (2.20)gives P(t)x  (t) = ξ t ϕ 1  u(t),t  , Q(t)x(t) = ζ t ϕ 2  u(t),t  . (2.36) On the other hand, u  (t) =  P(t)x(t)   = P  (t)x(t)+P(t)x  (t) = P  (t)x(t)+ξ t ϕ 1  u(t),t  = P  (t)  P(t)x(t)+Q(t)x(t)  + ξ t ϕ 1  u(t),t  . (2.37) Therefore, the IVP (2.1), (2.16)isequivalentto u  (t) = P  (t)u(t)+P  (t)ζ t ϕ 2  u(t),t  + ξ t ϕ 1  u(t),t  , (2.38) u  t 0  = P  t 0  x 0 =: u 0 , (2.39) x(t) = u(t)+ζ t ϕ 2  u(t),t  . (2.40) Let ψ(u,t) = P  (t)u + P  (t)ζ t ϕ 2 (u,t)+ξ t ϕ 1 (u,t). (2.41) Clearly, ψ is continuously differentiable in u and continuous in t. Moreover, the partial derivative of ψ w.r.t. u is bounded, hence ψ is Lipschitz continuous in u. It follows that the IVP (2.38), (2.39) and hence, the IVP (2.1), (2.16) has a unique solution on J. Step 2. Now we return to the discretized IVP (2.6), (2.17). Arguing as in DAE case, we rewrite (2.6)as f  P n x n+1 − x n τ ,u n + Q n−1 x n ,t n  = 0, n = 0, N − 1, (2.42) where u n := P n−1 x n .Forafixedn ≥ 0weconsiderthemap F t n ;u n ;t n ,t n−1 (z) = f  ξ t n z 1 ,u n + ζ t n−1 z 2 ,t n  , (2.43) where t −1 := t 0 . Acting in the same manner as for (2.25), we realize that the equation F  t n ;u n ;t n ,t n−1 (z)w = q, (2.44) where w = (w T 1 ,w T 2 ) T and w 1 ∈ R r , w 2 ∈ R m−r ,hastheform ∂f ∂y  ξ t n z 1 ,u n + ζ t n−1 z 2 ,t n  ξ t n w 1 + ∂f ∂x  ξ t n z 1 ,u n + ζ t n−1 z 2 ,t n  ζ t n−1 w 2 = q. (2.45) 8 On discrete analogues of nonlinear implicit differential equations Since Q n−1 ζ t n−1 z 2 = ζ t n−1 z 2 and Q n−1,n Q n,n−1 = Q n−1 ,whereQ n,n−1 := V n  QV −1 n −1 ,wecan rewrite the last equation as ∂f ∂y  ξ t n z 1 ,u n + ζ t n−1 z 2 ,t n  ξ t n w 1 + ∂f ∂x  ξ t n z 1 ,u n + ζ t n−1 z 2 ,t n  Q n−1,n Q n,n−1 ζ t n−1 w 2 = q. (2.46) Using the relations  G −1 n  ξ t n z 1 ,u n + ζ t n−1 z 2  ∂f ∂y  ξ t n z 1 ,u n + ζ t n−1 z 2 ,t n  = P n ,  G −1 n  ξ t n z 1 ,u n + ζ t n−1 z 2  ∂f ∂x  ξ t n z 1 ,u n + ζ t n−1 z 2 ,t n  Q n−1,n = Q n , (2.47) where  G n (y,x):= (∂f/∂y)(y,x,t n )+(∂f/∂x)(y,x,t n )Q n−1,n ,wereduce(2.44)totheform P n ξ t n w 1 + Q n,n−1 ζ t n−1 w 2 =  G −1 n  ξ t n z 1 ,u n + ζ t n−1 z 2  q. (2.48) Multiplying both sides of the last equation by P n and Q n , respectively, and taking into account relations P n Q n,n−1 = O; P n ξ t n w 1 = ξ t n w 1 ; Q n,n−1 ζ t n−1 w 2 = V n V −1 n −1 Q n−1 ζ t n−1 w 2 = V n V −1 n −1 ζ t n−1 w 2 , (2.49) we get ξ t n w 1 = P n  G −1 n  ξ t n z 1 ,u n + ζ t n−1 z 2  q, ζ t n−1 w 2 = Q n−1,n  G −1 n  ξ t n z 1 ,u n + ζ t n−1 z 2  q. (2.50) Therefore, (2.44) has a unique solution. On the other hand, since G −1 (y,x,t)isuniformly bounded, we can prove that  G −1 n (ξ t n z 1 ,u n + ζ t n−1 z 2 ) is also uniformly bounded and hence, w  c 1 q,wherec 1 is a positive constant, therefore [F  t n ;u n ;t n ,t n−1 (z)] −1 is uniformly bounded. Using similar ideas as those employed to reduce (2.20) to the system (2.38)and (2.40),wecanapplytheH’adamardtheoremto(2.42)toget P n x n+1 − x n τ = ξ t n ϕ 1  u n ,t n  , Q n−1 x n = ζ t n−1 ϕ 2  u n ,t n  , (2.51) or P n x n+1 = P n x n + τξ t n ϕ 1  u n ,t n  ; Q n−1 x n = ζ t n−1 ϕ 2  u n ,t n  . (2.52) Using the identity P n x n =  P n − P n−1  P n−1 x n +  P n − P n−1  Q n−1 x n + P n−1 x n (2.53) P. K. Anh and L. C. Loi 9 we rewrite the IVP (2.6), (2.17)as u n+1 =  P n − P n−1  u n +  P n − P n−1  ζ t n−1 ϕ 2  u n ,t n  + u n + τξ t n ϕ 1  u n ,t n  , (2.54) u 0 = u 0 := P 0 x 0 , (2.55) x n = u n + ζ t n−1 ϕ 2  u n ,t n  , n = 0, N. (2.56) Step 3. Together with (2.54), (2.55) we consider the explicit Euler scheme for the inherent ODE (2.38) ¯ u n+1 − ¯ u n τ = P  n ¯ u n + P  n ζ t n ϕ 2  ¯ u n ,t n  + ξ t n ϕ 1  ¯ u n ,t n  , n = 0, N − 1, ¯ u 0 = u 0 := P 0 x 0 , (2.57) where P  n := P  (t n ), or ¯ u n+1 = τP  n ¯ u n + τP  n ζ t n ϕ 2  ¯ u n ,t n  + τξ t n ϕ 1  ¯ u n ,t n  + ¯ u n (n = 0, N − 1), (2.58) ¯ u 0 = u 0 . (2.59) From (2.40), (2.56), it follows that x(t n ) − x n = u  t n  + ζ t n ϕ 2  u  t n  ,t n  − u n − ζ t n−1 ϕ 2  u n ,t n  =  u  t n  − ¯ u n  + ζ t n  ϕ 2  u  t n  ,t n  − ϕ 2  ¯ u n ,t n  +  ζ t n − ζ t n−1  ϕ 2  ¯ u n ,t n  +  ¯ u n − u n  + ζ t n−1  ϕ 2  ¯ u n ,t n  − ϕ 2  u n ,t n  (n = 0, N). (2.60) Clearly, the explicit Euler method for the IVP (2.38), (2.39) is convergent, that is,   ¯ u n − u  t n    = O(τ), n = 0,N. (2.61) Further, the partial derivative of ϕ 2 w.r.t. u is uniformly bounded and ζ t is continuous on J, therefore we get   ζ t n  ϕ 2  u  t n  ,t n  − ϕ 2  ¯ u n ,t n    = O(τ)(n = 0,N). (2.62) On the other hand, since ¯ u n is bounded, ϕ 2 is continuous and ζ t is uniformly continuous on J, we come to the conclusion that if   u n − ¯ u n   −→ 0(τ −→ 0) (2.63) then   x  t n  − x n   −→ 0(τ −→ 0). (2.64) 10 On discrete analogues of nonlinear implicit differential equations From (2.54), (2.58)wehave u n+1 − ¯ u n+1 =  u n − ¯ u n  +  P n − P n−1  u n − ¯ u n  + τξ t n  ϕ 1  u n ,t n  − ϕ 1  ¯ u n ,t n  +  P n − P n−1  ζ t n−1  ϕ 2  u n ,t n  − ϕ 2  ¯ u n ,t n  +  P n − P n−1 − τP  n  ¯ u n +  P n − P n−1 − τP  n  ζ t n−1 ϕ 2  ¯ u n ,t n  + τP  n  ζ t n−1 − ζ t n  ϕ 2  ¯ u n ,t n  , (2.65) this implies that   u n+1 − ¯ u n+1      u n − ¯ u n   +   P n − P n−1     u n − ¯ u n   + τ  L 1   u n − ¯ u n   +  L 2   P n − P n−1     u n − ¯ u n   +   P n − P n−1 − τP  n     ¯ u n   +   P n − P n−1 − τP  n     ζ t n−1 ϕ 2  ¯ u n ,t n    + τ   P  n      ζ t n−1 − ζ t n  ϕ 2  ¯ u n ,t n    , (2.66) where  L 1 ,  L 2 are positive constants satisfying   ξ t n  ϕ 1  u n ,t n  − ϕ 1  ¯ u n ,t n      L 1   u n − ¯ u n   ,   ζ t n−1  ϕ 2  u n ,t n  − ϕ 2  ¯ u n ,t n      L 2   u n − ¯ u n   . (2.67) Putting α n :=u n − ¯ u n , a n := 1+P n − P n−1  + τ  L 1 +  L 2 P n − P n−1  and observing that γ n :=   P n − P n−1 − τP  n     ¯ u n   +   P n − P n−1 − τP  n     ζ t n−1 ϕ 2  ¯ u n ,t n    + τ   P  n      ζ t n−1 − ζ t n  ϕ 2  ¯ u n ,t n    = o(τ)(n = 0,N), α 0 = 0, (2.68) we get the estimate α n+1  n−1  k=0  n  i=k+1 a i  γ k + γ n (n ≥ 0). (2.69) Since a i  1+τL,whereL is a positive constant, we have n  i=k+1 a i  (1 + τL) n−k  (1 + τL) n  e nτL  e L(T−t 0 ) . (2.70) Thus we come to the estimate α n+1  ne L(T−t 0 ) max k γ k + γ n = o(τ) τ , that is,   u n − ¯ u n   −→ 0(τ −→ 0), (2.71) as desired. Theorem 2.3 is proved.  Theorems 2.2 and 2.3 forlinearDAEswereprovedin[1]. [...]... ∈ Rk : B(x,t)ξ ∈ ImA(x,t) (3.6) Then the nonsingularity of G(x,t) is equivalent to the condition ᏿(x,t) ∩ ᏺ(t) = {0} ∀(x,t) ∈ J1 × J2 (3.7) 12 On discrete analogues of nonlinear implicit differential equations Since condition (3.7) is independent of the choice of Q(t), the nonsingularity of G(x,t) does not depend on the choice of Q(t) For simplicity, we can choose orthogonal projections Q(t) on ᏺ(t)... Explicit schemes applied to nonlinear DAEs and PDAEs lead to nonlinear IDEs In this paper the unique solvability of discretized problems obtained via discretization of nonlinear implicit differential equations is studied The compatibility between index notions for nonlinear index-1 DAEs and nonlinear index-1 IDEs as well as the convergence of the explicit Euler method for nonlinear index-1 DAEs were... number of iterations and the initial approximation for the fixed-point iteration ¯ (3.23) in both examples are μ = 50 and u(0) = 0, respectively 16 On discrete analogues of nonlinear implicit differential equations ×10−14 6 5 Errn 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 tn Figure 4.2 (Example 4.2) h = 0.05, τ = 0.0005 Figures 4.1 and 4.2 show the maximal values of the local error between the exact solution of (3.1)... ∀t ∈ J Next, calculating G(x,t) := A(x,t) + B(x,t)Q(t), we find ⎛ t+1 ⎜ G(x,t) = ⎜ 0 ⎝ −x2 −x 3t 2 + 1 / t 2 + 1 t2 + t / t2 + 1 xt ⎞ ⎟ t − t3 / t2 + 1 ⎟ ⎠ (t + 1)/ t 2 + 1 (4.7) 18 On discrete analogues of nonlinear implicit differential equations and detG(x,t) = (t + 1)2 + 2x3 t > 0 ∀(x,t) ∈ [0,1]2 It follows that G(x,t) is nonsingular for all (x,t) ∈ J × J Further, f (ξ,x,t) − f (ζ,x,t) = = cosξ1... solution, which can be approximated by iterations ¯ ¯ Eu(ν+1) = F u(ν) , ν = 0, ,μ − 1, ¯ where u(0) ∈ Rk(M −1) is an arbitrary vector and μ is a certain positive integer (3.23) 14 On discrete analogues of nonlinear implicit differential equations ¯ For this purpose we endow the space Rk(M −1) by the Euclidean norm, that is, if z = M −1 ¯ ∈ Rk(M −1) then z = ( m=1 zm 2 )1/2 Using the Gaussian elimination... between implicit difference equations and differential-algebraic equations, Acta Mathematica Vietnamica 29 (2004), no 1, 23–39 [2] P K Anh and H T N Yen, On the solvability of initial-value problems for nonlinear implicit difference equations, Advances in Difference Equations 2004 (2004), no 3, 195–200 [3] N S Bakhvalov, N P Zhidkov, and G M Kobel’kov, Numerical Methods, Nauka, Moscow, 1987 [4] M S Berger, Nonlinearity... Matha ematics 18 (1995), no 1–3, 267–292 [10] J M Ortega and W C Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970 Pham Ky Anh: Department of Mathematics, Vietnam National University, Hanoi, Vietnam E-mail address: anhpk@vnu.edu.vn Le Cong Loi: Department of Mathematics, Vietnam National University, Hanoi, Vietnam E-mail address: loilc@vnu.edu.vn... Theorem 3.1 Under the hypotheses (H1)–(H3) the discretized problem (3.4)–(3.5) has a unique solution, provided h is sufficiently small and τ = αh2 , α = const Proof From the above mentioned argument we see that the problem of finding solution of system (3.4)–(3.5) when τ, h are sufficiently small is reduced to the fixed-point problem ¯ ¯ u = H(u), (3.25) where H := E−1 F ¯ ¯ For any u, z ∈ Rk(M −1) , we... Lectures on Nonlinear Problems in Mathematical Analysis, Pure and Applied Mathematics, Academic Press, New York; Harcourt Brace Jovanovich, London, 1977 [5] E Griepentrog and R M¨ rz, Differential-Algebraic Equations and Their Numerical Treatment, a Teubner Texts in Mathematics, vol 88, BSB B G Teubner, Leipzig, 1986 [6] L C Loi, N H Du, and P K Anh, On linear implicit non-autonomous systems of difference... convergence of an explicit scheme for degenerated parabolic equations is now in progress Besides, the Floquet theory for linear index-1 IDEs and its applications to the stability theory for nonlinear IDEs has been established Acknowledgment The authors would like to express their gratitude to the referees, whose careful reading and numerous comments led to an improvement in style and presentation of the . 10.1155/ADE/2006/43092 2 On discrete analogues of nonlinear implicit differential equations convergence of the explicit Euler method for nonlinear index-1 DAEs is established. The results of this section are a nonlinear. classes of nonlinear implicit difference equations obtained via discretization of nonlinear differential-algebraic or partial differential-algebraic equa- tions. The unique solvability of discretized. ON DISCRETE ANALOGUES OF NONLINEAR IMPLICIT DIFFERENTIAL EQUATIONS PHAM KY ANH AND LE CONG LOI Received 16 February 2005;