ON DISCRETE ANALOGUES OF NONLINEAR IMPLICIT DIFFERENTIAL EQUATIONS

ON DISCRETE ANALOGUES OF NONLINEAR IMPLICIT DIFFERENTIAL EQUATIONS

PHAM KY ANH AND LE CONG LOI

Received 16 February 2005; Revised 26 September 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 forth.

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. InSection 2we 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, Pages1–19 DOI10.1155/ADE/2006/43092

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 results in [1].Section 3 deals with the unique solvability of a discretized problem for degenerated parabolic equations. InSection 4two numerical examples are given and finallySection 5summa- 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

fx(t),x(t),t=0, t∈J:=

t0,T, (2.1)

where the function f :R^m×R^m×J→R^m is continuous intand 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 projectionQ∈C¹(J,R^m×m) such that

Q²(t)=Q(t); ImQ(t)=ᏺ(t) ∀t∈J. (2.2) (ii) the matrixG(y,x,t) :=(∂ f /∂y)(y,x,t) + (∂ f /∂x)(y,x,t)Q(t) is nonsingular∀y,

x∈R^mand∀t∈J.

Together with (2.1) we consider a nonlinear IDE fn

xn+1,xn

=0 (n≥0), (2.3)

where the functions fn:R^m×R^m→R^mare supposed to be continuously differentiable.

We recall the following definition.

Definition 2.1 ([2, Definition 3.2]). Equation (2.3) is called an index-1 IDE if

(i) the subspacesᏺn:=Ker(∂ fn/∂y)(y,x) are independent ofy,x∈R^mand have the same dimension, that is, dimᏺn=m−rfor some integerrbetween 1 andm−1, (ii) the matricesG_n(y,x) :=(∂ f_n/∂y)(y,x) + (∂ f_n/∂x)(y,x)Q_n−1,nare nonsingular for all y,x∈R^m andn≥0, where the so-called connecting operatorsQn−1,nare de- fined as follows.

LetQn−1 andQnbe arbitrary projections onto subspacesᏺn−1 andᏺn, respectively.

Then Q_n−1=V_n−1QV _n⁻₋¹₁ and Q_n=V_nQV _n⁻¹, where V_n−1,V_n are nonsingular matri- ces andQ=diag(Or,Im−r). HereOr andIm−r stand for zero and identity matrices, re- spectively. We define an operator connecting two subspaces ᏺn−1 and ᏺnas Qn−1,n= V_n−1QV _n⁻¹. For definiteness, we putᏺ−1:=ᏺ0;Q₋1:=Q0andV₋1:=V0.

It can be verified (cf. [2]) that the matricesGn(y,x) are nonsingular if and only if Sn(y,x)∩ᏺn−1= {0} ∀y,x∈R^m;∀n≥0, (2.4) where, as in the DAE case,Sn(y,x) denotes the set

ξ∈R^m:∂ fn

∂x(y,x)ξ∈Im∂ fn

∂y(y,x)

. (2.5)

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 xn+1−xn

τ ,x_n,t_n

=0, n=0,N−1, (2.6)

wheretn=t0+nτ;τ:=(T−t0)/N,n=0,N. The following theorem ensures the compat- ibility of index notions for DAE (2.1) and IDE (2.6).

Theorem 2.2. Suppose the DAE (2.1) is of index-1 and the matricesG⁻¹(y,x,t) and (∂ f /

∂x)(y,x,t) are uniformly bounded. Then for sufficiently smallτ, the discretized equation (2.6) is also an index-1 IDE.

Proof. For the proof of the theorem we first reduce (2.6) to its normal form (2.3). Then we will show that (2.3) is of index-1 by verifying all the conditions ofDefinition 2.1. Let the DAE (2.1) be of index-1. Then the null-spaceᏺ(t)=Ker(∂ f /∂y)(y,x,t) does not depend on y,x∈R^m and is smooth int. In particular, dimᏺ(t)≡m−rfor some integerrbe- tween 1 andm−1. Further, the matrixG(y,x,t) :=(∂ f /∂y)(y,x,t) + (∂ f /∂x)(y,x,t)Q(t), whereQ(t)₌V(t)QV ⁻¹(t) is a smooth projection onᏺ(t), is nonsingular.

To reduce (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

∂ fn

∂y(y,x)=1 τ

∂ f

∂y

(y−x)/τ,x,tn ,

∂ fn

∂x(y,x)=∂ f

∂x

(y−x)/τ,x,tn

−1 τ

∂ f

∂y

(y−x)/τ,x,tn

.

(2.7)

Clearly, Ker(∂ f_n/∂y)(y,x)=Ker(∂ f /∂y)((y−x)/τ,x,t_n)=ᏺ(tn)≡ᏺn. Let P(t) :=I− Q(t);Qn:=Q(tn);Pn:=P(tn);Vn:=V(tn),n≥0 andᏺ−1:=ᏺ0;Q₋1:=Q0;V₋1:=V0. We define connecting operatorsQn−1,n=Vn−1QV _n⁻¹,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) :=∂ fn

∂y (y,x) +∂ fn

∂x(y,x)Q_n−1,n≡1

τH_n(y,x), (2.8)

where

Hn(y,x) :=∂ f

∂y((y−x)/τ,x,tn) +

τ∂ f

∂x

(y−x)/τ,x,tn

−∂ f

∂y

(y−x)/τ,x,tn Qn−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)

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 follows G¯⁻¹(y,x,t)=1

τ

τP(t) +Q(t)G⁻¹(y,x,t). (2.12) Now we will expressHn(y,x) in terms ofG((y−x)/τ,x,tn) and projectionsPn,Qn. Ob- serving that (∂ f /∂y)((y−x)/τ,x,tn)Qn=0 andQn−1,n−Qn=(Vn−1−Vn)QV _n⁻¹, after a short computation we find

Hn(y,x)=G¯(y−x)/τ,x,tn

×

I+τP_n+Q_nG⁻¹(y−x)/τ,x,t_n∂ f

∂x

(y−x)/τ,x,t_n−P_n

×

Vn−1−Vn QV_n⁻¹

.

(2.13)

By assumption, the matricesG⁻¹((y−x)/τ,x,tn) and (∂ f /∂x)((y−x)/τ,x,tn) are uni- formly bounded. Further, P(t), Q(t) and V⁻¹(t) are continuous, hence they are uni- formly bounded onJ. The smoothness ofV(t) on the compact segmentJimplies that

Vn−1−Vn c1τ, wherec1=maxt∈J V(t) . Thus the norm of the matrix Mn(y,x) :=

τPn+Qn

G⁻¹(y−x)/τ,x,tn∂ f

∂x

(y−x)/τ,x,tn

−Pn

Vn−1−Vn QV_n⁻¹ (2.14) will be bounded bycτ, where the constantcis determined by the bounds ofG⁻¹,∂ f /∂x, P,Q,V⁻¹andV. This fact implies the nonsingularity of the matrix

H¯n(y,x)=1

τG¯(y−x)/τ,x,tn

I+Mn(y,x), (2.15) providedτ < τ0:=1/c. The proof ofTheorem 2.2is complete.

Now for finding a solution of (2.1), satisfying the initial condition Pt0

xt0

−x⁰=0, (2.16)

we use the explicit Euler method, that is, we seek for a solution of (2.6) satisfying condi- tion

P0

x0−x⁰=0. (2.17)

Theorem 2.3. Under the assumptions ofTheorem 2.2, the explicit Euler method applied to the IVP (2.1), (2.16) does converge.

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 devoted to the similar decomposition for the discretized equation (2.6).

The last step deals with the convergence of the explicit Euler method.

Step 1. SinceQ(t) is a projection ontoᏺ(t)=Ker(∂ f /∂y)(y,x,t) it implies f(y,x,t)−fP(t)y,x,t=

1 0

∂ f

∂y

sy+(1−s)P(t)y,x,tQ(t)y ds=0, ∀y,x∈R^m;∀t∈J.

(2.18) Thus the IVP (2.1), (2.16) is reduced to the problem

fP(t)x(t),P(t)x(t) +Q(t)x(t),t=0, t∈J, Pt0

xt0

=Pt0

x⁰. (2.19)

Puttingu(t) :=P(t)x(t) we come to the equivalent IVP

fP(t)x(t),u(t) +Q(t)x(t),t=0, t∈J, (2.20) ut0

=Pt0

x⁰. (2.21)

To establish the convergence of the explicit Euler method, we will treat the DAE (2.20) in a slightly different way than that of [5,9]. SinceP(t) andQ(t) are smooth projections and dim(ImP(t))=r, dim(ImQ(t))=m−r,∀t∈J, there exist linear homeomorphisms

ξt:R^r−→ImP(t), ζt:R^m−r−→ImQ(t), (2.22) such thatξtandζtdepend continuously ont∈J.

For fixed ¯u∈R^mandt,t1,t2∈Jwe consider an operatorF_{t; ¯u;t}1,t2:R^m→R^mmapping everyz=(z^T₁,z^T₂)^T∈R^m, wherez1∈R^randz2∈R^m−r, into f(ξt1z1, ¯u+ζt2z2,t). For the sake of simplicity we denoteFt; ¯u:=Ft; ¯u;t,t. Thus

Ft; ¯u(z)= fξtz1, ¯u+ζtz2,t, (2.23) and the Frechet derivative ofFt; ¯u(z) is determined by

F_{t; ¯u}(z)w=∂ f

∂y

ξtz1, ¯u+ζtz2,tξtw1+∂ f

∂x

ξtz1, ¯u+ζtz2,tζtw2, (2.24)

wherew=(w₁^T,w^T₂)^T∈R^m,w1∈R^r,w2∈R^m−r. Consider an equation

F_{t; ¯u}(z)w=q, (2.25)

whereq∈R^m, or equivalently,

∂ f

∂y

ξtz1, ¯u+ζtz2,tξtw1+∂ f

∂x

ξtz1, ¯u+ζtz2,tζtw2=q. (2.26)

Observing thatξtw1∈ImP(t),ζtw2∈ImQ(t), henceP(t)ξtw1=ξtw1,Q(t)ζtw2=ζtw2, from (2.26) we find

∂ f

∂y

ξtz1, ¯u+ζtz2,tξtw1+∂ f

∂x

ξtz1, ¯u+ζtz2,tQ(t)ζtw2=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) we get

P(t)ξtw1+Q(t)ζtw2=G⁻¹ξtz1, ¯u+ζtz2,tq. (2.29) Multiplying both sides of (2.29) byP(t) andQ(t), respectively we find

ξ_tw1=P(t)G⁻¹ξ_tz1, ¯u+ζ_tz2,tq,

ζtw2=Q(t)G⁻¹ξtz1, ¯u+ζtz2,tq. (2.30) Thus (2.25) has a unique solutionw=(w^T₁,w^T₂)^T. Moreover,

w cw1+w2c q (2.31)

for some positive constantcsinceG⁻¹(ξ_tz1, ¯u+ζ_tz2,t),P(t),Q(t),ξ_t⁻¹,ζ_t⁻¹are uniformly bounded. It follows that

F_{t; ¯u}(z)⁻¹c. (2.32)

By the H’adamard theorem on homeomorphism (see [4,10]),F_{t; ¯u}is a homeomorphism betweenR^mandR^m. For fixed ¯u∈R^mandt∈J, the equation

Ft; ¯u(z)=0 (2.33)

has a unique solution

z=ϕ( ¯u,t)=

ϕ^T₁( ¯u,t),ϕ^T₂( ¯u,t)^T, (2.34) whereϕ1( ¯u,t)∈R^r,ϕ2( ¯u,t)∈R^m−r. Moreover, by the implicit function theorem,ϕ( ¯u,t) is continuously differentiable in ¯uand continuous intand

ϕ_u¯( ¯u,t)_{= −}F_{t; ¯u}(z)⁻¹∂ f

∂x

ξ_tz1, ¯u+ζ_tz2,t. (2.35)

The last relation shows that ϕu¯( ¯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_tϕ1

u(t),t, Q(t)x(t)=ζtϕ2

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) +ξtϕ1

u(t),t

=P(t)P(t)x(t) +Q(t)x(t)+ξ_tϕ1

u(t),t. (2.37)

Therefore, the IVP (2.1), (2.16) is equivalent to u(t)=P(t)u(t) +P(t)ζ_tϕ2

u(t),t+ξ_tϕ1

u(t),t, (2.38) ut0

=Pt0

x⁰=:u⁰, (2.39)

x(t)=u(t) +ζ_tϕ2

u(t),t. (2.40)

Let

ψ(u,t)=P(t)u+P(t)ζtϕ2(u,t) +ξtϕ1(u,t). (2.41) Clearly,ψ is continuously differentiable inuand continuous int. Moreover, the partial derivative ofψw.r.t.uis bounded, henceψ is Lipschitz continuous inu. It follows that the IVP (2.38), (2.39) and hence, the IVP (2.1), (2.16) has a unique solution onJ.

Step 2. Now we return to the discretized IVP (2.6), (2.17). Arguing as in DAE case, we rewrite (2.6) as

f Pnxn+1−xn

τ ,un+Qn−1xn,tn

=0, n=0,N−1, (2.42)

whereu_n:=P_n−1x_n. For a fixedn≥0 we consider the map

F_{tn;un;tn,tn−1}(z)=fξ_tnz1,u_n+ζ_tn−1z2,t_n, (2.43) wheret₋1:=t0.

Acting in the same manner as for (2.25), we realize that the equation

F_{tn;un;tn,tn−1}(z)w=q, (2.44) wherew₌(w₁^T,w^T₂)^T andw1∈R^r,w2∈R^m−r, has the form

∂ f

∂y

ξtnz1,un+ζtn−1z2,tn

ξtnw1+∂ f

∂x

ξtnz1,un+ζtn−1z2,tn

ζtn−1w2=q. (2.45)

Since Qn−1ζtn−1z2=ζtn−1z2 and Qn−1,nQn,n−1=Qn−1, whereQn,n−1:=VnQV _n⁻₋¹₁, we can rewrite the last equation as

∂ f

∂y

ξtnz1,un+ζtn−1z2,tn

ξtnw1+∂ f

∂x

ξtnz1,un+ζtn−1z2,tn

Qn−1,nQn,n−1ζtn−1w2=q. (2.46)

Using the relations

G⁻_n¹ξtnz1,un+ζtn−1z2

∂ f

∂y

ξtnz1,un+ζtn−1z2,tn

=Pn, G⁻_n¹ξtnz1,un+ζtn−1z2

∂ f

∂x

ξtnz1,un+ζtn−1z2,tn

Qn−1,n=Qn,

(2.47)

whereGn(y,x) :=(∂ f /∂y)(y,x,tn) + (∂ f /∂x)(y,x,tn)Qn−1,n, we reduce (2.44) to the form P_nξ_tnw1+Q_n,n−1ζ_tn−1w2=G⁻_n¹ξ_tnz1,u_n+ζ_tn−1z2

q. (2.48)

Multiplying both sides of the last equation byPnandQn, respectively, and taking into account relations

P_nQ_n,n−1=O; P_nξ_tnw1=ξ_tnw1;

Qn,n−1ζtn−1w2=VnV_n⁻₋¹₁Qn−1ζtn−1w2=VnV_n⁻₋¹₁ζtn−1w2, (2.49) we get

ξtnw1=PnG⁻_n¹ξtnz1,un+ζtn−1z2 q, ζ_tn−1w2=Q_n−1,nG⁻_n¹ξ_tnz1,u_n+ζ_tn−1z2

q. (2.50)

Therefore, (2.44) has a unique solution. On the other hand, sinceG⁻¹(y,x,t) is uniformly bounded, we can prove thatG⁻_n¹(ξtnz1,un+ζtn−1z2) is also uniformly bounded and hence, w c1 q , where c1 is a positive constant, therefore [F_{tn;un;tn,tn−1}(z)]⁻¹ is uniformly bounded. Using similar ideas as those employed to reduce (2.20) to the system (2.38) and (2.40), we can apply the H’adamard theorem to (2.42) to get

P_nxn+1−xn

τ ⁼ξ_tnϕ1

u_n,t_n, Qn−1xn=ζtn−1ϕ2

un,tn ,

(2.51) or

Pnxn+1=Pnxn+τξtnϕ1

un,tn

; Qn−1xn=ζtn−1ϕ2

un,tn

. (2.52)

Using the identity P_nx_n=

P_n−P_n−1

P_n−1x_n+P_n−P_n−1

Q_n−1x_n+P_n−1x_n (2.53)

we rewrite the IVP (2.6), (2.17) as un+1=

Pn−Pn−1

un+Pn−Pn−1 ζtn−1ϕ2

un,tn

+un+τξtnϕ1 un,tn

, (2.54)

u0=u⁰:=P0x⁰, (2.55)

x_n=u_n+ζ_tn−1ϕ2

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_nu¯n+P_nζtnϕ2

u¯n,tn +ξtnϕ1

u¯n,tn

, n=0,N−1,

¯

u0=u⁰:=P0x⁰,

(2.57)

whereP_n:=P(t_n), or

¯

u_n+1=τP_nu¯_n+τP_nζ_tnϕ2

u¯_n,t_n+τξ_tnϕ1

u¯_n,t_n+ ¯u_n (n=0,N−1), (2.58)

¯

u0=u⁰. (2.59)

From (2.40), (2.56), it follows that x(tn)−xn=utn

+ζtnϕ2

utn

,tn

−un−ζtn−1ϕ2

un,tn

=

ut_n−u¯_n+ζ_tnϕ2

ut_n,t_n−ϕ2

u¯_n,t_n+ζ_tn−ζ_tn−1

ϕ2

u¯_n,t_n +u¯n−un

+ζtn−1

ϕ2

¯ un,tn

−ϕ2 un,tn

(n=0,N).

(2.60)

Clearly, the explicit Euler method for the IVP (2.38), (2.39) is convergent, that is, u¯n−utn=O(τ), n=0,N. (2.61)

Further, the partial derivative ofϕ2w.r.t.uis uniformly bounded andζ_tis continuous on J, therefore we get

ζtn

ϕ2

utn ,tn

−ϕ2

u¯n,tn=O(τ) (n=0,N). (2.62) On the other hand, since ¯unis bounded,ϕ2is continuous andζtis uniformly continuous onJ, we come to the conclusion that if

u_n−u¯_n−→0 (τ−→0) (2.63)

then

xt_n−x_n−→0 (τ−→0). (2.64)

From (2.54), (2.58) we have u_n+1−u¯_n+1=

u_n−u¯_n+P_n−P_n−1

u_n−u¯_n +τξtn

ϕ1

un,tn

−ϕ1

u¯n,tn

+P_n−P_n−1

ζ_tn−1

ϕ2

u_n,t_n−ϕ2

u¯_n,t_n +Pn−Pn−1−τP_nu¯n+Pn−Pn−1−τP_nζtn−1ϕ2

u¯n,tn

+τP_nζtn−1−ζtn

ϕ2

¯ un,tn

,

(2.65)

this implies that

un+1−u¯n+1un−u¯n+Pn−Pn−1un−u¯n+τL1un−u¯n +L2P_n−P_n−1u_n−u¯_n+P_n−P_n−1−τP_nu¯_n +Pn−Pn−1−τP_nζtn−1ϕ2

u¯n,tn +τP_nζtn−1−ζtn

ϕ2

u¯n,tn,

(2.66)

whereL1,L2are positive constants satisfying ξtn

ϕ1

un,tn

−ϕ1

u¯n,tnL1un−u¯n, ζ_tn−1

ϕ2

u_n,t_n−ϕ2

u¯_n,t_nL2u_n−u¯_n. (2.67) Puttingα_n:= u_n−u¯_n ,a_n:=1 + P_n−P_n−1 +τL1+L2 P_n−P_n−1 and observing that

γn:=Pn−Pn−1−τP_nu¯n+Pn−Pn−1−τP_nζtn−1ϕ2

u¯n,tn +τP_nζ_tn−1−ζ_tnϕ2

u¯_n,t_n

=o(τ) (n=0,N), α0=0,

(2.68)

we get the estimate

αn+1ⁿ⁻ 1 k=0

_n

i=k+1

ai

γk+γn (n≥0). (2.69)

Sinceai1 +τL, whereLis a positive constant, we have n

i=k+1

ai(1 +τL)^n−k(1 +τL)ⁿe^nτLe^L(T−t0). (2.70) Thus we come to the estimate

α_n+1ne^L(T−t0)max

k γ_k+γ_n=o(τ)

τ , that is,u_n−u¯_n−→0 (τ−→0), (2.71)

as desired.Theorem 2.3is proved.

Theorems2.2and2.3for linear DAEs were proved in [1].