A finite difference method on layer-adapted mesh for singularly perturbed delay differential equations

Fevzi Erdogan 1 , Mehmet Giyas Sakar 1  and Onur Saldır 1
  • 1 Department of Mathematics, Faculty of Science, Van, Turkey
Fevzi Erdogan
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  • Department of Mathematics, Faculty of Science, Van Yuzuncu Yil University, Van, Turkey
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, Mehmet Giyas Sakar and Onur Saldır

Abstract

The purpose of this paper is to present a uniform finite difference method for numerical solution of a initial value problem for semilinear second order singularly perturbed delay differential equation. A numerical method is constructed for this problem which involves appropriate piecewise-uniform Shishkin mesh on each time subinterval. The method is shown to uniformly convergent with respect to the perturbation parameter. A numerical experiment illustrate in practice the result of convergence proved theoretically.

1 Introduction

Consider an initial value problem for the semilinear second order singularly perturbed delay differential equation [12] in the interval Ī = [0,T]:

εu(t)+a(t)u(t)+f(t,u(t),u(tr))=0,tI,
u(t)=φ(t),tI0,
u(0)=A/ε,
where I=(0,T]=p=1mIp , Ip = {t : rp−1 <trp}, 1 ≤ pm and rs = sr, for 0 ≤ sm and I0 = (−r,0]. 0 < ɛ ≤ 1 is the perturbation parameter, a(t) ≥ α > 0, b(t), c(t), f (t) and ϕ(t) are given sufficiently smooth functions satisfying certain regularity conditions to be specified and r is a constant delay, which is independent of ɛ. Moreover
|fu|b*and|fv|c*.
Delay differential equations play an important role in the mathematical modelling of various practical phenomena in the biosciences and control theory. Any system involving a feedback control will almost always involve time delays. These arise because a finite time is required to sense information and then react to it. A singularly perturbed delay differential equation is an ordinary differential equation in which the highest derivative is multiplied by a small parameter and involving at least one delay term [1,2,3,4]. Such problems arise frequently in the mathematical modelling of various practical phenomena, for example, in the modelling of several physical and biological phenomena like the optically bistable devices [5], description of the human pupil-light reflex [6], a variety of models for physiological processes or diseases and variational problems in control theory where they provide the best and in many cases the only realistic simulation of the observed phenomena [7]. An overview of numerical treatment for first and second order singularly perturbed delay differential equations, may be obtained in [8,9,10,11,12,13,14,15,16] (see, also references therein).

The numerical analysis of singular perturbation cases has always been far from trivial because of the boundary layer behavior of the solution. Such problems undergo rapid changes within very thin layers near the boundary or inside the problem domain. It is well known that standard numerical methods for solving singular perturbation problems do not give satisfactory result when the perturbation parameter is sufficiently small. Therefore, it is important to construct suitable numerical methods for these problems, whose accuracy does not depend on the perturbation parameter, i.e. methods that are uniformly convergent with respect to the perturbation parameter [17,18,19,20,21,22].

In a singularly perturbed delay differential equation, one encounters with two difficulties, one is because of occurrence of the delay term and another one is due to presence of perturbation parameter. To overcome the first difficulty, we employed the numerical method of steps [2] for the delay argument which converted the problem to a initial value problem for a singularly perturbed differential equation. Then we constructed a numerical scheme based on finite difference method on non uniform Shishkin mesh for the numerical solution.

In the present paper we discretize (1)(2) using a numerical method, which is composed of an exponentially fitted difference scheme on piecewise uniform Shishkin mesh on each time-subinterval. In section 2, we state some important properties of the exact solution. In section 3, we describe the finite difference discretization and introduce the piecewise uniform mesh. In section 4, we present convergence analysis for approximate solution. Uniform convergence is proved in the discrete maximum norm. Some numerical results are being presented in section 5. The technique to construct discrete problem and error analysis for approximate solution is similar to those in [8,9,23,24]

Throughout the paper, C will denote a generic positive constant independent of ɛ and of the mesh parameter.

2 The Continuous Problem

Here we show some properties of the solution of (1)(3), which are needed in later sections for the analysis of the appropriate numerical solution. For any continuous function g(t), we use ‖g for the continuous maximum norm on the corresponding interval.

Lemma 2.1

Let δ (t) be nonnegative and continuous function such that

δ(t)δ0+0t{c0δ(τ)+c1δ(τr)}dτ,t>0,
δ(t)=δ0,rt0,
where δ0, c0 and c1 are nonnegative constants. Then it holds that
δ(t)γ0s=0p{γ1s(trp1)ss!},tIp,p=1,2,...,m,
where γ0 = δ0exp(c0T) and γ1 = c1exp(c0T).

Proof

See [11].

Lemma 2.2

Let a,b,c, fC1 (Ī),ϕC1 (Ī0). Then for the solution u(t) of problem (1)(3) the following estimates hold

u(t),I¯C,
|u(t)|C{1+1εexp(αtε)},tI,
|u(t)|C{1+(trp1)p1ε2exp(α(trp1)ε)},tIp,p=1,2,
|u(t)|C,tIp,3pm,

Proof

The semilinear equation (1) can be written in the form

εu(t)+a(t)u+b(t)u(t)+c(t)u(tr)=f(t,0,0),tI,
where
b(t)=fu(t,u˜,v˜),c(t)=fv(t,u˜,v˜),u˜=γu,v˜=γu(tr),(0<γ<1)intermediatevalue.

We rewrite (11) in the form

εu(t)+a(t)u(t)=F(t),
where
F(t)=f(t,0,0)b(t)u(t)c(t)u(tr).
From (12), we have the following relation for u(t)
u(t)=u(0)exp(1ε0ta(s)ds)+1ε0tF(τ)exp(1ετta(s)ds)dτ.
Integrating (13) from 0 to t, we have
u(t)=φ(0)+Aε10texp(1ε0sa(τ)dτ)ds++1ε0tds0sF(τ)exp(1ετta(λ)dλ)dτ.
By the change of integral bounds, we get
|u(t)||φ(0)|+|Aε10texp(1ε0sa(τ)dτ)ds|++1ε|0tdτ{F(τ)}τtexp(1ετsa(λ)dλ)ds|
Then
|u(t)||φ(0)|+α1(|A|+f1)+α10t(b|u(τ)|+c|u(τr)|)dτ

Applying Lemma 2.1, we obtain (7).

Due to ‖FC, it now follows from (13) that

|u(t)||u(0)|exp(1ε0ta(s)ds)+1ε0t|F(τ)|exp(1ετta(s)ds)dτ.C{|A|ε1exp(αtε)+α1(1exp(αtε)}
which proves (8)

Now, differentiating the Eq.(1) we have

εu′″(t)+a(t)u(t)=Φ(t),tIk+1,
where
Φ(t)=f(t,0,0)(a(t)+b(t))(t)u(t)b(t)u(t)c(t)u(tr)c(t)u(tr).
From (14) we have the following relation for u(t)
u(t)=u(rk)exp(1εrkta(s)ds)+1εrktΦ(τ)exp(1ετta(s)ds)dτ.
and it is easy to see that
|Φ(t)||f(t)|+|(a(t)+b(t))u(t)|+|b(t)u(t)|+|c(t)u(tr)|+|c(t)u(tr)|C(1+|(a(t)+b(t))u(t)|+|c(t)u(tr)|).

It follows from (1.1) that

|u(0)|Cε2.

Using the estimates (8), (16) and (17) in (21) for k = 0, we have

|u(t)|Cε2exp(αtε)+1εC0t(1+1εexp(ατε))exp(α(tτ)ε)dτC(1+1ε2exp(αtε)),tI1,
i.e., the inequality (9) is valid for p = 1.

Next, from the last inequality for t = r, we have

|u(r)|C
and thereby from (15) it follows that
|u(t)||u(r)|exp(αtε)+1εC0t(1+1εexp(α(τr)ε))exp(α(tτ)ε)dτC(1+1ε2(tr)exp(αtε)),tI2.
i.e. the inequality (9) is also valid for p = 2.

Since

|Φ(t)|C,tIp,p=3,...,m,
the estimate (10) follows immediately from (15).

3 Discretization and Mesh

In this section, we construct a numerical scheme for solving the initial value problem (1)(2).

Let ω¯N0 be any non-uniform mesh on Ī:

ω¯N0={0=t0<t1<...<tN0=T,hi=titi1}
which contains by N mesh point at each subinterval Ip(1 ≤ pm) :
ωN,p={ti:(p1)N+1ipN},1pm,
and consequently
ωN0=p=1mωN,p.

To simplify the notation we set wi = w(ti) for any function w(t), and moreover yi denotes an approximation of u(t) at ti. For any mesh function {wi} defined on ω¯N0, we use

wt¯,i=(wiwi1)/hi,wt,i=(wi+1wi)/hi+1,wt,i0=(wt¯,i+wt,i)/2,wt¯t,i=(wt,iwt¯,i)/hi,w,N,p=w,ωN,p:=max1iN|wi|.

For the difference approximation of (1), we are using the following identity

hi1ti1ti+1Lu(t)ψi(t)dt=0,1iN01,
where exponential basis functions
ψi(t)={ψ1i(t)eai(tti1)/ε1eaih/ε1,ti1<t<tiψ2i(t)1eai(ti+1t)/ε1eaih/ε,ti<t<ti+1,0,t(ti1,ti+1),
and
hi1ti1ti+1ψi(t)dt=1.

We note that functions ψ1i(t) and ψ2i(t) are the solutions of the following problems, respectively:

εψ(t)aiψ(t)=0,ti1<t<tiψ(ti1)=0,ψ(ti)=1,εψ(t)aiψ(t)=0,ti<t<ti+1ψ(ti)=1,ψ(ti+1)=0.

Using interpolating quadrature rules with the weight and remainder terms in integral form on subintervals [ti−1, ti+1], consistent with [23,24], after a simple calculation, we have the following relation:

εθiut¯t,i+aiut0,i+f(ti,ui,uiN)+Ri=0,i=1,2,...,N01,
where the factor
θi=ρai2coth(aiρ/2),ρ=h/ε
and remainder term
Ri=hi1ti1ti+1[a(t)a(ti)]u(t)ψi(t)dt+hi1ti1ti+1ddtf(τ,u(τ),u(τr))K0,i*(t,τ)dτK0,i*(t,τ)=T0(tiτ)T0(tiτ),T0(t)=1,t>0;T0(t)=0,t<0.

To define an approximation for the boundary condition (3), we proceed our discretization process by

t0t1Lu(t)φ0(t)dt=0
where the exponential basis function
ψ0(t)={1ea0(t1t)/ε1ea0h/ε,t0<t<t10,t(t0,t1)

We note that functions ϕ0(t) is the solution of the following problem:

εψ(t)a0ψ(t)=0,t0<t<t1,ψ(t0)=0,ψ(t1)=1.

Whence, as similar above we can write the following difference relation:

εσut,0A+r(0)=0
where the coefficient and the remainder term
σ=1+a0ε1t0t1ψ0(t)dt=a0ρ1ea0ρ
r(0)=t0t1[a(t)a0]u(t)ψ0(t)dt+t0t1f(t,u(t),u(tr))ψ0(t)dt.

Neglecting Ri and r(0) in (19) and (22), we propose the following difference scheme for approximation (1)(2):

εθiyt¯t,i+aiyt0,i+f(ti,yi,yiN)=0,i=1,2,...,N01,
yi=φi,Ni0, εσyt,0A=0,
where θi and σ are defined by (20),(23), respectively.

The difference scheme (25)(26), in order to be ɛ- uniform convergent, we will use the non uniform Shishkin mesh. For the even number N, the piecewise uniform mesh ωN, p divides each of the interval [rp−1, σp] and [σp, rp] into N/2 equidistant subintervals, where the transition point σp, which separates the fine and coarse portions of the mesh is obtained by

σp=rp1+min{r/2,α1βpεlnN},
where β1 ≥ 1 and βp > 1 (2 ≤ pm) are some constants. Hence, if hp(1) and hp(2) denote the stepsizes in [rp−1, σp] and [σp, rp] respectively, we have
hp(1)=2(σprp1)N1,hp(2)=2(rpσp)N1,1pm,
so
ω¯N,p={ti=rp1+(i(p1)N)hp(1),i=(p1)N,...,(p1/2)Nti=σp+(i(p1/2)N)hp(2),i=(p1/2)N+1,...,pN,1pm.

In the rest of the paper we only consider this type mesh.

4 Convergence Analysis

We now estimate the approximate error zi = yiui, 0 ≤ iN0, which satisfies the discrete problem

εθizt¯t,i+aizt0,i+f(ti,yi,yiN)f(ti,ui,uiN)=Ri,i=1,...,N01,
zi=0,Ni0,εσzt,0A+r(0)=0,
where the truncation error Ri and r(0) are given by (21) and (24).

Lemma 4.1

Let zi be the solution of the problem (27)(28). Then the following estimate holds

z,N,pγ((θ*/σ)|r(0)|+Ck=1pR,ωN,k,1pm.
where θ*=ρα2coth(αρ/2) , γ = 4α−1 exp(4α−1 (b* + c*))

Proof

Let zt, i = vi. Then the relation (27) can be rewritten as

εθivt¯,i+ai2(vi+vi1)=Fi,
where
Fi=Rif(ti,yi,yiN)+f(ti,ui,uiN).

Solving the first order difference equation with respect to vi, we get

vi=v0Qi+hk=1iF𝓁εθk+hak/2Qik,
where
Qik={1,k=i,j=k+1i(1ajρ/(2θj)1+ajρ/(2θj)),0ki1.

Then, since

zp=hi=0p1vi=hi=1pvi1
and taking into consideration (5), from (30) after some manipulations we have the following inequality
|zp|4α1(εθ*|zt,0|+hi=1p1(|Ri|+b*|zi|+c*|ziN|)),1pN01.

Replacing iN = j, we get

hi=1p|ziN|=hj=1NpN|zj|hj=1N0|zj|+hj=1p1|zj|=ψL1(wN,0)+hj=1p1|zj|.

Therefore

|zp|C(εθ*|zt,0|+ψ1+i=1p(|Ri|+b*|zi|+c*|zi1|)

Taking also into account (28), and using difference analogue of Gronwall’s inequality this leads to (29).

Lemma 4.2

Let a,b,c, fC1 (Ī), ψC1 (Ī0). Then for the truncation errors Ri and r(0), the following estimates hold

R,ωN,pCN1lnN,1pm,|r(0)|CN1lnN.

Proof

From explicit expression (21) for Ri on an arbitrary mesh, we have

|Ri|=hi1ti1ti+1|(a(t)a(ti))||u(t)|ψi(t)dt+hi1ti1ti+1|ddtf(t,u(t),u(tr))|dt,1iN0.

This inequality together with (7), enable us to write

|Ri|C{hi+ti1ti+1(|u(t)|+|u(tr)|)dt},1iN0.

From here, in view of (8), it follows that

|Ri|C{hi+1εti1ti+1eαtεdt},1iN0
in which
hi={hp(1),(p1)Ni(p1/2)Nhp(2),(p1/2)N+1ipN.

At the each submesh ωN, p, we estimate the truncation error R as follows. We consider first the case σp = rp−1 + r/2, and so r/2 ≤ α−1θpɛ lnN and hp(1)=hp(2)=τp=r/N .

Hereby, since

hi1ε1ti1ti+1eαtεdtε1hp2βplnNαrrN=2β1βpN1lnN,(p1)NipN,1pm1
it follows from (32) that
|Ri|CN1lnN,(p1)NipN,1pm1.

We now consider the case σp = rp−1 + α−1βpɛ lnN and estimate Ri on [rp−1, σp] and [σp, rp] separately. In the layer region [rp−1, σp], the inequality (32) reduces to

|Ri|C(1+ε1)hp(1)=C(1+ε1)α1βpεlnNN/2,(p1)Ni(p1/2)N,1pm1.

Hence

|Ri|CN1lnN,(p1)Ni(p1/2)N,1pm1.

Next we estimate Ri for (p−1/2)N +1 ≤ ipN. In this case, recalling that ti=σp+(i(p1/2)N)hp(2)=rp1+α1βpεlnN+(i(p1/2)N)hp(2) , we obtain from (32)

|Ri|C{hp(2)+α1βp(eα(ti1)βpεeα(ti)βpε)}
and this implies that
|Ri|CN1.

Then, we estimate the remainder term r(0). From the explicit expression (32) and |ϕ0 (t)| ≤ 1, after similar calculation as above, it follows that

|r(0)|CN1lnN.

Combining the two previous lemmas gives us the following main convergence result.

Theorem 4.1

Let aC1 (Ī), ϕC1 (I0). The continuously differentiable function f (t,u,v) satisfies the conditions (1.4) and the derivative tf(t,u,v) is bounded for 0 ≤ tT and |u|, |v| ≤ C. Then the following estimate holds

|yiui|CN1lnN,0iN0.

5 Numerical Results

In this section, a simple numerical example is devised to verify the validity for the proposed method. Consider the test problem with

a=1,f=u(t1)+t2,T=2,φ=1+t,A=1.

We use the double mesh principle to estimate the errors and compute the experimental rates of convergence in our computed solution, i.e. We compare the computed solution with the solution on a mesh that is twice as fine [17]. The error estimate eεN,p and the computed convergence rate rεN,p obtained in this way are denoted by

eεN,p=maxωN,p|yε,Nyε,2N|,p=1,2;rεN,p=ln(eεN,p/eε2N,p)/ln2.
The resulting errors eεN,p and the corresponding numbers rεN,p for particular values of ɛ, N, for first and second subinterval are listed in the Tables 1 and 2.

Table 1

Maximum errors and rates of convergence on ωN,1

ɛN = 64N = 128N = 256N = 512N = 1024
2−10.00851160.00428320.00214850.00107600.0005384
0.991.001.001.00
2−20.01400070.00709300.00356930.00170900.0008906
0.980.980.991.00
2−40.02411810.01436030.00831660.00471460.0026331
0.750.780.810.84
2−60.02305410.01372670.00794970.00450660.0025149
0.750.780.810.84
2−80.02278810.01356840.00785800.00445460.0024859
0.750.780.810.84
2−100.02272160.01352880.00783500.00444160.0024786
0.750.780.810.84
2−120.02270500.01351890.00782930.00443820.0024768
0.750.780.810.84
2−140.02270080.01351640.00782790.00443610.0024763
0.750.780.810.84
2−160.02269980.01351580.007827560.00443740.0024762
0.750.780.810.84
Table 2

Maximum errors and rates of convergence on ωN,2

ɛN = 64N = 128N = 256N = 512N = 1024
2−10.00798140.00401030.00201010.00100630.0005033
0.980.991.001.00
2−20.00121380.00613920.00308750.00154820.0007775
0.980.991.001.00
2−40.02026000.01207540.00698850.00396090.0022110
0.750.790.810.84
2−60.01942430.01154260.00668020.00378620.0021126
0.750.780.820.84
2−80.01914270.01140940.00660310.00374250.0020882
0.750.780.820.84
2−100.01908680.01137610.00658380.00373150.0020821
0.750.780.820.84
2−120.01907290.01136780.00657900.00372880.0020806
0.750.780.820.84
2−140.01906960.01136570.00657780.00372810.0020668
0.750.780.820.84
2−160.01906850.01136520.00657540.00372800.0020801
0.750.780.820.84

The values of ɛ for which we solve the test problem are ɛ = 2i,i = 1,2,4,...,16. These convergence rates are increasing as N increases for any fixed ɛ. Tables 1 and 2 verify the ɛ-uniform convergence of the numerical solution on both subintervals and computed rates are essentially in agreement with our theoretical analysis.

6 Conclusions

A numerical method is developed to solve singularly perturbed initial value problem for semilinear second-order delay differential equation. This method is based on exponentially fitted difference scheme on an equidistant mesh, which gives first order uniform convergence in the discrete maximum norm. The method is shown to be uniformly convergent to the continuous solution i.e., independent of mesh parameter and perturbation parameter. Efficiency of the present method is demonsrated by a numerical example and also by comparing the results with exact solution of the problem. Presented numerical results are essentially in agreement with our theoretical analysis.

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