AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/304071 304071 Research Article Stepsize Restrictions for Nonlinear Stability Properties of Neutral Delay Differential Equations Gu Wei 1 http://orcid.org/0000-0002-1128-530X Wang Ming 2 http://orcid.org/0000-0002-6234-2714 Li Dongfang 3 Bhrawy Ali H. 1 School of Statistics and Mathematics Zhongnan University of Economics and Law Wuhan 430073 China znufe.edu.cn 2 School of Mathematics and Physics China University of Geosciences Wuhan 430074 China cug.edu.cn 3 School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China hust.edu.cn 2014 2072014 2014 22 05 2014 09 07 2014 21 7 2014 2014 Copyright © 2014 Wei Gu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The present paper is concerned with the relationship between stepsize restriction and nonlinear stability of Runge-Kutta methods for delay differential equations. We obtain a special stepsize condition guaranteeing global and asymptotical stability properties of numerical methods. Some confirmations of the conditions on Runge-Kutta methods are illustrated at last.

1. Introduction

Neutral delay differential equations (NDDEs) are widely used in various kinds of applied disciplines such as biology, ecology, electrodynamics, and physics and hence intrigue lots of researchers in numerical simultation and analysis (see, e.g., ). Up to now, many researchers have discussed nonlinear stability properties for NDDEs. In 2000, Bellen et al.  studied BNf-stable continuous Runge-Kutta methods for NDDEs. They extended the contractivity requirements to the numerical stability analysis. Vermiglio and Torelli further pointed out that the numerical solution produced by the methods can preserve the contractivity property of the theoretical solution in . In 2002, Zhang  derived nonlinear stability properties for theoretical and numerical solutions of NDDEs based on natural Runge-Kutta schemes. After that, Wang et al. [7, 8] first introduced the concepts of GS(l)- and GAS(l)-stability for nonautonomous nonlinear problems. They proved that (k,l)-algebraically stable Runge-Kutta methods and (k,p,0)-algebraically stable general linear methods lead to GS(l)- and GAS(l)-stability for NDDEs, respectively. Recently, Bhrawy et al.  studied several kinds of collocation method for some NDDEs. For more analogues results, we refer readers to . Useful as these stability results are, however, no conclusions have been found to develop the relationship between nonlinear stability analysis and stepsize restriction with some numerical schemes for NDDEs, especially for some Runge-Kutta methods.

The present paper was in part inspired by the work of Spijker et al. With stepsize restriction to some numerical schemes, they revealed to us some monotonicity and stability properties for ODEs, respectively (see, ). We extend their study to nonlinear NDDEs in the present paper. With stepsize restriction to Runge-Kutta schemes, global and asymptotical stability results for NDDEs are obtained, respectively.

The rest of the paper is organized as follows. In Section 2, we consider Runge-Kutta schemes with linear interpolation procedure for NDDEs. Some concepts, such as global and asymptotical stability, are also collected. Section 3 is devoted to stability analysis. The given results set up a relationship between the stepsize restriction and nonlinear stability for nonlinear NDDEs. Some examples of Runge-Kutta schemes are presented in Section 4. Finally, we end up with some conclusions and extension in the last section.

2. Runge-Kutta Methods for NDDEs

In the present paper, we consider the following nonlinear NDDEs: (1)    ddt[y(t)-Ny(t-τ)]=f(t,y(t),y(t-τ)),t>0,    y(t)=ψ(t),-τt0, and the perturbed problem (2)    ddt[z(t)-Nz(t-τ)]=f(t,z(t),z(t-τ)),t>0,z(t)=ϕ(t)-τt0. Here, τ denotes a positive delay term, NCd×d is a constant matrix with N<1, ψ(t) and ϕ(t) are continuous, and f: [0,+]×X×XX, such that (1) and (2) own a unique solution, respectively, where X is a real or complex Hilbert space. As in [20, 21], we assume there exist some inner product ·,· and the induced norm · such that (3)Re(y-z)-N(u-v),f(t,y,u)-f(t,z,v)αy-z2+βu-v2+δf(t,y,u)-f(t,z,v)2, where α0,β0, and δ<0 are real constants.

When N=0, the problem (1) degenerates into nonlinear DDEs of the following type: (4)    y(t)=f(t,y(t),y(t-τ)),t>0,    y(t)=ψ(t)-τt0. Nonlinear stability analysis for such systems can be found in [6, 2225]. Condition (3) can be equivalent to the assumptions in these literatures (see , Remark 2.1).

Now, let us consider s-stage Runge-Kutta methods for (1); the coefficients of the schemes may be organized in Buther tableau as follows: (5)cAbT, where c=[cl,,cs]T, b=[b1,,bs]T, and A=(aij)i,j=1s.

According to Liu in , Runge-Kutta methods for NDDEs can be written as (6)yn+1-Ny~n+1=yn-Ny~n+hj=1sbjf(tn+cjh,yj(n),y~j(n)),  yi(n)-Ny~i(n)=yn-Ny~n+hj=1saijf(tn+cjh,yj(n),y~j(n))  hhhhhhhhhhhhhhhhhhhhhhhhi=1,2,,s, where h is stepsize and tn=nh,yn,y~n,yi(n) and y~i(n) are approximations to the analytic solutions y(tn), y(tn-τ), y(tn+cih), and y(tn+cih-τ), respectively. We set τ=(m-θ)h with θ[0,1), and the arguments y~n and y~j(n) are determined by (7)y~n=θyn-m+1+(1-θ)yn-m,y~j(n)=θyj(n-m+1)+(1-θ)yj(n-m), where yi=ψ(ti) for ti0 and yj(i)=ψ(ti+cjh) for ti+cjh0.

Now, let yn and zn be two sequences of approximations to problems (1) and (2), respectively. Following Definitions 9.1.1 and 9.1.2 in  for delay systems, we introduce some stability concepts.

Definition 1.

A numerical method for DDEs or NDDEs is called globally stable, if there exists a constant C such that (8)yn-znCmax-τt0ψ(t)-ϕ(t) holds when the method is applied to (1) and (2) under some assumptions.

Definition 2.

A numerical method for DDEs or NDDEs is said to be asymptotically stable, if (9)limnyn-zn=0 holds when the method is applied to (1) and (2) under some assumptions.

3. Stability Analysis

In the section, we will discuss the relationship between the stepsize restriction and nonlinear stability of the method.

Theorem 3.

Assume condition (3) holds, α+β0, and there exists a positive real number r, such that the matrix (10)M=diag(b)A+ATdiag(b)-bbT+1rdiag(b) is nonnegative definite, where bi0, i=1,2,,s. Then the Runge-Kutta method with linear interpolation procedure for NDDEs (1) is globally stable under the stepsize restriction (11)hr-2δ.

Proof.

Let {yn,yi(n),y~i(n)} and {zn,zi(n),z~i(n)} be two sequences of approximations to problems (1) and (2), respectively, and write (12)Ui(n)=yi(n)-zi(n),U~0(n)=y~i(n)-z~i(n),U0(n)=yn-zn,U~0(n)=y~n-z~n,Wi=h[f(tn+cih,yi(n),y~i(n))-f(tn+cih,zi(n),z~i(n))]. With the notation, Runge-Kutta methods with the same stepsize h for (1) and (2) yield (13)U0(n+1)-NU~0(n+1)=U0(n)-NU~0(n)+j=1sbjWj,Ui(n)-NU~i(n)=U0(n)-NU~0(n)+j=1saijWj,i=1,2,,s. Thus, we have (14)U0(n+1)-NU~0(n+1)2=U0(n)-NU~0(n)+j=1sbjWj,U0(n)-NU~0(n)+i=1sbiWi=U0(n)-NU~0(n)2+2i=1sbiReU0(n)-NU~0(n),Wi+i,j=1sbibjWi,Wj=U0(n)-NU~0(n)2+2i=1sbiReUi(n)-NU~i(n)-j=1saijWj,Wi+i,j=1sbibjWi,Wj=U0(n)-NU~0(n)2+2i=1sbiReUi(n)-NU~i(n),Wi-i,j=1s(biaij+ajibj-bibj)Wi,Wj. Now, in view of the nonnegative definite matrix M, we obtain (15)-i,j=1s(biaij+ajibj-bibj)Wi,Wj1ri=1sbiWi,Wi. On the other hand, in terms of condition (3), we find (16)ReUi(n)-NU~i(n),WiαhUi(n)2+βhU~i(n)2+δhWi2.

Then, together with (14), (15), and (16) and using the conditions h/r-2δ, we get (17)U0(n+1)-NU~0(n+1)2U0(n)-NU~0(n)2+2i=1shbi(αhUi(n)2+βhU~i(n)2+(δh+12r)Wi2)U0(n)-NU~0(n)2+2i=1shbi(αUi(n)2+βU~i(n)2). Noting that (18)U~i(n)2=[θUi(n-m+1)+(1-θ)Ui(n-m)]2θ2Ui(n-m+1)2+(1-θ)2Ui(n-m)2+θ(1-θ)(Ui(n-m+1)2+Ui(n-m)2)=θUi(n-m+1)2+(1-θ)Ui(n-m)2 and α+β0, we have (19)U0(n+1)-NU~0(n+1)2U0(n)-NU~0(n)2+2i=1shβbi(U~i(n)2-Ui(n)2)U0(n)-NU~0(n)2+2i=1shβbi(θUi(n-m+1)2+(1-θ)Ui(n-m)2-Ui(n)2)U0(0)-NU~0(0)2+2i=1shβbi(j=-m+1-1θUi(j)2+j=-m-1(1-θ)Ui(j)2)U0(0)-NU~0(0)2+2i=1smhβbimax-mj-1Ui(j)22U0(0)2+2NU~0(0)2+2i=1smhβbimax-mj-1Ui(j)2(2+2N2+2τi=1sβbi)max-τt0ψ(t)-ϕ(t)2=(2+2N2+2τβ)max-τt0ψ(t)-ϕ(t)2.

This implies that (20)U0(n+1)-NU~0(n+1)C~max-τt0ψ(t)-ϕ(t), where C~=(2+2N2+2τβ).

Note that N<1; we have (21)U0(n+1)NU~0(n+1)+C~max-τt0ψ(t)-ϕ(t)=NθU0n-m+2+(1-θ)U0n-m+1+C~max-τt0ψ(t)-ϕ(t)max(U0n-m+2,U0n-m+1)+C~max-τt0ψ(t)-ϕ(t). An induction to (21) yields (22)U0(n+1)(1+C~)max-τt0ψ(t)-ϕ(t). Therefore, the conclusion is proven.

Corollary 4.

Assume condition (3) holds; α+β0. Then an algebraically stable Runge-Kutta method with linear interpolation procedure for DDEs (4) or NDDEs (1) is globally stable.

Remark 5.

A Runge-Kutta method is algebraically stable if the matrix (23)diag(b)A+ATdiag(b)-bbT is nonnegative definite and bi0(i=1,2,,s). For example, the s-stage Gauss, Radau IA, Radau IIA, and Lobatto IIIC methods are algebraically stable. Corollary 4 can be derived for r=. This implies that the stepsize restriction for DDEs disappears.

Corollary 6.

Assume condition (3) holds, α+β0, and there exists a positive real number r, such that the matrix (24)M=diag(b)A+ATdiag(b)-bbT+1rdiag(b) is nonnegative definite, where bi0,  i=1,2,,s. Then the Runge-Kutta method with linear interpolation procedure for DDEs (4) is globally stable under the stepsize restriction (25)hr-2δ.

Theorem 7.

Assume condition (3) holds, α+β<0, the function f(t,u,v) is uniformly Lipschitz continuous with constant L in variables u and v, and there exists a positive real number r, such that the matrix (26)M=diag(b)A+ATdiag(b)-bbT+1rdiag(b) is nonnegative definite, where bi0,  i=1,2,,s. Then the Runge-Kutta method with linear interpolation procedure for NDDEs (1) is asymptotically stable under the stepsize restriction (27)hr-2δ.

Proof.

Like in the proof of Theorem 3, let σ=α+β<0, and we can easily find (28)U0(n+1)-NU~0(n+1)2U0(n)-NU~0(n)2+2i=1shbi(αUi(n)2+βU~i(n)2)2(U0(n)2+N2U~0(n)2)+2i=1shbi((σ-β)Ui(n)2+βU~i(n)2)=2(U0(n)2+N2U~0(n)2)+2i=1shβbi(U~i(n)2-Ui(n)2)+2i=1shbiσUi(n)2(2+2N2+2τi=1sβbi)max-τt0ψ(t)-ϕ(t)2+2j=1ni=1shbiσUi(j)2. Note σ<0 and bi0; we have (29)limni=1sbiUi(n)=0. On the other hand, (30)Wi=h[f(tn+cih,yi(n),y~i(n))-f(tn+cih,zi(n),z~i(n))]hL(Ui(n)+U~i(n)). Now, in view of (13), (29), and (30), we obtain (31)limnU0(n)-NU~0(n)=0. Since (32)U0(n)=U0(n)-NU~0(n)+NU~0(n)U0(n)-NU~0(n)+NU~0(n)U0(n)-NU~0(n)+Nmax(U0n-m+2,U0n-m+1) and N<1, an induction to (32) gives (33)limnU0(n)=0 which completes the proof.

Corollary 8.

Assume condition (3) holds, α+β<0, the function f(t,u,v) is uniformly Lipschitz continuous with constant L in variables u and v. Then an algebraically stable Runge-Kutta method with linear interpolation procedure for DDEs (4) or NDDEs (1) is asymptotically stable.

Corollary 9.

Assume condition (3) holds, α+β<0, the function f(t,u,v) is uniformly Lipschitz continuous with constant L in variables u and v, and there exists a positive real number r, such that the matrix (34)M=diag(b)A+ATdiag(b)-bbT+1rdiag(b) is nonnegative definite, where bi0, i=1,2,,s. Then the Runge-Kutta method with linear interpolation procedure for DDEs (4) is asymptotically stable under the stepsize restriction (35)hr-2δ.

4. Some Examples

As it is shown in the theorems, the parameters δ and r in the matrix M play a key role in the stability analysis. The larger the existed parameter r is, the larger stepsize we could choose. In this section, we will show some examples.

Consider the following case, like the conditions in  or , if f(t,y,u)=f~(t,y-Nu) and (36)ρ((y-Nu)-(z-Nv))+f~(t,y-Nu)-f~(t,z-Nv)ρ((y-Nu)-(z-Nv)) with ρ>0, we have the following form in an inner product norm: (37)Re(y-Nu)-(z-Nv),f~(t,y-Nu)-f~(t,z-Nv)δf~(t,y-Nu)-f~(t,z-Nv)2 with δ=-1/(2ρ)<0.

In particular, let f(t,y,u)=-a(My-Nu), where a>0,  M<1 are constants independent of t, respectively. We have (38)Rey-Nu,f(t,y,u)=ReMy-Nu-(M-1)y,-a(My-Nu)=-1aa(My+Nu)2+aM(M-1)y2+Rea(1-M)y,Nu-1aa(My+Nu)2+aM(M-1)y2+12a(1-M)N(y2+u2)=(aM(M-1)+12a(1-M)N)y2+12a(1-M)Nu2-1af(t,y,u)2.

Next, we give some examples on how to calculate the parameter r.

Example 1.

Consider s-stage 1-order Runge-Kutta methods (see , section 2.7) (39)001s1s02s1s1s0s-1s1s1s1s1s01s1s1s1s1s1s and we have (40)M=diag(b)A+ATdiag(b)-bbT+1rdiag(b)=(1rs-1s2)Is. Thus, the matrix M is nonnegative definite for 0<rs. They imply that these methods for DDE with interpolation are stable with stepsize restriction h-2δs.

Example 2.

Consider 2-stage 2-order Runge-Kutta method: (41)0001101212 and we obtain (42)M=[12r-14141412r-14]. Therefore, the matrix M is nonnegative definite for 0<r1. They imply that the stepsize h-2δ is feasible under the assumptions (3) for NDDEs (1).

For more Runge-Kutta methods with the nonnegative definite matrix M, we refer readers to Section 2.2.4 in . Higueras revealed to us how to find the largest r such that the matrix M is nonnegative definite. He pointed that if the matrix diag(b) is positive definite, the largest r can be determined by (43)r=-λmin-1([diag(b)]-1/2[ATdiag(b)A+ATdiag(b)-bbT]×[diag(b)]-1/2), where λmin(·) denotes the smallest eigenvalue of the matrix (·).

5. Conclusions and Discussions

In this study, we show that the Runge-Kutta methods with stepsize restrictions can preserve global and asymptotical stability of the continuous DDEs or NDDEs. An algebraically stable Runge-Kutta method with linear interpolation procedure for DDEs or NDDEs is globally stable and asymptotically stable. These results can be easily extended to the following equation with several delays: (44)ddt[y(t)-i=1lNiy(t-τi)]=f(t,y,y(t-τ1),,y(t-τl)),t0,y(t)=ψ(t),t0, under the following assumption: (45)Re(y-z)-i=1lNi(yi-zi),f(t,y,y1,,yN)-f(t,z,z1,,zN)i=1lNi(yi-zi)αy-z2+i=1N  βiyi-zi2+δf(t,y,y1,,yN)-f(t,z,z1,,zN)2, where τi>0,i=1,2,,l, yi=y(t-τi), and zi=z(t-τi). We do not list them here for the sake of brevity.

Moreover, the present results have certain instructive effect in numerical simulation. In the future, we hope to apply the results to some real-world problems, for example, reaction-diffusion dynamical systems with time delay  and non-Fickian delay reaction-diffusion equations [25, 29].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by NSFC (Grant nos. 11201161, 11171125, and 91130003).

Bellen A. Zennaro M. Numerical Methods for Delay Differential Equations 2003 Oxford, UK Clarendon Press 10.1093/acprof:oso/9780198506546.001.0001 MR1997488 Brunner H. Collocation Methods for Volterra Integral and Related Functional Differential Equations 2004 Cambridge, UK Cambridge University Press 10.1017/CBO9780511543234 MR2128285 Enright W. H. Hayashi H. Convergence analysis of the solution of retarded and neutral delay differential equations by continuous numerical methods SIAM Journal on Numerical Analysis 1998 35 2 572 585 10.1137/S0036142996302049 MR1618854 2-s2.0-0001080457 Bellen A. Guglielmi N. Zennaro M. Numerical stability of nonlinear delay differential equations of neutral type Journal of Computational and Applied Mathematics 2000 125 1-2 251 263 10.1016/S0377-0427(00)00471-4 MR1803194 ZBL0980.65077 2-s2.0-0034516813 Vermiglio R. Torelli L. A stable numerical approach for implicit non-linear neutral delay differential equations BIT Numerical Mathematics 2003 43 1 195 215 10.1023/A:1023613425081 MR1981648 2-s2.0-2942666323 Zhang C. Nonlinear stability of natural Runge-Kutta methods for neutral delay differential equations Journal of Computational Mathematics 2002 20 6 583 590 MR1938638 ZBL1018.65101 2-s2.0-0036880750 Wang W. Li S. Su K. Nonlinear stability of Runge-Kutta methods for neutral delay differential equations Journal of Computational and Applied Mathematics 2008 214 1 175 185 10.1016/j.cam.2007.02.031 MR2391681 2-s2.0-38949086564 Wang W. Li S. Su K. Nonlinear stability of general linear methods for neutral delay differential equations Journal of Computational and Applied Mathematics 2009 224 2 592 601 10.1016/j.cam.2008.05.050 MR2492892 ZBL1167.65046 2-s2.0-58149330778 Bhrawy A. H. AlZahrani A. Baleanu D. Alhamed Y. A modified generalized Laguerre-Gauss collocation method for fractional neutral functional differential equations on the half-line Abstract and Applied Analysis 2014 2014 7 692193 10.1155/2014/692193 MR3224319 Bhrawy A. H. Alghamdi M. A. A shifted Jacobi-Gauss collocation scheme for solving fractional neutral functional-differential equations Advances in Mathematical Physics 2014 2014 8 595848 10.1155/2014/595848 MR3198299 Bhrawy A. H. Assas L. M. Tohidi E. Alghamdi M. A. A Legendre-Gauss collocation method for neutral functional-differential equations with proportional delays Advances in Difference Equations 2013 2013, article 63 10.1186/1687-1847-2013-63 Doha E. H. Bhrawy A. H. Baleanu D. Hafez R. M. A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations Applied Numerical Mathematics 2014 77 43 54 10.1016/j.apnum.2013.11.003 MR3145364 Doha E. H. Baleanu D. Bhrawy A. H. Hafez R. M. A pseudospectral algorithm for solving multipantograph delay systems on a semi-infinite interval using legendre rational functions Abstract and Applied Analysis 2014 2014 9 816473 10.1155/2014/816473 Li D. Zhang C. Nonlinear stability of discontinuous Galerkin methods for delay differential equations Applied Mathematics Letters 2010 23 4 457 461 10.1016/j.aml.2009.12.003 MR2594863 2-s2.0-76749166248 Niu Y. L. Zhang C. J. Exponential stability of nonlinear delay differential equations with multidelays Acta Mathematicae Applicatae Sinica 2008 31 4 654 662 MR2516466 Ferracina L. Spijker M. N. Stepsize restrictions for total-variation-boundedness in general Runge-Kutta procedures Applied Numerical Mathematics 2005 53 2-4 265 279 10.1016/j.apnum.2004.08.024 MR2128526 2-s2.0-14844314172 Higueras I. On strong stability preserving time discretization methods Journal of Scientific Computing 2004 21 2 193 223 10.1023/B:JOMP.0000030075.59237.61 MR2069949 ZBL1074.65095 2-s2.0-3042714776 Spijker M. N. Stepsize restrictions for stability of one-step methods in the numerical solution of initial value problems Mathematics of Computation 1985 45 172 377 392 10.2307/2008131 MR804930 ZBL0579.65092 Spijker M. N. Stepsize conditions for general monotonicity in numerical initial value problems SIAM Journal on Numerical Analysis 2007 45 3 1226 1245 10.1137/060661739 MR2318810 2-s2.0-46949088290 Zhang C. He Y. The extended one-leg methods for nonlinear neutral delay-integro-differential equations Applied Numerical Mathematics 2009 59 6 1409 1418 10.1016/j.apnum.2008.08.006 MR2510501 2-s2.0-62549146665 Wen L. Wang S. Yu Y. Dissipativity of Runge-Kutta methods for neutral delay integro-differential equations Applied Mathematics and Computation 2009 215 2 583 590 10.1016/j.amc.2009.05.039 MR2561516 ZBL1177.65196 2-s2.0-68949149770 Dekker K. Verwer J. G. Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations 1984 Amsterdam, The Netherlands North-Holland Publishing MR774402 In 't Hout K. J. Stability analysis of Runge-Kutta methods for systems of delay differential equations IMA Journal of Numerical Analysis 1997 17 1 17 27 10.1093/imanum/17.1.17 MR1427613 2-s2.0-0031506225 Li D. Zhang C. Qin H. LDG method for reaction-diffusion dynamical systems with time delay Applied Mathematics and Computation 2011 217 22 9173 9181 10.1016/j.amc.2011.03.153 MR2803982 2-s2.0-79957526635 Li D. Zhang C. Wang W. Long time behavior of non-Fickian delay reaction-diffusion equations Nonlinear Analysis: Real World Applications 2012 13 3 1401 1415 10.1016/j.nonrwa.2011.11.005 MR2863967 2-s2.0-84655170161 Huang C. Fu H. Li S. Chen G. Nonlinear stability of general linear methods for delay differential equations BIT Numerical Mathematics 2002 42 2 380 392 10.1023/A:1021955126558 MR1912593 2-s2.0-0041878541 Liu Y. K. Numerical solution of implicit neutral functional-differential equations SIAM Journal on Numerical Analysis 1999 36 2 516 528 10.1137/S003614299731867X MR1668211 2-s2.0-0041625708 Higueras I. Monotonicity for Runge-Kutta methods: inner product norms Journal of Scientific Computing 2005 24 1 97 117 10.1007/s10915-004-4789-1 MR2219365 2-s2.0-23244435121 Li D. Tong C. Wen J. Stability of exact and discrete energy for non-Fickian reaction-diffusion equations with a variable delay Abstract and Applied Analysis 2014 2014 9 840573 10.1155/2014/840573 MR3178896