- Research
- Open Access
Numerical algorithm for nonlinear delayed differential systems of nth order
- Josef Rebenda^{1}Email authorView ORCID ID profile and
- Zdeněk Šmarda^{2}
https://doi.org/10.1186/s13662-019-1961-3
© The Author(s) 2019
- Received: 12 October 2018
- Accepted: 15 January 2019
- Published: 24 January 2019
Abstract
The purpose of this paper is to propose a semi-analytical technique convenient for numerical approximation of solutions of the initial value problem for p-dimensional delayed and neutral differential systems with constant, proportional and time varying delays. The algorithm is based on combination of the method of steps and the differential transformation. Convergence analysis of the presented method is given as well. Applicability of the presented approach is demonstrated in two examples. A system of pantograph type differential equations and a system of neutral functional differential equations with three types of delays are considered. The accuracy of the results is compared to those obtained by the Laplace decomposition algorithm, the residual power series method and Matlab package DDENSD. A comparison of computing time is presented, too, showing reliability and efficiency of the proposed technique.
Keywords
- Differential transformation
- Method of steps
- Delayed differential system
- Multiple delays
MSC
- 34K28
- 34K07
- 34K40
- 65L03
1 Introduction
Systems of functional differential equations (FDEs), in particular delayed or neutral differential equations, are often used to model processes in the real world. To give some examples, we mention models in population dynamics [1], neuromechanics [2], machine tool vibrations [3], etc. Further models and details can be found, for instance, in monographs [4] and [5].
Semi-analytical methods expressing solutions to problems with delays in a series form have been studied in the last two decades. Methods such as the variational iteration method (VIM) [6], Adomian decomposition method (ADM) [7], homotopy perturbation method (HPM) [8], homotopy analysis method (HAM) [9] and also methods based on the Taylor theorem such as the differential transformation (DT) [10], Taylor collocation method [11] and Taylor polynomial method [12] have been developed to approximate solutions to different problems for FDEs. Other ways to use the series approach in solving FDEs are, e.g., the method of polynomial quasisolutions [13, 14], finite difference methods [15, 16], and the functional analytic technique (FAT) [17, 18].
The main aim of the work is to apply a combination of the method of steps and DT as a convenient tool for finding an approximate solution to the initial value problem for functional differential systems used in dynamical models. Convergence analysis and error estimates of the method are investigated as well. We give some experimental results in Sect. 4 to show that the algorithm produces reliable results with the same or better efficiency than the reference methods.
2 Methods
The main idea of our approach is to combine the differential transformation and general method of steps.
The differential transformation has been, and still is, an active research topic during the last years. As examples of recently published results, we mention research papers [19–23]. These papers among other publications contain new algorithms and their applications to solving different problems involving differential equations.
Definition 1
Definition 2
Lemma 1
Remark 1
Transformation formulas for shifted arguments \(w(t)=u(t-a)\) are often proved and applied in papers. However, using these formulas when solving initial value problems for delayed differential equations is not convenient since the uniqueness of solutions is violated. The reason is that the values of the initial vector function for \(t < 0\) are not taken into account.
One of the drawbacks of the common approach to the differential transformation is that there is no use of direct transformation formulas for equations with nonlinear terms containing unknown function \(u(t)\), for instance, \(f(u)= \operatorname {e}^{\cos {u}}\) or \(f(u) = \sqrt{1+u^{4}}\).
The second method, namely the method of steps, enables us to replace the terms involving constant or time-dependent delays by the initial vector function and its derivatives. Then the original initial value problem for a system of delayed or neutral differential equations is simplified to the initial problem for a system of ordinary differential equations. Details on the method of steps can be found, e.g., in monographs [4, 5, 27].
3 Results
- 1.
\(\alpha _{i}(t) = q_{i} t\), where \(q_{i} \in (0,1)\) (proportional delay).
- 2.
\(\alpha _{i}(t) = t-\tau _{i}\), where \(\tau _{i}>0\) is a real constant (constant delay).
- 3.
\(\alpha _{i}(t)=t-\tau _{i}(t)\), where \(\tau _{i}(t) \geq \tau _{i0}>0\) for \(t>0\) is a real function (time-dependent or time-varying delay).
Let \(t^{*}= \min_{1\leq i \leq r} \{\inf_{t>0} (\alpha _{i}(t) ) \} \leq 0\), \(m= \max \{m_{1},m_{2},\ldots,m_{r}\} \leq n\). In the case \(m=n\), system (5) is a neutral system, otherwise it is a delayed differential system.
If \(t^{*}<0\), an initial vector function \(\varPhi (t) = (\phi _{1}(t),\ldots,\phi _{p}(t))^{T}\) must be assigned to system (5) on the interval \([t^{*}, 0]\). Moreover, we assume that \(\phi _{j}(t) \in C ^{n}([t^{*},0],\mathbb{R})\) for \(j=1,\ldots,p\).
- (H1)
The functions \(f_{j}\), \(j=1,\ldots, p\) are analytic in \([0,T^{*}] \times \mathbb{R}^{np} \times \mathbb{R}^{\omega p}\).
- (H2)
Remark 2
Hypothesis (H2) is valid, for example, if the delay functions \(\alpha _{i}\) are Lipschitz continuous on \([0,T^{*}]\), the functions \(\phi _{j}, \phi '_{j},\ldots, \phi _{j}^{(n)}\) are Lipschitz continuous on \([t^{*},0]\), and the functions \(f_{j}\) are continuous with respect to t on \([0,T^{*}]\) and Lipschitz continuous with respect to the rest of the variables on \(\mathbb{R}^{np} \times \mathbb{R}^{\omega p}\). More details and other types of sufficient conditions for existence of a unique solution can be found in [5, Sects. 3.2 and 3.3], or [27, Sect. 2.2].
We start with the method of steps. We substitute the initial vector function \(\varPhi (t)\) and its derivatives in all places where the unknown functions with constant or time-dependent delays and derivatives of those functions take place. This turns the delayed system (5) into a system of ordinary differential equations or differential equations with proportional delays in the case when system (5) contains proportional delays.
Now we formulate and prove two theorems on convergence and an error estimate of the approximate solution to the studied problem obtained using the differential transformation.
Theorem 1
Let hypotheses (H1) and (H2) be valid and denote \(\mathbf{F}_{k}(t) = \mathbf{U}(k)t^{k}\). If there exist a constant δ, \(0<\delta < 1\), and \(k_{0} \in \mathbb{N}\) such that \(\Vert \mathbf{F}_{k+1}(t)\Vert \leq \delta \Vert \mathbf{F}_{k}(t)\Vert \) for all \(k \geq k_{0}\), then the series \(\sum_{k=0}^{\infty } \mathbf{F}_{k}(t)\) converges to a unique solution on the interval \(J=[0,\gamma ]\), \(\gamma \leq T^{*}\).
Proof
Theorem 2
Proof
Remark 3
Recent results on error estimates and convergence of Taylor series can be found, e.g., in [28].
4 Applications and discussion
As the first application, we have chosen the initial value problem, which has been solved in [29] using the Laplace decomposition method (LDM) and in [30] using the residual power series method (RPSM).
Example 1
Error analysis of \(u_{1}\) on \([0,1]\)
t | Exact solution −cost | DT \(u_{1}\) | Abs. errors DT | Abs. errors LDM | Abs. errors RPSM |
---|---|---|---|---|---|
0.2 | −0.9800665 | −0.9800666 | 1.0E−7 | 8.904E−5 | 1.0E−7 |
0.4 | −0.9210609 | −0.9210666 | 5.7E−6 | 1.511E−3 | 5.7E−6 |
0.6 | −0.8253335 | −0.8254000 | 6.65E−5 | 8.051E−3 | 6.65E−5 |
0.8 | −0.6967067 | −0.6970666 | 3.599E−4 | 2.665E−2 | 3.599E−4 |
1.0 | −0.5403023 | −0.5416666 | 1.3642E−3 | 6.766E−2 | 1.3642E−3 |
Error analysis of \(u_{2}\) on \([0,1]\)
t | Exact solution tcost | DT \(u_{2}\) | Abs. errors DT | Abs. errors LDM | Abs. errors RPSM |
---|---|---|---|---|---|
0.2 | 0.1960133 | 0.1960133 | 0.0 | 5.496E−6 | 0.0 |
0.4 | 0.3684243 | 0.3684266 | 2.3E−6 | 1.808E−4 | 2.3E−6 |
0.6 | 0.4952013 | 0.4952400 | 3.87E−5 | 1.408E−3 | 3.87E−5 |
0.8 | 0.5573653 | 0.5576533 | 2.89E−4 | 6.069E−3 | 2.89E−4 |
1.0 | 0.5403023 | 0.5416666 | 1.3643E−3 | 1.890E−2 | 1.3643E−3 |
Error analysis of \(u_{3}\) on \([0,1]\)
t | Exact solution sint | DT \(u_{3}\) | Abs. errors DT | Abs. errors LDM | Abs. errors RPSM |
---|---|---|---|---|---|
0.2 | 0.1986693 | 0.1986693 | 0.0 | 6.4558E−5 | 0.0 |
0.4 | 0.3894183 | 0.3894186 | 3.0E−7 | 9.9595E−4 | 3.0E−7 |
0.6 | 0.5646424 | 0.5646480 | 5.60E−6 | 4.8397E−3 | 5.60E−6 |
0.8 | 0.7173561 | 0.7173973 | 4.12E−5 | 1.4613E−2 | 4.12E−5 |
1.0 | 0.8414709 | 0.8416666 | 1.957E−3 | 3.3917E−2 | 1.957E−3 |
Comparison of computing time for \(u_{1}\)
t | DT | LDM | RPSM |
---|---|---|---|
0.2 | 6.3E−5 | 8.7E−4 | 6.3E−5 |
0.4 | 6.5E−5 | 6.7E−4 | 6.5E−5 |
0.6 | 6.5E−5 | 6.9E−4 | 6.5E−5 |
0.8 | 6.4E−5 | 7.2E−4 | 6.4E−5 |
1.0 | 6.6E−5 | 8.8E−4 | 6.6E−5 |
Comparison of computing time for \(u_{2}\)
t | DT | LDM | RPSM |
---|---|---|---|
0.2 | 6.6E−5 | 6.7E−4 | 6.6E−5 |
0.4 | 6.4E−5 | 6.7E−4 | 6.4E−5 |
0.6 | 6.7E−5 | 6.7E−4 | 6.7E−5 |
0.8 | 6.5E−5 | 8.5E−4 | 6.5E−5 |
1.0 | 6.5E−5 | 8.3E−4 | 6.5E−5 |
Comparison of computing time for \(u_{3}\)
t | DT | LDM | RPSM |
---|---|---|---|
0.2 | 6.6E−5 | 7.7E−4 | 6.6E−5 |
0.4 | 6.7E−5 | 6.6E−4 | 6.7E−5 |
0.6 | 6.6E−5 | 6.7E−4 | 6.6E−5 |
0.8 | 6.6E−5 | 8.3E−4 | 6.6E−5 |
1.0 | 6.7E−5 | 8.5E−4 | 6.7E−5 |
Remark 4
In [29], the authors used LDM and obtained only approximate solutions of the initial value problem (12), (13). Applying RPSM, the authors were able to find closed-form solutions in [30]. However, the calculations are too complicated, and the residual functions (RPSM) and initial guesses (LDM) contain analytical forms of functions sin and cos, which means that these methods are not convenient for use in a purely numerical software.
As the second application, we have chosen a system with all three types of delays considered to show reliability and efficiency of the proposed approach in solving difficult tasks.
Example 2
Comparison of values of solution components obtained by DT and Matlab
t | Method | |||
---|---|---|---|---|
DT | Matlab | |||
\(u_{1}\) | \(u_{2}\) | \(u_{1}\) | \(u_{2}\) | |
0.00 | 1.0000 | 0.0000 | 1.0000 | 0.0000 |
0.05 | 1.0513 | 0.0024 | 1.0513 | 0.0024 |
0.10 | 1.1051 | 0.0093 | 1.1050 | 0.0093 |
0.15 | 1.1618 | 0.0203 | 1.1614 | 0.0203 |
0.20 | 1.2209 | 0.0348 | 1.2204 | 0.0348 |
0.25 | 1.2832 | 0.0524 | 1.2822 | 0.0524 |
0.30 | 1.3481 | 0.0726 | 1.3469 | 0.0726 |
0.35 | 1.4160 | 0.0951 | 1.4146 | 0.0951 |
Comparison of computing time
t | Method | |||
---|---|---|---|---|
DT | Matlab | |||
\(u_{1}\) | \(u_{2}\) | \(u_{1}\) | \(u_{2}\) | |
0.05 | 7.9E−5 | 6.5E−5 | 6.2E−2 | 6.2E−2 |
0.10 | 6.9E−5 | 7.0E−5 | 8.1E−2 | 6.2E−2 |
0.15 | 7.1E−5 | 6.8E−5 | 6.2E−2 | 6.2E−2 |
0.20 | 7.0E−5 | 6.9E−5 | 6.3E−2 | 6.2E−2 |
0.25 | 7.0E−5 | 7.0E−5 | 6.2E−2 | 6.2E−2 |
0.30 | 6.3E−5 | 6.9E−5 | 6.3E−2 | 6.3E−2 |
0.35 | 6.9E−5 | 6.8E−5 | 6.2E−2 | 6.3E−2 |
Remark 5
System (15) contains all three types of delay which were considered in this paper. Moreover, it contains a term which is nonlinear (nonpolynomial) in the dependent variable \(u_{1}\). In this sense, the present paper contains more complicated systems in applications than papers about other semi-analytical methods like VIM [6], ADM [7] or HPM [8].
5 Conclusions
The approach presented in this paper is an effective semi-analytical technique, convenient for numerical approximation of a unique solution to the initial value problem for systems of functional differential equations, in particular delayed and neutral differential equations. Considering systems of equations with three types of delays brings a generalization with respect to the problem studied in [20]. The comparison of results was done against the Laplace decomposition method, residual power series method and Matlab package DDENSD. The need of computational work is reduced compared to the other methods. The differential transformation algorithm gives an approximate solution which is in good concordance with reference results produced by Matlab. Under certain circumstances, it is possible to identify the unique solution to the initial value problem in closed form. Further steps can be done in the development of the presented technique for systems with distributed and state dependent delays.
Declarations
Funding
The first author was supported by the Grant CEITEC 2020 (LQ1601) with financial support from the Ministry of Education, Youth and Sports of the Czech Republic under the National Sustainability Programme II. The work of the second author was supported by the Grant FEKT-S-17-4225 of Faculty of Electrical Engineering and Communication, Brno University of Technology.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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