An approximate solution for a neutral functional–differential equation with proportional delays

doi:10.1016/j.amc.2015.03.040

Applied Mathematics and Computation

Volume 260, 1 June 2015, Pages 27-34

https://doi.org/10.1016/j.amc.2015.03.040 Get rights and content

Abstract

In this paper, a novel algorithm based on reproducing kernel theory for neutral functional–differential equation with proportional delays is proposed. The advantages of the presented method are the establishment of complete ɛ-approximate solution theory and high precision of absolute error. Some examples are given to demonstrate the validity and applicability of the new method and some comparisons are made with the existing results.

Introduction

Functional–differential equations with proportional delays are usually referred as pantograph equations or generalized pantograph equations, the name of which was originated by Ockendon and Tayler in connection with the dynamics of a current collection system on an electric locomotive [1]. The pantograph equations occur in a wide variety of many phenomena in applied sciences, such as number theory, electrodynamics, astrophysics, nonlinear dynamical systems, probability theory on algebraic structure, quantum mechanics and cell growth [1], [2], [3].

In recent years, pantograph equations have been studied by many authors who have investigated both their analytical and numerical aspects. Here, we briefly review a limited number of them. In [4], [15], the multi-pantograph equations were investigated using Runge–Kutta method. The papers [5], [6], [16], [17] studied the numerical solutions of delay differential equations by collocation method. Authors of [7] considered using Chebyshev polynomials method to obtain the numerical solution of the pantograph equation. Sezer et al. [8], [9] obtained the approximate solution of the pantograph equation based on the Taylor method. In [10], Alomari et al. used homotopy analysis method to solve a class of delay differential equations. Recently, the authors in [11], [12] developed Variational iteration method and Homotopy perturbation method to solve neutral functional–differential equation with proportional delays, respectively. Moreover, Heydari et al. [13] solved pantograph equation with neutral term successfully by the method which consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions.

In this paper, we employ a novel method named optimal residual method to solve the following neutral functional–differential equation with proportional delays, $\begin{matrix} {(u (t) + a (t) u (p_{m} t))}^{(m)} = β u (t) + \sum_{k = 0}^{m - 1} b_{k} (t) u^{(k)} (p_{k} t) + f (t), 0 \leq t \leq T \end{matrix}$ with the initial conditions $\begin{matrix} \sum_{k = 0}^{m - 1} c_{i k} u^{(k)} (0) = λ_{i}, i = 0, 1, \dots, m - 1, \end{matrix}$ where a and b_k(k = 0, 1, …, m − 1) are given analytical functions, and β, p_k, c_ik, λ_k denote given constants with 0 < p_k < 1(k = 0, 1, …, m). Differing from the previous methods, here, we will build a complete ɛ-approximate solution theory based on reproducing kernel theory, and the ɛ-approximate solution can be easily obtained by solving normal equations.

The rest of the paper is organized as follows. In Section 2, we introduce some mathematical preliminaries about the dense subset of $W_{2}^{n}$ . Section 3 is devoted to applying optimal residual method for solving neutral functional-differential equation with proportional delays base on reproducing kernel theory. In Section 4, some numerical results are given to clarify the method, and comparisons are made with methods that were reported in other published works in the literature. Finally, a conclusion is given in Section 5.

Section snippets

Preliminaries

In this section, we will prove that {1, x, x², …, x^{m + 1}, cos kx, sin kx, k = 1, 2, ⋅⋅⋅} is a completely independent system of $W_{2}^{m + 1}$ .

Lemma 2.1

The space W₁ = span{1, cos kx, sin kx, k = 1, 2, 3, ⋅⋅⋅} is a dense subset of L²[0, T]. (see [18]).

Put $\begin{matrix} W_{2}^{m + 1} ≜ W_{2}^{m + 1} [0, T] = {u (x) | u^{(m)} (x) i s a n a b s o l u t e l y c o n t i n u o u s r e a l - v a l u e d \\ f u n c t i o n o n [0, T] a n d u^{(m + 1)} (x) \in L^{2} [0, T]}, \\ (u, v) = \sum_{i = 0}^{m} u^{(i)} (0) v^{(i)} (0) + \int_{0}^{T} u^{(m + 1)} (x) v^{(m + 1)} (x) d x, u, v \in W_{2}^{m + 1}, \\ W = s p a n {1, x, x^{2}, \dots, x^{m + 1}, \cos k x, \sin k x, k = 1, 2, 3, \dots} . \end{matrix}$

Lemma 2.2

If any w ∈ W₁, then $\int_{0}^{x} {(x - t)}^{m} w (t) d t \in W$ .

Proof

Assume that $w = a_{0} + \sum_{k}$

The ɛ-approximate solution of Eq. (1) with (2)

Put $\begin{matrix} (A u) (t) = {(u (t) + a (t) u (p_{m} t))}^{(m)} - β u (t) - \sum_{k = 0}^{m - 1} b_{k} (t) u^{(k)} (p_{k} t) \\ (B_{i} u) (t) = \sum_{k = 0}^{m - 1} c_{i k} u^{(k)} (0), i = 0, 1, \dots, m - 1 . \end{matrix}$ Then Eq. (1) with (2) turns into $\begin{matrix} A u = f \end{matrix}$ subject to $\begin{matrix} B_{i} u = λ_{i}, i = 0, 1, \dots, m - 1, \end{matrix}$ where $u \in W_{2}^{m + 1} [0, T],$ $f \in W_{2}^{1} [0, T],$ $λ_{i} \in R$ . $A : W_{2}^{m + 1} [0, T] ⟶ W_{2}^{1} [0, T],$ $B_{i} : W_{2}^{m + 1} [0, T] ⟶ R$ are all bounded linear operators. And $W_{2}^{m + 1} [0, T],$ $W_{2}^{1} [0, T]$ are both reproducing kernel spaces defined in Section 2.

Assume Eq. (5) with (6) has an unique solution. In this section, we assume that ${ϕ_{j}}_{j = 1}^{\infty}$ is a completely independent system of $W_{2}^{m + 1}$ . For

Numerical experiments

In this part, some examples are provided to illustrate performance of the proposed method. For the sake of comparing purposes, we consider the same examples as used in [11].

Example 1

Consider the following first-order neutral functional–differential equation with proportional delay: $\begin{matrix} {\begin{matrix} u^{'} (t) = - u (t) + \frac{1}{2} u (\frac{t}{2}) + \frac{1}{2} u^{'} (\frac{t}{2}), 0 < t < 1, \\ u (0) = 1 . \end{matrix} \end{matrix}$ The exact solution is u(t) = e^−t. In Table 1, we compare the absolute errors of the optimal residual method (for n = 4, 5, 6) with the ones for the two-stage order-one Runge–Kutta

Conclusion

In this paper, the optimal residual method based on reproducing kernel theory is employed successfully to study neutral functional–differential equations with proportional delays. Comparing with Runge–Kutta method and variational iteration method, the results of numerical examples demonstrate that the presented method is more accurate.

Acknowledgments

The author would like to express thanks to the unknown referees for their careful reading and helpful comments. The work was supported by the National Natural Science Foundation of China (Grant No. 11401139).

References (18)

D. Li et al.
Runge–kutta methods for the multi-pantograph delay equation
Appl. Math. Comput.
(2005)
Y. Muroya et al.
On the attainable order of collocation methods for pantograph integro-differential equations
J. Comput. Appl. Math.
(2003)
E. Ishiwata et al.
Rational approximation method for delay differential equations with proportional delay
Appl. Math. Comput.
(2007)
S. Sedaghat et al.
Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials
Commun. Nonlinear Sci.
(2012)
M. Sezer et al.
A taylor method for numerical solution of generalized pantograph equations with linear functional argument
J. Comput. Appl. Math.
(2007)
M. Sezer et al.
Approximate solution of multi-pantograph equation with variable coefficients
J. Comput. Appl. Math.
(2008)
X. Chen et al.
The variational iteration method for solving a neutral functional-differential equation with proportional delays
Comput. Math. Appl.
(2010)
J. Biazar et al.
The homotopy perturbation method for solving neutral functional-differential equations with proportional delays
J. King Saud Univ.
(2012)
M. Heydari et al.
Operational matrices of Chebyshev cardinal functions and their application for solving delay differential equations arising in electrodynamics with error estimation
Appl. Math. Modell.
(2013)

There are more references available in the full text version of this article.

Cited by (6)

On solving systems of multi-pantograph equations via spectral tau method
2018, Applied Mathematics and Computation
Citation Excerpt :
As an important kind of delay differential equations, the pantograph equation has been used in modeling many real-life phenomena, such as economy, biology, astrophysics, control and electrodynamics, see [1–3]. Various numerical approaches were utilized for pantograph equations; Bota and Cãruntu [4] applied the ϵ-Approximate polynomial method for solving multi-pantograph equations with variable coefficients, Akkaya et al. [5] applied a numerical approach based on the First Boubaker polynomials for solving pantograph delay differential equations, Doha et al. [6] applied the Jacobi rational-Gauss collocation method for solving generalized pantograph equations, Ahmed and Mukhtar [7] applied a stochastic approach for solving multi-pantograph equations, Cheng et al. [8] applied the reproducing kernel method for neutral functional-differential equation, Reutskiy [9] used the spectral collocation method for approximating the solution of pantograph functional differential equations, Akyuz-Dascioglu and Sezer [10] applied the Bernoulli collocation method for solving high-order generalized pantograph equations and Javadi et al. [11] introduced a numerical method based on Bernstein polynomials to solve generalized pantograph equations. As a spectral approach for solving numerically some kinds of differential equations, the tau method is classified as one of the most important used methods due to its high convergence and accuracy.
The current manuscript focuses on solving systems of multi-pantograph equations. The spectral tau method is applied for solving systems of multi-pantograph equations with shifted Jacobi polynomials as basis functions. The convergence analysis of the proposed technique is also investigated. We introduced the numerical solutions of some test problems and compared the obtained numerical solutions of such problems with those given using different numerical methods.
Robust spectral treatment for time-fractional delay partial differential equations
2023, Computational and Applied Mathematics
Haar Wavelet Series Method for Solving Simultaneous Proportional Delay Differential Equations
2023, Lecture Notes in Networks and Systems
A new stable collocation method for solving a class of nonlinear fractional delay differential equations
2020, Numerical Algorithms
Legendre wavelet solution of neutral differential equations with proportional delays
2019, Journal of Applied Mathematics and Computing
A novel hybrid method for solving combined functional neutral differential equations with several delays and investigation of convergence rate via residual function
2019, Computational Methods for Differential Equations

View full text

An approximate solution for a neutral functional–differential equation with proportional delays

Abstract

Introduction

Section snippets

Preliminaries

The ɛ-approximate solution of Eq. (1) with (2)

Numerical experiments

Conclusion

Acknowledgments

Appl. Math. Comput.

J. Comput. Appl. Math.

Appl. Math. Comput.

Commun. Nonlinear Sci.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

Comput. Math. Appl.

J. King Saud Univ.

Appl. Math. Modell.