An approximate solution for a neutral functional–differential equation with proportional delays
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, with the initial conditions where a and bk(k = 0, 1, …, m − 1) are given analytical functions, and β, pk, cik, λk denote given constants with 0 < pk < 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 . 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, …, xm + 1, cos kx, sin kx, k = 1, 2, ⋅⋅⋅} is a completely independent system of .
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
Lemma 2.2 If any w ∈ W1, then. Proof Assume that
The ɛ-approximate solution of Eq. (1) with (2)
Put Then Eq. (1) with (2) turns into subject to where . are all bounded linear operators. And are both reproducing kernel spaces defined in Section 2.
Assume Eq. (5) with (6) has an unique solution. In this section, we assume that is a completely independent system of . 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:
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).
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2018, Applied Mathematics and ComputationCitation 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.
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