Piecewise reproducing kernel method for linear impulsive delay differential equations with piecewise constant arguments

doi:10.1016/j.amc.2018.12.054

Applied Mathematics and Computation

Volume 349, 15 May 2019, Pages 304-313

https://doi.org/10.1016/j.amc.2018.12.054 Get rights and content

Abstract

In this paper, we introduce a piecewise reproducing kernel method for impulsive delay differential equations with piecewise constant arguments. The method is an improved reproducing kernel method. Compared with the classical reproducing kernel method, the solutions obtained using the present method can give good approximations for a larger time interval. Some numerical examples are used to show the effectiveness and simplicity of the method.

Introduction

Consider the following impulsive differential equations with piecewise constant argument ${\begin{matrix} u^{'} (t) + p (t) u (t) + q (t) u ([t - 1]) = f (t), t \geq 0, t \neq k, k = 1, 2, \dots, \\ ▵ u (k) = r u (k), k = 1, 2, \dots, \\ u (- 1) = A_{1}, u (0) = A_{0}, \end{matrix}$ where p(t), q(t) and f(t) are continuous functions, A₀ and A₁ are real constants, $▵ u (k) = u (k +) - u (k -)$ and [  ·  ] denotes the greatest integer function. Furthermore, we assume that (1.1) has a unique solution.

Impulsive differential equations have been applied successfully in ecology, population dynamic, optimal control, etc. Therefore, impulsive differential equations have attracted much attention. It is usually impossible to obtain the exact solution of such equations. Hence, it is important to develop numerical methods for solving such problems. Liu and Zeng [1], [2], [3] proposed the linear multistep methods for impulsive differential equations and impulsive delay differential equations. Zhang and Liang [4] obtained global superconvergence and local superconvergence of the collocation solution for linear impulsive differential equations. Zhang [5], [6] discussed the stability of Runge–Kutta methods for linear impulsive delay differential equations and linear impulsive delay differential equations with piecewise constant arguments.

Based on the reproducing kernel theory, a method called the reproducing kernel method (RKM) was developed by Geng and Cui [7], [8], [9]. The method has been widely applied to many fields [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34]. However, the direct application of the RKM to the considered impulsive delay differential equations with piecewise constant arguments can not produce accurate numerical solutions. Recently, based on the proposed RKM, Geng and Qian [13], [14] present the piecewise reproducing kernel method (PRKM) for singularly perturbed problems. Following the idea in [13], [14], in this paper, we will present the PRKM for impulsive delay differential equations with piecewise constant arguments.

The rest of the paper is organized as follows. In the next Section, some basic definitions and properties of the reproducing kernel theory is introduced. RKM for linear differential equations is introduced in Section 3. PEKM for impulsive differential equations with piecewise constant argument is presented in Section 4. Error analysis are introduced in Section 5. The numerical examples are provided in Section 6. Section 7 ends this paper with a brief conclusion.

Section snippets

Preliminaries

In this section, we provide the definition of solution to impulsive differential equations with piecewise constant argument, and some basic definitions and properties of the reproducing kernel Hilbert space which are useful in the following discussion.

Definition 2.1

[35] A function $u : R^{+} \cup {- 1} \to R$ is said to be a solution of (1.1) if it satisfies the following conditions:

(a) u(t) is continuous for t ∈ [0, ∞) with the possible exception of the points [t] ∈ [0, ∞),

(b) u(t) is right continuous and has left-hand

RKM for linear differential equations

Here introduce the RKM for linear differential operator equation ${\begin{matrix} L u (t) = u^{'} (t) + p (t) u (t) = g (t), t \in (a, b), \\ u (a) = 0 . \end{matrix}$ Here we only consider homogeneous initial condition $u (a) = 0$ since the inhomogeneous initial conditions $u (a) = α$ can be reduced to $u (a) = 0$ easily. Under the assumption that (3.1) has a unique solution, we shall give the representation of solution of (3.1) in RKHS W³[a, b] in which every function satisfies the homogenous initial condition of (3.1).

Put $ψ_{i} (t) = L_{s} {K (t, s) |}_{s = t_{i}}$ . The orthonormal system ${ψ$

PRKM for impulsive differential equations with piecewise constant argument

In this section, we illustrate the PRKM for impulsive differential equations with piecewise constant argument (1.1). We solve (1.1) by using the RKM in a piecewise fashion.

Consider (1.1) on [0, T]. We first divide [0, T] into M equidistant subintervals $[t_{j}, t_{j + 1}], j = 0, 1, \dots, M - 1,$ with $t_{0} = 0$ and $t_{M} = T$ . It is required that ${k : k \in N, k \leq T} \subset {t_{i}}_{i = 0}^{M}$ . Denote by h the length of the subinterval, i.e., $h = \frac{T}{M}$ . Then we get the approximate solutions on every interval $[t_{i - 1}, t_{i}]$ by using the RKM in the RKHS $W^{3} [t_{i - 1}, t_{i}]$ .

Error analysis

From Li and Wu [16], we have the following two theorems.

Theorem 5.1

If y(x) is the solution of ${\begin{matrix} L y (t) = g (t), t \in [a, b] \\ y (a) = α \end{matrix}$ and p(t) ≥ γ > 0, then $| y (x) | \leq \frac{1}{γ} \max {| y (a) |, max_{t \in [a, b]} | L y |} .$

Theorem 5.2

The approximation u_{1, N}(t) to the solution of (1.1) on [t₀, t₁] obtained by using the RKM in space W³[t₀, t₁] satisfies $∥ u (t) - u_{1, N} (t) ∥_{\infty} = max_{t \in [t_{0}, t_{1}]} | u (t) - u_{1, N} (t) | \leq c_{1} {\bar{h}}^{2},$ where $\bar{h} = \frac{h}{N},$ c₁ is a positive constant.

Proof

The proof follows directly from Li and Wu [15] and Theorem 5.1. □

Following we consider the error between u_{2, N}(t) and u(t) on

Numerical experiments

Example 6.1

Consider the following impulsive delay differential equations with piecewise constant arguments [6] ${\begin{matrix} u^{'} (t) - u (t) + u ([t - 1]) = 0, t \geq 0, t \neq k, k = 1, 2, \dots, \\ ▵ u (k) = 9 u (k), k = 1, 2, \dots, u (- 1) = 1, u (0) = 1, \end{matrix}$

Its exact solution can be obtained by Theorem 2.2. Using the present method, taking $T = 10,$ $M = 10,$ the numerical results are compared with [6] in Table 1. The exact solution and approximate solution are shown in Fig. 1. The absolute errors between the approximate solution and exact solution are shown in Fig. 2. Table 1 and

Conclusion

In this paper, a new effective method is proposed for solving impulsive delay differential equations with piecewise constant arguments. The comparison of numerical results show that the present method can give accurate approximate solutions.

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.11801044, No.11326237, No.11271100).

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Piecewise reproducing kernel method for linear impulsive delay differential equations with piecewise constant arguments

Abstract

Introduction

Section snippets

Preliminaries

RKM for linear differential equations

PRKM for impulsive differential equations with piecewise constant argument

Error analysis

Numerical experiments

Conclusion

Acknowledgments

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