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Keywords:

  • asymptotic stability;
  • θ-methods;
  • partial differential equation;
  • piecewise constant arguments

Abstract

  1. Top of page
  2. Abstract
  3. I Introduction
  4. II The stability of the analytic solution
  5. III The stability of the numerical solution
  6. IV Numerical Experiments
  7. References

This article is concerned with the stability analysis of the analytic and numerical solutions of a partial differential equation with piecewise constant arguments of mixed type. First, by means of the similar technique in Wiener and Debnath [Int J Math Math Sci 15 (1992), 781–788], the sufficient conditions under which the analytic solutions asymptotically stable are obtained. Then, the θ-methods are used to solve the above-mentioned equation, the sufficient conditions for the asymptotic stability of numerical methods are derived. Finally, some numerical experiments are given to demonstrate the conclusions.Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1-16, 2014


I Introduction

  1. Top of page
  2. Abstract
  3. I Introduction
  4. II The stability of the analytic solution
  5. III The stability of the numerical solution
  6. IV Numerical Experiments
  7. References

Recently, more and more researchers are concentrating on differential equations with piecewise constant arguments (EPCA), see for instance [1-6] and the references cited therein. It is well known that studies of EPCA were motivated by the fact that they represent a hybrid of continuous and discrete dynamical systems and combine the properties of both the differential and the difference equations. The theory of EPCA was initiated in [7, 8] and has been developed intensively in the last few decades. For a brief summary of the theory, the reader is referred to the book by Wiener [9]. This class of equations plays an important role in modeling phenomena of the real world, examples of the application of EPCA to the problems of biology, mechanics, and electronics can be seen in articles [7, 10-12]. So, it is valuable to investigate the properties of the solution of these equations.

We have found some results in the literature that directly deal with the qualitative investigation of EPCA, including the stability [13], the oscillations [14], and the periodicity [15, 16], and so forth. For further information of EPCA, the reader can see [17-21] and the references cited therein.

However, many EPCA cannot be solved analytically or their handling is more complicated. Therefore, taking numerical methods is a practical and nice choice. The original work for the numerical study of EPCA should be attributed to Liu et al. [22]. Subsequently, some works on the convergence, the stability, the oscillations, and the dissipativity of numerical methods for EPCA have been carried out. Song et al. [23] studied the stability of θ-methods for advanced EPCA. In the same year, Yang et al. [24] considered the numerical stability of Runge–Kutta methods for retarded EPCA. Later, Lv et al. [25] investigated the stability of Euler–Maclaurin methods for neutral EPCA. Recently, Song and Liu [26] discussed the convergence of the linear multistep methods for EPCA with one delay [t] and constructed an improved linear multistep method. Lv et al. [27] studied the analytical and numerical stability regions of Runge–Kutta methods for EPCA with complex coefficients. Moreover, The oscillations of θ-methods and Runge–Kutta methods were investigated in [28, 29] for retarded EPCA, respectively. In addition, Wang and Li [30] considered the dissipativity of Runge–Kutta methods for a class of neutral EPCA. There are also some authors who have considered the stability and the oscillations of numerical solutions for EPCA (see [31-34]). However, all of them are based on ODEs, up to now, the authors are not aware of any published results on the numerical solution of partial differential equations (PDEs) with piecewise constant arguments except for [35, 36]. In [35, 36], the authors studied the numerical stability of θ-methods and Galerkin methods for PDEs with one delay [t], respectively. Unlike [35, 36], the novelty of our article is that we will consider a more complicated equation and discuss the stability of analytic solution and numerical solution, respectively.

With respect to the PDEs with piecewise constant time, it has been shown in [37] that they naturally arise in the process of approximating PDEs using piecewise constant arguments. And it is important to investigate boundary value problems (BVP) and initial-value problems for EPCA in partial derivatives and explore the influence of certain discontinuous delays on the behavior of solutions to some typical problems of mathematical physics.

In this work, we shall consider the following problem

  • display math(1)

where inline image and inline image signifies the greatest integer function. The main goal of the work is to investigate the stability of analytic solution and numerical solution for (1), respectively.

This article is divided into four sections and is organized as follows. In Section II, we establish the stability conditions of the analytic solutions for (1). The approach we use is based on the theory of separation of variables. In Section III, we study the numerical stability of (1) in the θ-methods and state our main result. Several numerical experiments are presented in Section IV of this article which confirm our results.

II The stability of the analytic solution

  1. Top of page
  2. Abstract
  3. I Introduction
  4. II The stability of the analytic solution
  5. III The stability of the numerical solution
  6. IV Numerical Experiments
  7. References

In this section, we will give the conditions under which the analytic solution of (1) is asymptotically stable.

Definition 1. (see [9]) A solution of (1) is a function inline image satisfies the conditions

  • inline image is continuous in inline image,
  • The partial derivative inline imageexist and are continuous in Ω with the possible exception of the points inline image, where one-sided derivatives exist,
  • inline imagesatisfies inline imagein Ω with the possible exception of the points inline image, and conditions inline imageand inline image .

Definition 2. (see [9]) If any solution inline image of (1) satisfies

  • display math

then the zero solution of (1) is called asymptotically stable.

In order to determine the stability conditions of (1), we will make use of the similar methods in [38], consider the following problem

  • display math(2)

where inline image and inline image are real inline image matrices, inline image, and C are real constant inline image matrices and A is nonsingular.

By the method of separation of variables, we find the nonzero solution of (2) in the form

  • display math

substituting it into the equation in ((2)) yields

  • display math

that is

  • display math

so we have

  • display math

which generates the BVP

  • display math(3)

and equation

  • display math(4)

The general solution of equation in ((2)) is

  • display math

where

  • display math

and inline image and inline image are arbitrary constant matrices.

From inline image, we conclude that inline image, and inline image gives inline image. Moreover, we set

  • display math

in (3) and the following result is obtained.

Theorem 1. There exists an infinite sequence of matrix eigenfunctions for (3)

  • display math

which is complete and orthogonal in the space inline image of inline image matrices.

The solution of (4) will be given in the following theorem.

Theorem 2. Let inline image be the solution of the problem

  • display math

and let

  • display math

then, the problem

  • display math

has a unique solution

  • display math(5)

Proof. On the interval inline image, from (4) it follows that

  • display math
  • display math(6)

according to Theorem 2.2 in [38], the general solution of (6) is as follows

  • display math

at t = n

  • display math

thus

  • display math

then, we have

  • display math

which is equivalent to

  • display math

at inline image

  • display math

that is

  • display math

we arrive at

  • display math

and we further obtain that

  • display math
  • display math
  • display math(7)

So, we get the first main result of this article.

Corollary 1. If the following conditions

  • display math(8)

or

  • display math(9)

are satisfied, then the zero solution of (1) is asymptotically stable.

Proof. Let

  • display math

according to Theorems 1 and 2, we have

  • display math(10)

Then by Theorem 2.3 in [38] we know that the zero solution of (1) is asymptotically stable if and only if inline image, so we have

  • display math(11)

Therefore,

  • display math(12)

which is equivalent to

  • display math

thus, we have

  • display math(13)

or

  • display math(14)

Hence, from (13) it follows that

  • display math(15)

we can obtain (8) by using the monotonicity of function inline image. On the other hand, (9) can be got from (14) in the same way.

III The stability of the numerical solution

  1. Top of page
  2. Abstract
  3. I Introduction
  4. II The stability of the analytic solution
  5. III The stability of the numerical solution
  6. IV Numerical Experiments
  7. References

In this section, we focus on the numerical asymptotic stability of θ-methods with respect to (1).

A θ-Methods

Let inline image and inline image are step-sizes of spatial and time directions which satisfy inline image and inline image, respectively, where inline image are positive integers. Denote the spatial and time nodes as inline image and inline image, respectively, and inline image as an approximation to inline image. Applying the θ-methods to (1), we have

  • display math(16)

where inline image is an approximation to inline image.

If we denote inline image, then both inline image and inline image can be defined as inline image according to Definition 1. In a similar way, both inline image and inline image can be defined as inline image. Then, the first term of (16) can be written as

  • display math
  • display math(17)

Remark 1. For the θ-methods

  • The explicit finite difference scheme is given when inline image
  • The Crank–Nicolson implicit finite difference scheme is given when inline image
  • The fully implicit finite difference scheme is given when inline image

Let inline image, then (17) can be written as

  • display math
  • display math
  • display math(18)

and let inline image, (18) yields

  • display math

where inline image.

Denote inline image and inline image triple-diagonal matrix inline image. Then (16) can be written as

  • display math(19)

B Stability Analysis

Definition 3. If any solution of (17) satisfies

  • display math

then, the zero solution of (16) is called asymptotically stable.

Let inline image denote the eigenvalue of the matrix F. We have

Lemma 1. (see [39]) The matrix F is a positive definite matrix, with the eigenvalues

  • display math

From (19), we can get

  • display math(20)

where

  • display math
  • display math
  • display math

Iteration of (20) gives

  • display math
  • display math

and

  • display math

It is easy to know that

  • display math

thus,

  • display math

so, we have

  • display math(21)

where

  • display math

therefore, we obtain the other two main results of this article.

Theorem 3. Under the condition (8) and

  • display math(22)

and

  • display math(23)

for m is even or

  • display math(24)

for m is odd, the zero solution of (19) is asymptotically stable.

Proof. From (21), we know that the zero solution of (19) is asymptotically stable if and only if the eigenvalue of matrix G satisfies

  • display math(25)

For convenience, let

  • display math

obviously, inline image. Then, the inequality (25) gives

  • display math(26)

From (26), we have

  • display math

and so

  • display math(27)

If m is even, in view of (22) and (23), we can get inline image and inline image. Then together with (8) we know that (27) is satisfied. So (25) holds. Therefore, the zero solution of (19) is asymptotically stable. Similarly, the case of m is odd can be easily obtained.

Theorem 4. Under the condition (9) and

  • display math(28)

or

  • display math(29)

for m is odd or

  • display math(30)

for m is even, the zero solution of (19) is asymptotically stable, where inline image in (28) denotes that r does not exist.

Proof. Similar to the proof of Theorem 3.

IV Numerical Experiments

  1. Top of page
  2. Abstract
  3. I Introduction
  4. II The stability of the analytic solution
  5. III The stability of the numerical solution
  6. IV Numerical Experiments
  7. References

In this section, we give some examples to illustrate the conclusions in the article. Consider the following two problems

  • display math(31)
  • display math(32)

It is not difficult to verify that the coefficients in (31) and (32) satisfy the conditions (8) and (9), respectively. In Figs. 1-3, we draw the numerical solutions of (31) with different parameters. Moreover, in Figs. 4-7, we also draw the numerical solutions of (32) with different parameters. It is easy to see that the numerical solutions are asymptotically stable. A detailed description is as follows. In Fig. 1, let inline image, and inline image, it is easy to calculate that inline image according to (23). Obviously, inline image. Therefore, the numerical solution of (31) is asymptotically stable according to Theorem 3, which is in agreement with Fig. 1. For other cases, we can verify them in the similar line (see Figs. 2-7, Theorems 3 and 4). These numerical examples confirm our theoretical results in this article.

image

Figure 1. The numerical solution of (31) with inline image and inline image.

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image

Figure 2. The numerical solution of (31) with inline image and inline image.

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image

Figure 3. The numerical solution of (31) with inline image and inline image.

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image

Figure 4. The numerical solution of (32) with inline image and inline image.

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image

Figure 5. The numerical solution of (32) with inline image and inline image.

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image

Figure 6. The numerical solution of (32) with inline image and inline image.

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image

Figure 7. The numerical solution of (32) with inline image and inline image.

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The authors thank the reviewers, Prof. Mingzhu Liu and Prof. Hui Liang for their helpful comments and constructive suggestions to improve the quality of the article.

References

  1. Top of page
  2. Abstract
  3. I Introduction
  4. II The stability of the analytic solution
  5. III The stability of the numerical solution
  6. IV Numerical Experiments
  7. References
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