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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 642318, 21 pages
http://dx.doi.org/10.1155/2012/642318
Research Article

A Class of New Pouzet-Runge-Kutta-Type Methods for Nonlinear Functional Integro-Differential Equations

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received 2 October 2011; Accepted 7 February 2012

Academic Editor: Shaher Momani

Copyright © 2012 Chengjian Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents a class of new numerical methods for nonlinear functional-integrodifferential equations, which are derived by an adaptation of Pouzet-Runge-Kutta methods originally introduced for standard Volterra integrodifferential equations. Based on the nonclassical Lipschitz condition, analytical and numerical stability is studied and some novel stability criteria are obtained. Numerical experiments further illustrate the theoretical results and the effectiveness of the methods. In the end, a comparison between the presented methods and the existed related methods is given.

1. Introduction

In the last ten years the numerical analysis and computational solution of various types of functional-integrodifferential equations (FIDEs) have received considerable attention. Many of these numerical schemes were derived by suitably adapting classical numerical methods for ordinary differential equations (ODEs) or integrodifferential equations (IDEs) to FIDEs, and there is a growing literature on their convergence and stability properties. Of these papers, Zhang and Vandewalle [1, 2] dealt with nonlinear numerical stability of FIDEs of the form Yu et al. [3] extended the analysis to FIDEs of neutral-type, while Zhang et al. [4, 5] derived several improved numerical stability results for neutral FIDEs of the form (1.2). For the class of neutral FIDEs: Brunner and Vermiglio [6] made an insight into the analytical and numerical stability of continuous Runge-Kutta methods. Moreover, in the papers [7, 8], Brunner presented superconvergence results of collocation methods for several classes of FIDEs with constant or variable (vanishing) delays.

The reader may wish to consult Baker’s survey paper [9] and Brunner’s monograph [10] for details on related earlier work and for additional references.

However, up to now, no numerical investigation appears to have been carried out for general nonlinear FIDEs of the form: in which the integral term on the left-hand side is no longer pure delay type: in contrast to the FIDEs (1.1)–(1.3), it contains information on the solution on the interval . Hence, the numerical analysis and computational solution of (1.4) is rather more complex than is the case for (1.1)–(1.3).

In the present paper, with an adaptation of the underlying Pouzet-Runge-Kutta methods (cf. Brunner and van der Houwen [11]), we obtain a class of new numerical methods for nonlinear FIDEs (1.4) and study analytical and numerical stability of the equations. The paper is organized as follows. In Section 2 we derive results on the asymptotic stability of analytical solutions, under the assumption of nonclassical Lipchitz conditions. Section 3 describes the adaptation of the Pouzet-Runge-Kutta method to the FIDE (1.4). In Section 4 some lemmas are given which will play a key role in the analysis of the global and asymptotical stability properties of the Pouzet-Runge-Kutta solutions (Section 5). Here, we also state stability results for a number of concrete methods. Some numerical experiments are given in Section 6 to illustrate the theoretical results and the effectiveness of the methods. Finally, in Section 7, a comparison between the presented methods and the existed related methods is given.

2. Stability Results for Exact Solutions

Let and denote a given inner product and its induced norm on the -dimensional complex space , respectively. The functions ,, are assumed to be continuous and possess the properties where are nonnegative constants. We will refer to the class of FIDEs of the form (1.4) which satisfies (2.1)-(2.2) as FIDEs of class .

In order to study the stability of solutions to (1.4), we need to consider the system with a different initial function , which also belongs to the class .

Definition 2.1. System (1.4) is called globally stable if there exists a constant such that Moreover, system (1.4) is called asymptotically stable if
In order to gain insight into the global and asymptotical stability of system (1.4), we will use the generalized Halanay inequality as presented in Wang [12], compare also to [13].

Lemma 2.2 (see [12]). Assume that the functions satisfy the inequalities Here, are given nonnegative continuous functions on for which there exist constants such that Then the following inequalities hold: where

With this lemma, we will be able to obtain an analytical stability result for the system (1.4). In order to do so, we introduce some notations:

Theorem 2.3. Assume that the system (1.4) belongs to the class with Then this system is globally and asymptotically stable.

Proof. It follows from (2.1), (1.4), and (2.3) that By (2.2), it holds that Substituting (2.13) into (2.12) yields Note that This, together with and (2.13), implies that Hence, by combining (2.14) and (2.16) we are led to On the other hand, we have where we have used (2.13). Therefore, under the condition (2.11), an application of Lemma 2.2 to (2.17)-(2.18) yields the conclusion.

As an application of Theorem 2.3, we present several examples as follows.

Example 2.4. Consider the -dimensional system of linear functional-integrodifferential equation where is a known continuous function such that (2.19) has a unique solution. It is easy to check that system (2.19) belongs to the class if, provided there exist constants such that, for all and , in which the matrix norm and the logarithmic norm are induced by the vector inner-product norm. A direct application of Theorem 2.3 shows that the system (2.19) is globally and asymptotically stable whenever condition (2.20) and the following condition hold:

Example 2.5. Consider the system of partial functional-integrodifferential equations where is a continuous function chosen such that this system has the exact solution . By discretizing the spatial variable by a uniform mesh , the system (2.22) can be transformed into a system of ordinary functional-integrodifferential equations, where When the standard inner product and its induced norm are used, one easily verifies that the system (2.23) belongs to the class . Moreover, in light of Theorem 2.3, we know that the system (2.23) is globally and asymptotically stable.

3. The Pouzet-Runge-Kutta Discretization

In order to obtain a class of effective numerical schemes for FIDEs of the form (1.4), we first recall some related concepts and results on the underlying Runge-Kutta methods for ordinary differential equations (see, e.g., [14]). An -stage Runge-Kutta method is described by the Butcher tableau where

A Runge-Kutta method (3.1) is called algebraically stable if

where the notation “≥” means that a matrix is nonnegative definite. It is said to be strictly stable at infinity if exists and satisfies , where

and denotes the identity matrix; it is said DJ-irreducible if there is no nonempty index set such that

S-irreducible if there is no partition of with such that for all and

and irreducible if it is both DJ-irreducible and S-irreducible.

Proposition 3.1 (cf. [15]). A DJ-irreducible, algebraically stable Runge-Kutta methods satisfies  for all .

Proposition 3.2 (cf. [14]). Assume that a Runge-Kutta method (3.1) with distinct and positive satisfies the simplifying condition . Then this method is algebraically stable if and only if  .

The class of extended Pouzet-Runge-Kutta methods for FIDEs (1.4) with underlying Runge-Kutta method (3.1) is given by Here, the stepsize is chosen as ,  , , , and are approximations to , , and , respectively, with denoting the memory term

The integral approximations and are given by the Pouzet quadrature rules Moreover, it is assumed that

4. Some Basic Lemmas

In the subsequent numerical analysis we will imply the following notations:

Moreover, when the method (3.7)–(3.10) is applied to the problem (2.3), we set

and denote the corresponding approximations of , and , by , and , respectively. Similarly, it is assumed that

As mentioned in Section 1, the following lemmas will play key roles in derivation of our main results.

Lemma 4.1. Assume that the underlying Runge-Kutta method (3.1) is algebraically stable, and the conditions (2.1)-(2.2) hold. Then the extended Pouzet-Runge-Kutta scheme (3.7) satisfies

Proof. A straightforward computation and the assumption of algebraic stability lead to where . Hence, it holds that By (2.1), we further have for , An induction argument shows that the inequality (4.7) implies in which Also, the assumptions and allow us to write A combination of (4.8), (4.9), and (4.10) yields (4.4). Hence the lemma is proven.

Lemma 4.2. Under the condition (2.2), the Pouzet quadrature rule (3.9) satisfies

Proof. By (2.2), and the Cauchy-Schwarz inequality, we have Also, it follows from (3.11), (4.3), and that A combination of (4.13)–(4.15) yields that for all , Inserting (4.16) into (4.12) generates (4.11). This completes the proof.

Lemma 4.3 (cf. [5]). Under the condition (2.2), the Pouzet quadrature rule (3.10) satisfies

5. Numerical Stability

This section will involve the analysis of the global and asymptotical stability properties of the Pouzet-Runge-Kutta method (3.7)–(3.10). The relevant numerical stability concepts are defined as follows.

Definition 5.1. The Pouzet-Runge-Kutta method (3.7)–(3.10) is called globally stable for the problems of class if there exists a stability constant , which depends only on and the method, such that Moreover, the method (3.7)–(3.10) is called asymptotically stable for the problems of class if
We first establish a result on the global stability of Pouzet-Runge-Kutta methods. An analogous result on their asymptotical stability will be given in Theorem 5.7.

Theorem 5.2. Assume that the underlying Runge-Kutta method (3.1) with positive is algebraically stable. Then the corresponding Pouzet-Runge-Kutta method (3.7)–(3.10) is globally stable for the class , with stability constant whenever where .

Proof. It follows from Lemma 4.3 and that Also, we have that and, by the inequality (3.13) in [1], Inserting both (5.6) and (5.7) into (5.5) yields With , (3.11) and (4.3), the left side of (5.8) can be bounded by Substituting this bound into (4.4) yields which, together with (4.11), (5.10), and , implies that for , Observing the condition (5.4) to (5.11), we readily derive the global stability conclusion. Hence, the theorem is proven.

Applying Proposition 3.1 and Proposition 3.2 to Theorem 5.2, respectively, we can obtain the following corollaries.

Corollary 5.3. Assume that an underlying Runge-Kutta method is DJ-irreducible and algebraically stable. Then the corresponding Pouzet-Runge-Kutta method (3.7)–(3.10) is globally stable for the class whenever (5.4) holds.

Corollary 5.4. Assume that an underlying Runge-Kutta method, with distinct and positive , satisfies the simplifying condition , and . Then the corresponding Pouzet-Runge-Kutta method (3.7)–(3.10) is globally stable for the class whenever (5.4) holds.

Remark 5.5. In [14, Chapters IV.5 and IV.12], Hairer and Wanner have shown that the Runge-Kutta methods of type Gauss, Radau IA, Radau IIA, and Lobatto IIIC are all algebraically stable and have invertible matrices , distinct , and for all . Hence, by Theorem 5.2, we have a more concrete stability result for the Pouzet-Runge-Kutta schemes based on these important Runge-Kutta methods.

Corollary 5.6. Assume that an underlying Runge-Kutta method (3.1) is of type Gauss, Radau IA, Radau IIA, or Lobatto IIIC. Then the corresponding Pouzet-Runge-Kutta method (3.7)–(3.10) is globally stable for the class whenever (5.4) holds.

Next, we will address the asymptotic stability of the extended Pouzet-Runge-Kutta methods. The notation,

will subsequently be used, and we define a vector norm on the space by

Moreover, we will employ the Kronecker product and its well-known properties (cf. [16]).

Theorem 5.7. Assume that an underlying Runge-Kutta method (3.1) is irreducible, algebraically stable, and strictly stable at infinity. Then the corresponding Pouzet-Runge-Kutta method (3.7)–(3.10) is asymptotically stable for the class whenever (5.4) holds.

Proof. By (3.7) we have When the matrix is invertible, it follows from the first equation of (5.14) that Insertion of (5.15) into the second equation of (5.14) yields When is singular, we adopt a technique, proposed by Hairer and Wanner [14]. It consists in replacing by the regular matrix everywhere, followed by considering the limit , whose existence is assured by irreducibility and algebraic stability of the method (see Lemma  3.2 in [14]). Thus, we set Hence, it holds that where . Applying Lemma 4.6 in [2] to (5.18) we obtain if and only if
Next, we prove (5.19). We need only to show that since is a known condition. In fact, by Proposition 3.1 we know that for all . According to this and (5.4), we derive from (5.11) that This implies that Combining (4.17) and (5.22) yields for all , which leads to A combination of (5.23) and (5.24) gives (5.21). Hence, (5.19) is true.
Finally, we prove that . By (3.9), (2.2), and (5.22), we have which means that . This, together with (5.19), implies that Accordingly, the theorem is proven.

The proof of the above theorem reveals that the irreducibility of the method is used only for the case where the Runge-Kutta matrix is singular. Hence, for Pouzet-Runge-Kutta methods whose underlying Runge-Kutta has an invertible the matrix , the irreducibility condition in Theorem 5.7 can be dropped. This is made precise in the following theorem.

Theorem 5.8. Assume that an underlying Runge-Kutta method (3.1) with invertible matrix and positive is algebraically stable and strictly stable at infinity. Then the corresponding Pouzet-Runge-Kutta method (3.7)–(3.10) is asymptotically stable for the class whenever the condition (5.4) holds.

In light of Theorem 5.7, Propositions 3.1, and Propositions 3.2, we obtain the following analogues of Corollaries 5.3 and 5.4.

Corollary 5.9. Assume that an underlying Runge-Kutta method (3.1) with distinct is irreducible, algebraically stable, and satisfies and the simplifying condition . Then the corresponding Pouzet-Runge-Kutta method (3.7)–(3.10) is asymptotically stable for the class whenever (5.4) holds.

Corollary 5.10. Assume that an underlying Runge-Kutta method (3.1) with distinct and positive is irreducible, strictly stable at infinity, and satisfies the simplifying condition . Then the corresponding Pouzet-Runge-Kutta method (3.7)–(3.10) is asymptotically stable for the class whenever (5.4) holds.

Similar to Theorem 5.8, the irreducibility condition can be dropped in Corollary 5.10 when the matrix is invertible. Moreover, by Remark 5.5, Theorem 5.8, and the fact that the stability functions of Radau IA, Radau IIA, and Lobatto IIIC methods all satisfy (cf. [14]), one can establish an asymptotic stability result analogous to the one in Corollary 5.6.

Corollary 5.11. Assume that the underlying Runge-Kutta method (3.1) is of type Radau IA, Radau IIA, or Lobatto IIIC. Then the corresponding Pouzet-Runge-Kutta method (3.7)–(3.10) is asymptotically stable for the class whenever (5.4) holds.

6. Numerical Illustration

In order to illustrate the effectiveness of the extended Pouzet-Runge-Kutta methods (3.7)–(3.10), we will apply the two-stage methods of type Gauss, Radau IA, Radau IIA, or Lobatto IIIC to the system (2.23), respectively, where the solution domain is chosen as . These methods produce a series of high-precision numerical solutions for the (2.22) on .

Let

The corresponding values of the above methods applied to the system (2.23) are listed in the Table 1(a), indicating that the methods satisfy the condition (5.4). Hence, the methods are globally stable by Corollary 5.6, and asymptotically stable by Corollary 5.11, except for the Gauss-type method.

tab1
Table 1: (a) The -values of the methods for (2.23), (b) The errors of the numerical solutions of (2.22) by methods (3.7)–(3.10).

The excellent stability properties of the methods lead us to expect good numerical results. In order to confirm this, we use the Newton-Raphson iteration technique to implement the above numerical schemes. Taking the following four groups of space-time stepsizes: and then applying the above Pouzet-Runge-Kutta schemes to the system (2.23) on , respectively, we obtain sixteen sets of numerical solutions. The numerical solution generated by the extended two-stage Gauss-type method with space-time stepsizes is plotted in Figure 1. The solution figures for the other methods are quite similar to Figure 1, and hence we omit them here. In order to show the computational precision of the obtained numerical solutions, we use

642318.fig.001
Figure 1: Numerical solution of (2.22) by 2-stage Gauss-type method.

to characterize the errors of the methods, where

is the vector whose entries consist of the system (2.22) true solutions at the meshpoints , . The errors of the above four methods with different stepsizes are displayed in Table 1(b); they confirm again that the methods are rather effective.

7. Concluding Remarks

In paper [1], with a combination of Runge-Kutta methods and Pouzet quadrature rules, the authors also obtained an alternative type of numerical methods for (1.1), namely, where the meanings of the notations are similar to those indicated in method (3.7) and are determined by (3.10). But such methods cannot be applied directly to systems (1.4) unless the integrated function is continuously differentiable on its domain. When the latter holds, the system (1.4) can be transformed into where

This is just of the form (1.1) and implies, under the condition that g is continuously differentiable, that systems (1.4) can be solved by methods {(7.1), (3.10)}. Thus, the numerical stability theory in [1] is applicable and hence a series of global and asymptotical stability results can be followed immediately. However, it is difficult to compare the theoretical results in this way with those in previous sections since both discretization schemes are different and there is no direct relationship between the condition (2.1)-(2.2) and the condition imposed on (1.1) (see [1]). Moreover, we have noted that a system (1.4) can be solved by scheme {(7.1), (3.10)} only when continuously differentiable, which shows that schemes (3.7)–(3.10) have a wider applicable range than schemes {(7.1), (3.10)} do.

In the following, we give a comparison between methods (3.7)–(3.10) and methods {(7.1), (3.10)} with some numerical experiments. It is evident that the system (2.23) can be changed into the form (1.1). Hence this system also can be solved by methods {(7.1), (3.10)}. Similarly, we take the space-time stepsizes in (6.2) and apply the two-stage methods {(7.1), (3.10)} of type Gauss, Radau IA, Radau IIA, and Lobatto IIIC to the system (2.23) on , respectively, then a series of high-precision numerical solutions can be worked out, whose errors are displayed in Table 2(a). It follows from Tables 1(b) and 2(a)–2(c) that the numerical precisions and the computational times of the both methods based on the same type of underlying Runge-Kutta method are almost similar under the same stepsize. This implies that methods (3.7)–(3.10) are comparable.

tab2
Table 2: (a) The errors of the numerical solutions of (2.22) by methods {(7.1), (3.10)}, (b) The CPU times (in second) of methods (3.7)–(3.10) for (2.23) and (c) The CPU times (in second) of methods {(7.1), (3.10)} for (2.23).

Acknowledgments

This work is supported by NSFC (Grant no. 11171125, 91130003) and NSFH (Grant no. 2011CDB289), and its initial version was completed at Hong Kong Baptist University when Co Zhang visited Professor Hermann Brunner. The author is grateful to Hermann Brunner for his valuable discussion and aborative correction to the paper.

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