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Abstract and Applied Analysis
Volume 2010, Article ID 704367, 10 pages
http://dx.doi.org/10.1155/2010/704367
Research Article

A Refinement of Quasilinearization Method for Caputo's Sense Fractional-Order Differential Equations

Department of Mathematics, Gebze Institute of Technology, Gebze, Kocaeli 141-41400, Turkey

Received 29 March 2010; Revised 15 May 2010; Accepted 22 June 2010

Academic Editor: Dumitru Baleanu

Copyright © 2010 Coşkun Yakar and Ali Yakar. 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

The method of the quasilinearization technique in Caputo's sense fractional-order differential equation is applied to obtain lower and upper sequences in terms of the solutions of linear Caputo's sense fractional-order differential equations. It is also shown that these sequences converge to the unique solution of the nonlinear Caputo's sense fractional-order differential equation uniformly and semiquadratically with less restrictive assumptions.

1. Introduction

The well-known quasilinearization method [1, 2] in differential equation has been employed to obtain a sequence of lower and upper bounds which are the solutions of linear differential equations that converge quadratically. However, the convexity and concavity assumption that is demanded by the method of quasilinearization has been a stumbling block for further development of the theory. Recently, this method has been generalized, refined, and extended in several directions so as to be applicable to a much larger class of nonlinear problems by not demanding convexity and concavity property [1, 37]. Moreover, other possibilities that have been explored make the method of generalized quasilinearization universally useful in applications [3, 6, 7].

The theory of nonlinear fractional-order dynamic systems has been investigated depending on the development in the theory of fractional-order differential equations. In this context, generalized quasilinearization method has been reconsidered, and similar results parallel to classical theory of differential equations have been obtained [1, 2, 8].

In this work, the quasilinearization technique coupled with lower and upper solutions is employed to study Caputo’s fractional-order differential equation for which particular and general results that include several special cases are obtained. Moreover, one gets monotone sequences whose iterates are the solutions of corresponding linear problems and the sequences converge to the solutions of the original nonlinear problems. Instead of imposing the convexity assumption on the function involved, we assume weaker conditions as well as for the concave functions. This is a definite advantage of this constructive technique. Furthermore, these monotone flows are shown to converge semiquadratically.

Consider the following initial value problem: where and is Caputo’s sense fractional-order derivative. Let and be the lower and upper solutions of (1.1) satisfying the following inequalities (1.2) and (1.3), respectively, on :

The corresponding Volterra fractional integral equation is

Caputo’s sense fractional-order differential equation is given by (1.1), and the corresponding Volterra fractional integral equation is given by (1.4). Here, we consider the function on the right-hand side of (1.1) and split it into three parts as , , and , where satisfies a weaker condition than convexity, satisfies a weaker condition than concavity, and is two-sided Lipschitzian.

2. Preliminaries

In this section, we state a comparison result and a corollary. For the proof, please see [2].

Theorem 2.1. Let be locally Hölder continuous for an exponent and , where , and
(i)(ii), where is Riemann-Liouville fractional-order derivative and is such that
Suppose further that the standard Lipschitz condition is satisfied; that is, Then, , where and imply that

Corollary 2.2. The function , where , is admissible in Theorem 2.1 to yield on .

We note that Theorem 2.1 and Corollary 2.2 also hold for Caputo’s fractional derivative; see [2].

3. Monotone Technique and Method of Quasilinearization

In monotone iterative technique that we have used an existence result of nonlinear fractional-order differential equations with Caputo’s derivative in a sector based on theoretical considerations and described a constructive method which implies monotone sequences of functions that converge to the solution of (1.1). Since each member of these sequences is the solution of a certain linear fractional-order differential equation with Caputo’s derivative which can be explicitly computed, the advantage and the importance of the technique need no special emphasis. Moreover, these methods can successfully be employed to generate two-sided pointwise bounds on solutions of initial value problems of fractional-order differential equations with Caputo’s derivatives from which qualitative and quantitative behaviors can be investigated.

The idea of relating the study of nonlinear fractional-order differential equations with Caputo’s derivative through its related linear fractional-order differential equations with Caputo’s derivative finds further extension in the method of quasilinearization. In this case, again, we obtain existence of solutions of (1.1) under certain restrictions after formulating sequences of solutions of related linear fractional-order differential equations with Caputo’s derivative. These sequences converge quadratically in the constructive methods. The method involves the formulation of upper and lower solutions.

Due to some advantages of Caputo’s derivative, we have applied the quasilinearization technique to the given nonlinear fractional-order differential equations with Caputo’s derivative not Riemann-Liouville (R-L) derivative. The main advantage of Caputo’s derivative is that the initial conditions for fractional-order differential equations are of the same form as those of ordinary differential equations with integer derivatives. Another difference is that Caputo’s derivative for a constant is zero, while the Riemann-Liouville fractional-order derivative for a constant is not zero but equals to , which is not zero Table 1 depicts the correspondence between the features of quasilinearization in the context of the integer order and fractional-order with Caputo’s derivative. Therefore, under the suitable assumptions but different conditions, we have Table 1.

tab1
Table 1

4. Main Result

In this section, we will prove the main theorem that gives several different conditions to apply the method of generalized quasilinearization to the nonlinear fractional-order differential equations with Caputo’s derivative and state four remarks for special cases.

Theorem 4.1. Assume that(i), , and on , where and (ii) Assume also that exists and is nondecreasing in for each as Furthermore, exists and is nonincreasing in for each as (iii) Moreover assume that is two-sided Lipschitzian in such that , where is the Lipschitz constant.
Then, there exist monotone sequences and which converge uniformly and monotonically to the unique solution of (1.1) and the convergence is semiquadratic.

Proof. Consider the following linear fractional-order initial value problems with Caputo’s derivatives order : Since the right-hand sides of the equations satisfy a Lipschitz condition, it is obvious that unique solutions exist. We will show that First, we will prove that Put on . Then, and Hence, applying Corollary 2.2, we get Let us set ; then, using and and the fact that , we have This implies that which because of Corollary 2.2 yields on Thus, we have on Similarly, one can prove that on We now prove that on . For this purpose we set and note that
Then, Since by using nondecreasing property of and nonincreasing property of , we obtain which shows that This proves that Therefore, on Hence, (4.6) is proved.
Using mathematical induction with we obtain We must prove that To do so, we set Then, where we have used the inequalities in , and the fact that is nondecreasing in and is nonincreasing in Thus, we have Again, from Corollary 2.2, we get on Similarly, it can be shown that on
Next we need to show that on
Set ; then,
Thus, we have   and Consequently, as before, it follows from Corollary 2.2 we get that   on
Employing the standard arguments [2], one can easily show that and converge uniformly and monotonically to the unique solution of (1.1).
To prove the semiquadratic convergence, we set and Note that and where Now using the nondecreasing property of and nonincreasing property of , we get Thus, we have where , , , , and
Then, we obtain where is the Mittag-Leffler function.
Let ; then, where and
Thus, we reach the desired result which shows the semiquadratic convergence.
Similarly, using suitable computation, we arrive at where and and

Remark 4.2. Let ; then, we have the monotone method, and the convergence is linear.

Remark 4.3. Let ; then, Theorem 4.1 reduces to Theorem of [4], and the convergence is quadratic.

Remark 4.4. Let and be concave; then, Theorem 4.1 reduces to Theorem of [4], and the convergence is quadratic.

Remark 4.5. Let ; then, Theorem 4.1 reduces to the theorem in [5].

5. An Example

The following example illustrates how the main result of the theorem may be applied for the nonlinear fractional differential equation order and

Example 5.1. Let us consider the following nonlinear fractional-order initial value problem with Caputo's derivative order : where and
Let and for be lower and upper solutions of the fractional-order differential equation with Caputo's derivative order , respectively. Then, and satisfy the inequalities in assumption as
On the other hand, , exist, and is nondecreasing and is nonincreasing in for each in assumption Also, it can be shown that these three functions , and hold in the correspondence inequality in assumptions and with the nonnegative constants , , and
Therefore, we can construct the monotone sequences and whose elements are solutions of linear fractional-order differential equations with Caputo's derivatives order of (5.3) and (5.4), respectively, as Since the right-hand sides of the equations satisfy a Lipschitz condition, it is obvious that unique solutions exist such that, for all
Employing the standard techniques [2], sequences and converge uniformly and monotonically to the unique solution of , since using the fact that satisfies a Lipschitz condition that is is bounded on the sector Moreover, the convergence is semiquadratic.

Acknowledgment

This work has been supported by The Scientific and Technological Research Council of Turkey and the Department of Mathematics at Yeditepe University.

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