Rapid Convergence of Approximate Solutions for Fractional Differential Equations

Abstract

In this paper, we develop a generalized quasilinearization technique for a class of Caputo’s fractional differential equations when the forcing function is the sum of hyperconvex and hyperconcave functions of order (), and we obtain the convergence of the sequences of approximate solutions by establishing the convergence of order ().

1. Introduction

Fractional differential equations have received attention from some researchers because they have extensive application in mechanics, biochemistry, electrical engineering, medicine, and many other fields (see [1–6]). For more information about the basic theory of fractional differential equations, we can refer to the monographs [7–9] and references cited therein. It is well known [10] that the monotone iterative technique offers an approach for obtaining approximate solutions to a wide variety of nonlinear differential equations. Recently, there are some results on the monotone sequences of approximate solutions converging uniformly to a solution of fractional differential equations by employing monotone iterative technique and generalized monotone iterative method coupled with the method of upper and lower solutions, which can be found in [11–21].

In view of applications, it is very significant to study the rate of convergence of solutions. The quasilinearization method [22] is one of the effective methods to obtain a sequence of approximate solutions with quadratic convergence, and it is extremely useful in scientific computations due to its accelerated rate of convergence as in [23, 24]. A few results of quadratic convergence for fractional differential equations were also obtained by applying quasilinearization, such as the initial value problem of Caputo’s fractional differential equations [13, 25, 26], fractional differential equations via initial time different lower and upper solutions [27], and the system of fractional differential equations [28]. However, to the best of our knowledge, there are few results of rapid convergence of fractional differential equations. Recently, Wang and others obtained the results on rapid convergence of solutions for various differential equations [29–33]. Inspired and motivated by [34, 35], in the present paper, we will discuss the rapid convergence of approximate solutions of fractional differential equations when the forcing function is the sum of hyperconvex and hyperconcave functions with coupled lower and upper solutions, and construct sequences of approximate solutions that converge rapidly to the extremal solutions of (1) by using an improved quasilinearization method (rate of convergence ).

2. Preliminaries

Consider the initial value problem of Caputo’s fractional differential equations (IVP): where are continuous functions, and .

A function is called a solution of IVP (1) if it satisfies (1).

Firstly, we give the following definitions and lemmas.

Definition 1. The locally Hӧlder continuous functions are coupled lower and upper solutions of type I of IVP (1) if the following inequalities hold:

Definition 2. The locally Hölder continuous functions are coupled lower and upper solutions of type II of IVP (1) if the following inequalities hold:

Let denote the th partial of with respect to . .

Definition 3. A function is called -hyperconvex, , if ; is called -hyperconcave if the inequality is reversed.

The linear IVP of Caputo’s fractional differential equation is given by where is a locally Hölder continuous function, . The unique solution of (4) can be expressed in the following form [7]: where are Mittag-Leffler’s functions, and denotes the Gamma function.

Remark 4. We note that (5) remains valid if and are functions mapping from , , and is an matrix.

We need the following lemmas to prove our main results, which proofs can be found in literature [26].

Lemma 5. Assume that is a locally Hӧlder continuous function such that for , and for . Then .

Lemma 6. Assume that is a family of continuous functions on . For each and the Caputo’s fractional differential equations, we have the following: the functions satisfy for . Then, the family is equicontinuous on .

In our further investigations, we need the following comparison results.

Lemma 7. Assume that one of the following conditions holds:
H₁ are coupled lower and upper solutions of type I of (1) and where is a constant
H₂ are coupled lower and upper solutions of type II of (1) and where is a constant
Then, implies on .

Proof. Firstly, we prove that the conclusion is valid when H₁ holds. To do this, let and for any small so that , , and . Then, in view of H₁, we have Similarly, we have .
We next show that on , which proves the conclusion as . Letting , suppose that is not true on . Then, there exists a such that and for . In view of Lemma 5, , that is, . Thus, we arrive at the following contradiction:

Similar to the proof process above, we can obtain the result of Lemma 7 when H₂ holds.

Lemma 8. Assume that , where , and one of the following conditions holds:
H₃ are coupled lower and upper solutions of type I of (1) such that on , and is monotone nonincreasing in for
H₄ are coupled lower and upper solutions of type II of (1) such that on , and is monotone nonincreasing in for
Then, there exists a solution of (1) satisfying on .

Proof. Suppose that H₃ holds. Consider the mapping defined by Then and it has a solution on with .
Firstly, we prove that on . Letting , for any small such that on . We can prove that on , which shows that as . Setting , suppose is not true on , then there exists a such that and for . From Lemma 5, it follows that , that is, . Therefore, On the other hand, we have and it contradicts with (14). This contradiction proves the claim.

Similarly, letting , we can find that .

It is easy to construct the proofs of the results relative to H₄. We omit the details.

To obtain the results of this paper, we need to consider two-dimensional Caputo’s fractional differential systems: where and is a locally Hölder continuous function.

Lemma 9. Assume that are locally Hölder continuous functions satisfying the following: and whenever , where is a constant, . Then implies on .

Proof. Let and for any small and so that , , and . Consequently, we obtain, for each Similarly, we have We next prove that on , which shows the required conclusion as . Suppose that is not true on , then there exists an indexj, and a such that and for . Set it then follows from Lemma 5 that that is . Furthermore, which leads to a contradiction. This completes the proof.

3. Main Results

In this section, we consider that and are hyperconvex and hyperconcave in of order , respectively. We first give some inequalities depending on whether is even or odd [34]: (i)(ii)

Based on the above inequalities, we have the following result which is relative to the coupled lower and upper solutions of type I in Definition 1 when is even.

Theorem 10. Consider the following assumptions:
A₁ are coupled lower and upper solutions of type I of (1) with on
A₂ such that and are the hyperconvex and hyperconcave in of the order , respectively
A₃ satisfies Then, there exist monotone sequences and converging uniformly to the solution of (1) on and the convergence is of the order .

Proof. It follows from assumption A₂ that the inequalities (22), (23), (24), and (25) hold. Consider the following fractional differential equations: Firstly, applying (31) and (32) and taking , we obtain Condition A₁ and inequalities (22), (23), (24), and (25) imply Employing assumption A₃ and the Taylor series expansion with the Lagrange remainder, we get where and . Therefore, and are nonincreasing in and , respectively. By Lemma 8, there exist solutions and of (33), (34) on such that and . Furthermore, in view of the inequalities (22), (23), (24), and (25) and condition A₂, we have and in view of H₁, we have for . Hence, .
By induction, for all , we can obtain that where and are solutions of and According to (40) and (41), and are coupled lower and upper solutions of type I of (1). We have on from assumption A₃. It then follows from Lemma 8 that is a solution of (1) on satisfying . Hence, we have By (42), the sequences and are uniformly bounded on . From Lemma 6, the sequences and are equicontinuous on . Consequently, by employing the Ascoli-Arzela Theorem, sequences and are uniformly convergent on .
Finally, we prove that the convergence of and is of the order .
Set and for with . Using the known conditions and the mean value theorem, we obtain where , , and and it follows from assumption A₂ that and are bounded on , that is, .
Similarly, we have where For the following inequalities we can get where in which Formulas (4) and (5) and Lemma 9 imply where Hence, we have where , . Similarly, we get Employing the Binomial Theorem and the inequalities (52) and (53), we obtain that is, where , , .
The following theorem is relative to the coupled lower and upper solutions of type II in Definition 1 when is odd.

Theorem 11. Consider the following assumptions:
D₁ are coupled lower and upper solutions of type II of (1) with on .
D₂ such that and are hyperconvex and hyperconcave in of order , respectively.
D₃ satisfies Then, there exist monotone sequences and converging uniformly to the solution of (1) on and the convergence is of the order .

Proof. It follows from assumption D₂ that the inequalities (26), (27), (28), and (29) hold. In order to construct the sequences and , consider the following fractional differential equations: Similar to the proof of Theorem 10, the conclusion can be obtained. We omit the details.

Data Availability

Data sharing not applicable to this article as no data sets were generated or analysed during the current study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors completed the paper together. All authors read and approved the final manuscript.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (11771115 and 11271106).