The Green's function for Caputo fractional boundary value problem with a convection term

1.   Introduction

In this paper, we give a new approach to construct Green's function for the following Caputo two-point boundary value problems with a constant convection coefficient

where the constants $\lambda, \beta_0, \beta_1, \gamma_0, \gamma_1$ and the function $h \in C[a, b]$ are given.

In recent years, fractional differential equations are becoming a powerful tool to describe real-world phenomena, enormous numbers of very interesting and novel applications of fractional differential equations in physics, chemistry, engineering, finance, and other sciences have been developed. The Caputo derivative is especially suitable to describe real phenomena since in many ways it behaves like the usual derivative of integer order. In particular, the Caputo derivative of constant functions is zero, which is not true for the Riemann-Liouville derivative. However, the study of fractional differential equations with Caputo derivative is far from enough, there are still many basic problems to be solved, for example, the research on the expression of Green's function and its sign in the problem with a constant convection coefficient has aroused the interest of experts.

Papers such as ^[1,2,3] examine the non-negativity of Green's functions for Caputo two-point boundary value problems, but these papers contain no convection terms, Papers ^[4,5] consider Caputo two-point boundary value problems with convection, when the convection term is constant, the non-negative condition for Green's function is sufficient but not a necessary condition.

As a special case of problems (1.1)–(1.2), X. Meng and M. Stynes ^[6] consider the following Caputo two-point boundary value problems with a constant convection coefficient

an explicit formula for the associated Green's function is obtained by applying two-parameter Mittag-Leffler functions, and the necessary and sufficient conditions that ensure non-negativity of the Green's function can be deduce. This is the first derivation in the research literature of an explicit Green's function for a Caputo two-point boundary value problem with a convection term.

Recently, Z. Bai et al. ^[7] restudied problem (1.3)–(1.4), they constructed the Green's function by use of the Laplace transform.

Motivated by the works ^[6,7], in this paper, we will give the Green's function of boundary value problems (1.1)–(1.2) and generalize the results of ^[6,7]. Compared with these two articles, this paper includes the following features. Firstly, the operator theory is used in the process of solving the problem. By using operator theory, we generalize the conclusions of ^[6,7], these results cannot be obtained by using the methods provided in ^[6,7]. Secondly, the method provided in this paper may be more straightforward and easy to be generalized to solving other problems.

The paper is organized as follows. Some fundamental concepts and lemmas are described in Section 2 while Section 3 is devoted to the construction of Green's function for problem (1.1)–(1.2). In Section 4, we give the positive property of Green's function. Finally, the Green's function for multi-point boundary value problem is shown in Section 5.

2.   Basic definitions and preliminaries

In this section, we will recall some of the necessary definitions and results that will be used in the main results.

Definition 2.1. ^[8] Let $\alpha\geq 0$ and $f$ be a real function defined on $[a, b]$. The Riemann-Liouville fractional integral of order $\alpha$ is defined by $(I_{a^+}^0f)\equiv f$ and

Definition 2.2. ^[8] The Caputo fractional derivative of order $\alpha\geq 0$ is defined by $(^CD_{a^+}^{0}f)\equiv f$ and

for $\alpha > 0$, where $m$ is the smallest integer greater or equal to $\alpha$.

Lemma 2.3. If $\alpha \geq 0$ and $\beta > 0$, then

Lemma 2.4. ^[8] Let $\alpha > 0$ and $n = [\alpha]+1$, then

for some $c_{i} \in \mathbb{R}$, $i = 0, 1, 2, \cdots, n$.

Definition 2.5 ^[8] The Mittag-Leffler function is defined by:

The two-parameter Mittag-Leffler function is defined by:

For convenience, we denote $F_\beta(x) = x^{\beta-1}E_{\alpha-1, \beta}[\lambda x^{\alpha-1}]$, the following properties of $F_{\beta}$ have been deduced in ^[6].

$(P_1): [F_{\beta+1}(x)]' = F_\beta(x)$ for $\beta\geq 0, x\geq 0$;

$(P_2): F_1(0) = 1, F_\beta(0) = 0$ for $\beta > 1$;

$(P_3): F_1(x) > 0$ for $x > 0$, $F_2(x)$ is increasing for $x\geq 0$;

$(P_4): F_{\alpha-1}(x) > 0$ for $x > 0$, $F_\alpha(x)$ is increasing for $x > 0$.

Now, we prove the important result used in this paper.

Lemma 2.6. For any $\lambda\in R, \alpha > 0$, we have the following results.

(1) For any $r\in C([a, b], \mathbb{R})$, series $\sum_{k = 0}^{\infty}\lambda^kI_{a^+}^{k\alpha}r(t)$ is convergent andthe sum is

(2) The operator $I-\lambda I_{a^+}^{\alpha}:C([a, b], \mathbb{R})\to C([a, b], \mathbb{R})$ is reversible and:

Proof. (1) By the properties of Mittag-Leffler function and two-parameter Mittag-Leffler function, we have

and

Thus, (1) is proved.

(2) First, we show that $(I-\lambda I_{a^+}^{\alpha})\left(\sum_{k = 0}^{\infty}\lambda^kI_{a^+}^{k\alpha}r(t)\right) = r(t)$. In fact, it is easy to see

Similarly, we can easily prove the fact that:

3.   The Green's function of Problem (1.1)–(1.2)

Theorem 3.1. Assume that $\beta_0\geq 0, \beta_1\geq 0$. The boundary valueproblem (1.1)–(1.2) has a unique solution

where

Proof. Applying $I_{a^+}^{\alpha}$ to the both sides of the equation (1.1), we have:

From Lemmas 2.4,

let $t = a$, we obtain $u(a) = c_0$, so

or

by Lemma 2.6, we have

and

by the boundary conditions $u(a)-\beta_0u'(a) = \gamma_0$ and $u(b)+\beta_1u'(b) = \gamma_1$, we can get

thus,

therefore,

4.   The positive property of the Green's function

A function $h(x)$ is said to be log-concave if $\ln h(x)$ is concave, i.e., $(\ln h(x))^{''}\leq 0$. Similarly, a function $h(x)$ is said to be log-convex if $\ln h(x)$ is convex, i.e., $(\ln h(x))^{''}\geq0$.

Lemma 4.1. ^[6] Fix $\tau\in(0, 1]$. Then for $x > 0$, the functions $x^\tau E_{\tau, \tau+1}(x^\tau)$ and $E_{\tau, 1}(x^\tau)$ are log-concave.

Lemma 4.2. ^[6] Fix $\tau\in(0, 1]$. Then for $x > 0$, the functions $x^\tau E_{\tau, \tau+1}(-x^\tau)$ is log-concave; $E_{\tau, 1}(-x^\tau)$and $x^{\tau-1} E_{\tau, \tau}(-x^\tau)$ are log-convex.

Theorem 4.3. Fix $t\in[a, b]$. Then for $a\leq s\leq t$,

is a decreasing function of $s$.

Proof. $\bf{(I)}.$ If $\lambda = 0$, then

it is easy to check that

so the conclusion holds.

$\bf{(II)}.$ If $\lambda \neq 0$. By Lemma 4.1 and 4.2, the function $|\lambda|F_{\alpha}(x)$ is log-concave when $x > 0$. Thus, one can infer from the property $(P_1)$ that for $x > 0$,

This shows that $\frac{F_{\alpha-1}(x)}{F_{\alpha}(x)}$ is a decreasing function for $x > 0$. Consequently

or

That is, the numerator of $\frac{\partial}{\partial s}\left(\frac{F_{\alpha}(b-s)}{F_{\alpha}(t-s)}\right)$ is non-negative. So, $\frac{F_{\alpha}(b-s)}{F_{\alpha}(t-s)}$ is an increasing function of $s$. As $\frac{F_{\alpha-1}(x)}{F_{\alpha}(x)}$ is a decreasing function for $x > 0$, the function $\frac{F_{\alpha-1}(b-s)}{F_{\alpha}(b-s)}$ is an increasing function of $s$. Consequently, $\frac{F_{\alpha-1}(b-s)}{F_{\alpha}(b-s)}\cdot\frac{F_{\alpha}(b-s)}{F_{\alpha}(t-s)} = \frac{F_{\alpha-1}(b-s)}{F_{\alpha}(t-s)}$ is also increasing on $s$. By considering its derivative with respect to $s$, we obtain

By (4.1) and (4.2), the numerator of $f_t'(s)$ is

hence $f_t(s)$ is a decreasing function of $s\in [a, t]$.

Lemma 4.4. For $a\leq t\leq b$, the function

is an increasing function of $t$.

Proof. (I). If $\lambda = 0$, then for $\beta\geq 0, F_\beta(x) = \frac{1}{\Gamma(\beta)}x^{\beta-1}$, thus

it is easy to check that

we have

hence $g(t)$ is an increasing function of $t$.

(II). If $\lambda > 0$, then by Lemma 4.1 the function $F_1(x) = \lambda F_{\alpha}(x)+1 = x^{\alpha-1}E_{\alpha-1, \alpha}(\lambda x^{\alpha-1})+1$ is log-concave when $x > 0$, so

which imply

(III). If $\lambda < 0$, we can similarly prove that $\left(\frac{F_1(x)}{F_{\alpha-1}(x)}\right)'\geq 0$.

Suppose $t\in (a, b)$, By the Cauchy mean value theorem, there exists $\xi\in (t-a, b-a)$ such that

Equivalently,

Furthermore, $\left(\frac{F_1(x)}{F_{\alpha-1}(x)}\right)'\geq 0$ implies that

or

By (4.3) and (4.4), the sign of the numerator of $g'(t)$ is

Hence $g(t)$ is an increasing function of $t\in [a, b]$.

The main result of this paper is as follow.

Theorem 4.5. Assume $\beta_1\geq 0$. Then the Green's function $G(t, s)$ isnonnegative if and only if

Proof. The Green's function $G(t, s)$ is nonnegative on $[a, b]\times[a, b]$ if and only if

Equivalently,

by Theorem 4.3 and Lemma 4.4, inequality (4.5) is equivalent to

The proof is complete.

5.   The Green's function for multi-point boundary condition

In this section, we present the Green's function for the following multi-point boundary value problem

where the constants $\lambda, \beta_0, \beta_1, \gamma_i > 0 (i = 1, 2, \cdots, m-2), a < \xi_1 < \cdots < \xi_{m-2} < b$ and the function $h \in C[a, b]$ are given.

Theorem 5.1. Assume that $\beta_0\geq 0, \beta_1\geq 0$. The boundary valueproblem (5.1)–(5.2) has a unique solution

where

Proof. For convenience, we denote $\rho = \beta_0+\beta_1F_1(b-a)+F_2(b-a)$, then we have the relations

According to the proof of Theorem 3.1, $u(t)$ and $u'(t)$ satisfy

by the boundary conditions $u(a)-\beta_0u'(a) = 0$ and $u(b)+\beta_1u'(b) = \sum_{i = 1}^{m-2}\gamma_iu(\xi_i)$, we can get $c_0-\beta_0c_1 = 0$ and

thus,

therefore,

6.   Conclusions

In this paper, by use of the operator theory, the Green's function for a class of Sturm-Liouville fractional boundary value problems is obtained. Compare with other literature, our method provide some new ideas for the study of this kind of problems and easy to be generalized to solving other related problems.

Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions. This work is supported by the Tianjin Natural Science Foundation (grant no. 20JCYBJC00210).