On definition of solution of initial value problem for fractional differential equation of variable order

1.   Introduction

In this paper, we research a continuous approximate solution of the following variable order initial value problem

where $0 < p(t, x(t)) < 1$, $u_0\in\mathbb{R}$, $p(t, x(t))$ and $f(t, x(t))$ are given real-valued functions, $^{C}\!D_{0+}^{p(t, x(t))}$ denotes variable order Caputo fractional derivative defined by

and $I_{0+}^{1-p(t, x(t))}$ is variable order Riemann-Liouville fractional integral defined by

For details, please refer to ^[1,2].

Fractional calculus has been acknowledged as an extremely powerful tool in describing the natural behavior and complex phenomena of practical problems due to its applications in ^[3,4,5,6,7]. However, the constant fractional order calculus is not the ultimate tool to model the phenomena in nature. Therefore, variable order fractional calculus is proposed. Moreover, variable order fractional differential equations provide better descriptions for nonlocal phenomena with varying dynamics than constant order differential equations and are extensively researched in ^{[1,2,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]}. Among these, there are many works dealing with numerical methods for some class of variable fractional order differential equations, for instance, ^{[1,2,8,9,10,12,13,14,15,16,17,18,19,21,22,23,24,25,26,30,31]}. In particular, variable order fractional boundary value problems are considered by numerical method base on reproducing kernel theory in ^[30,31].

There are several definitions of variable order fractional integrals and derivatives in ^[1,2]. We notice that when the order $p(t)$ is a constant function $p$, variable order Riemann-Liouville fractional derivative and integral are exactly constant order fractional derivative and integral. It is well known that the Riemann-Liouville fractional integral has the law of exponents, i.e. $I_{0+}^\alpha I_{0+}^\beta(\cdot) = I_{0+}^{\beta}I_{0+}^\alpha(\cdot) = I_{0+}^{\alpha+\beta}(\cdot), \alpha > 0, \beta > 0$. Based on the law of exponents, we can obtain some properties which are associated with fractional derivative and integral. For this reason, fractional order differential equations are transformed into equivalent integral equations. Thus some results of nonlinear functional analysis (for instance, some fixed point theorems) have been applied to considering the existence of solution of fractional order differential equations (see, e.g. ^{[3,20,32,33,34]} and the references therein). However, the law of exponents doesn't hold for variable order fractional integral. For example, in ^{[21,22,23,24]},

where $h(t)$ and $g(t)$ are both general nonnegative functions.

Then we will consider the properties which are interrelated with variable order fractional integral and variable order fractional derivative by given some examples.

Example 1.1. Let $p(t) = \frac t4+\frac 14$, $q(t) = \frac 34-\frac t4$, $f(t) = t, 0\leq t\leq 2$. Now, we calculate $I_{0+}^{p(t)}I_{0+}^{q(t)}f(t)|_{t = 1}$, $I_{0+}^{q(t)}I_{0+}^{p(t)}f(t)|_{t = 1}$ and $I_{0+}^{p(t)+q(t)}f(t)|_{t = 1}$, where $I_{0+}^{p(t)}$ and $I_{0+}^{q(t)}$ is defined in (1.3).

For $1\leq t\leq 2$, we have

so

yet

and

Therefore,

Example 1.1 illustrates that the law of exponents of the variable order Riemann-Liouville fractional integral doesn't hold when the order is non-constant continuous function.

Example 1.2. Let $p(t) = \left\{ \begin{array}{ll} \frac 12\ \ 0\leq t\leq 1, \\ \frac 13, \ \ 1 < t\leq 6, \end{array} \right.$, $q(t) = \left\{ \begin{array}{ll} \frac 12, \ \ 0\leq t\leq 1, \\ \frac 23, \ \ 1 < t\leq 6, \end{array} \right.$ and $f(t) = t, 0\leq t\leq 6$. We'll consider $I_{0+}^{p(t)}I_{0+}^{q(t)}f(t)|_{t = 4}$, $I_{0+}^{q(t)}I_{0+}^{p(t)}f(t)|_{t = 4}$ and $I_{0+}^{p(t)+q(t)}f(t)|_{t = 4}$, where $I_{0+}^{p(t)}$ and $I_{0+}^{q(t)}$ are defined in (1.3).

For $1\leq t\leq 4$, we have

thus,

By the same way, we get

and

As a result, we deduce

Example 1.2 shows that the law of exponents of the variable order Riemann-Liouville fractional integral doesn't hold when the order is piecewise constant function defined in the same partition.

Example 1.3. Let $p(t) = \frac t4+\frac 14$, $f(t) = t, 0\leq t\leq 3$. Now, we consider $I_{0+}^{p(t)} {}^{C}\!D_{0+}^{p(t)}f(t)|_{t = 2}$ and ${}^{C}\!D_{0+}^{p(t)}I_{0+}^{p(t)}f(t)|_{t = 2}$.

By (1.2) and (1.3), we have

which implies that $I_{0+}^{p(t)}{}^{C}\!D_{0+}^{p(t)}$ is different with the result of constant order fractional derivative and integral, that is,

where $0 < \alpha < 1$, $g\in C^{1}[0, b], 0 < b < +\infty$.

On the other hand, we have

which illustrates that $^{C}\!D_{0+}^{p(t)}I_{0+}^{p(t)}$ is different with the result of constant order fractional derivative and integral, that is,

where $0 < \alpha < 1$, $h\in C^{1} [0, b], 0 < b < +\infty$.

Example 1.3 verifies that the properties (1.4) and (1.5) of constant order fractional calculus don't hold for variable order fractional calculus when the order is a continuous function.

Example 1.4. Let $p(t) = \left\{ \begin{array}{ll} \frac 12, \ \ 0\leq t\leq 1, \\ \frac 13, \ \ 1 < t\leq 6, \end{array} \right.$, $f(t) = t, 0\leq t\leq 6$. Now, we consider $I_{0+}^{p(t)}{}^{C}\!D_{0+}^{p(t)}f(t)|_{t = 4}$ and ${}^{C}\!D_{0+}^{p(t)}I_{0+}^{p(t)}f(t)|_{t = 4}$.

By (1.2) and (1.3), for $2\leq t\leq 6$, we have

so

On the other hand, for $2\leq t\leq 6$, we get

so

Example 1.4 verifies the properties (1.4) and (1.5) of constant order fractional calculus is impossible for $I_{0+}^{p(t)}{}^{C}\!D_{0+}^{p(t)}f(t)$ and ${}^{C}\!D_{0+}^{p(t)}I_{0+}^{p(t)}f(t)$ when the order is a piecewise function.

Hence, we can claim that variable order fractional integral defined by (1.3) has no law of exponents. Moreover, for general functions $p(t)$ and $f(t)$, the representations of ${}^{C}\!D_{0+}^{p(t)}I_{0+}^{p(t)}f(t)$ and $I_{0+}^{p(t)}{}^{C}\!D_{0+}^{p(t)}f(t)$ are not clear. These obstacles make it difficult for us to transform variable order fractional differential equation into equivalent integral equation. As a result, it is almost impossible that some nonlinear functional analysis classical methods such as fixed point theorems are applied to prove the existence of solution of the corresponding integral equation. To the best of our knowledge, there are few works (^[8,23,24,27]) to deal with the existence of solutions to variable order fractional differential equations.

In ^[18], authors considered the following numerical solutions for variable order fractional functional boundary value problem

where $D^{\alpha(x)}$ is the variable order Caputo fractional derivative defined by

$a(x), b(x), c(x)\in C^{2}[0, 1]$; $\alpha(x), \tau(x), f(x)\in C[0, 1]$; and $\lambda_{0}, \lambda_{1} \in \mathbb{R}$. When $\alpha(x) = \frac{5+\sin(x)}4$, $a(x) = cos(x)$, $b(x) = 4$, $c(x) = 5$, $\tau(x) = x^{2}$, $f(x) = \frac{2x^{2-\alpha(x)}}{\Gamma(3-\alpha(x))}+5x^4+4x^2+2x\cos(x)$ and $\lambda_{0} = 0, \lambda_{1} = 1$, the exact solution of problem (1.6) was given by

In (1.6), if we take $f(x) = 6x^6+5x^4+4x^2+2x\cos(x)$, it is almost impossible to obtain its exact solution. In fact, we don't even know whether the solution to problem (1.6) exists.

In ^[8], authors discussed the existence of solution for a generalized fractional differential equation with non-autonomous variable order operators

where $x\in\mathbb{R}^{n}$ is the state vector, $f: \mathbb{R} \times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is a vector field, $c\in\mathbb{R}$ is the lower terminal, $x_{0}$ is the initial value, $0 < q\left(t, x\left(t\right) \right) \leqslant 1$, and $_{c}\!D_t^{q(t, x(t))}$ is the variable order Riemann-Liouville fractional differential operator defined as follows

In ^[8], authors claimed that, by ^[35], the initial value problem (1.7) is equivalent to the integral equation

where $\Psi : \mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}^{n}$ is the intialization function ^[35] which is needed in the general form given in (1.8) and the scalar $a$ is used for characterizing the initial period such that for $a < t < c$, the initial information is given, and for $t < a$ we consider $f(t) = 0$. Given $a$, one can develop a closed analytical formula for $\Psi$ ^[35].

However, in our opinion, there is no theoretical basis for this assertion. Because there are not contents related to variable order fractional integral and derivative in ^[35].

In ^[27], authors considered the existence results of solution to the initial value problem (1.1), in which $f(t, x(t)): [0, T]\times\mathbb{R} \rightarrow \mathbb{R}$ and $p:[0, T]\times \mathbb{R}\rightarrow (0, 1)$ are both continuous functions, and $0 < p_1\leq p(t, x(t))\leq p_2 < 1$, $u_0\in \mathbb{R}$. By means of Arzela-Ascoli theorem, authors obtained that the following sequence

existed a subsequence still denoted by the sequence $\{x_n\}$ which uniformly converged to a continuous function $x^{*}$. Set $x(t) = \int_0^tx^{*}(s)ds+u_0$, then authors obtained the problem (1.1) existed one solution $x(t)$. But, we find that it has fatal errors in these analysis procedure, that is the result of uniformly bounded of sequence $\{x_n\}$. This is easy to be overlooked. In our opinion the sequence $\{x_n\}$ is non-uniformly bounded. As a result, the existence result of solution to the initial value problem (1.1) is not obtained by Arzela-Ascoli theorem. On the other hand, it is almost impossible to transform the initial value problem (1.1) into equivalent integral equation.

Base on these facts, how to deal with the existence of solution of variable order fractional differential equations is a principal problem to be solved. In this paper, according to the classical definition of solution of integer order(or constant fractional order) differential equation, we propose a new definition of continuous approximate solution to the problem (1.1) under two kinds of variable order $p(t, x(t))$ and $p(t)$.

The paper is organized as follows. In section 2, a new definition of approximate solution to the problem (1.1) for variable order $p(t, x(t))$ is proposed and we provide an example to demonstrate the definition. Section 3 is devoted to introduce another definition of approximate solution for $p(t)$, then an example is given to illustrate the theoretical result.

2.   Definition of approximate solution for $p(t, x(t))$

Throughout this section, we assume that

$(A_1)$: $p:[0, T]\times \mathbb{R}\rightarrow (0, 1)$ is a continuous function;

$(A_2)$: $f:[0, T]\times \mathbb{R}\rightarrow \mathbb{R}$ is a continuous differential function.

We begin with definitions and characters of solution of integer order and constant fractional order.

Remark 2.1. According to the definition of solution $x(t)$ of differential equation, it should be defined in the interval, in which the equation is satisfied. For instance, a function $x(t)$ is called a solution of the following initial value problem

if $x$ is defined in the interval $[0, T]$, satisfying the equation and the initial value condition $x(0) = 0$.

Remark 2.2. Fractional operators are typical nonlocal operators, which can well describe the memory and global correlation of physical processes. Hence, for initial value problem of fractional order differential equations, its solution in given interval is affected by the state of solution in the preceding intervals.

By Remark 2.1, a function $x:[0, T]\rightarrow \mathbb{R}$ is called a solution of the initial value problem (1.1) if $x(t)$ satisfies the equation (1.1) and $x(0) = u_0$. However, based on analysis above, we are faced with extreme difficulties in considering the existence of solution of initial value problem (1.1) in sense of this definition. Thus, a new continuous approximate solution of problem (1.1) is proposed.

We start off by analyzing the problem (1.1) based on the facts above.

Let

We consider the initial value problem defined in the interval $[0, T]$ as following

Let $x_1\in C^{1}(0, T ] \cap C[0, T]$ be a solution of the initial value problem (2.2)(by standard way, we know that the initial value problem (2.2) exists continuous solution in $[0, T]$ under some assumptions on $f$). Since $x_1(t)$ is right continuous at point $0$, then for arbitrary small $\varepsilon > 0$, there exists $\delta_{01} > 0$ such that

And because $p(t, x_1(t))$ is right continuous at point $(0, x_1(0)) = (0, u_0)$, then together with (2.3) and (2.1), for the above $\varepsilon$, there exists $\delta_0 > 0$ such that

If $\delta_0 < T$, we take $\delta_0\doteq T_1$ and continue next procedure. Otherwise, we take $T_1 = T$ and end this procedure.

We assume that $\delta_0 < T$, and then let

In order to consider the existence of solution to (1.1) in the interval $[T_1, T_2]$, we let

By (2.6) and integration by parts, we denote

Hence, we may consider the initial value problem defined in the interval $[T_1, T]$ as following

where $x_1(t)$ is the solution of the initial value problem (2.2) and $\varphi_{x_1}$ is the function defined by (2.7).

Let $x_2\in C^{1}(T_1, T ] \cap C[T_{1}, T]$ be a solution of the problem (2.8) (by standard way, we know that the problem (2.8) exists continuous solution in $[T_1, T]$ under some assumptions on $f$). Since $x_2(t)$ is right continuous at point $T_1$, then for the above $\varepsilon$, there exists $\delta_{11} > 0$ such that

And because $p(t, x_2(t))$ is right continuous at point $(T_1, x_2(T_1)) = (T_1, x_1(T_1))$, then together with (2.9), for the above $\varepsilon$, there exists $\delta_1 > 0$ such that

If $T_1+\delta_1 < T$, we take $T_1+\delta_1\doteq T_2$ and continue next procedure. Otherwise, we take $T_2 = T$ and end this procedure. Obviously, according to (2.10) and (2.5), we obtain

We assume that $T_1+\delta_1 < T$, and then let

We let

Thus, by (2.13) and integration by parts, we have

$i = 1, 2, T_0 = 0$.

We may consider the initial value problem in the interval $[T_2, T]$ as following

where $x_i(t)$ is the solution of the initial value problems (2.2) and (2.8) respectively, and $\phi_{x_i}(i = 1, 2)$ is the function defined by (2.14).

Let $x_3\in C^{1}(T_2, T ] \cap C[T_{2}, T]$ be a solution of the problem (2.15)(by standard way, we know that the problem (2.15) exists continuous solution in $[T_2, T]$ under some assumptions on $f$). Since $x_3(t)$ is right continuous at point $T_2$, then for the above $\varepsilon$, there exists $\delta_{21} > 0$ such that

And because $p(t, x_3(t))$ is right continuous at point $(T_2, x_3(T_2)) = (T_2, x_2(T_2))$, then together with (2.16), for the above $\varepsilon$, there exists $\delta_2 > 0$ such that

If $T_2+\delta_2 < T$, we take $T_2+\delta_2\doteq T_3$ and continue next procedure. Otherwise, we take $T_3 = T$ and end this procedure. Obviously, according to (2.17) and (2.12), it holds

We assume $T_2+\delta_2 < T$, and then let

We let

By (2.20) and integration by parts, we get

$i = 1, 2, 3, T_0 = 0$.

We may consider the initial value problem in the interval $[T_3, T]$ as following

where $x_i(t)$ are solutions of the initial value problems (2.2), (2.8) and (2.15) respectively, and $\omega_{x_i}(i = 1, 2, 3)$ is the function defined by (2.21).

Since $[0, T]$ is a finite interval, we continue this procedure and could finish it by finite steps. That is, there exists $\delta_{n^{*}-2} > 0$, $\delta_{n^{*}-1} > 0$ ($n^{*}\in N)$ such that $T_{n^{*}-2}+\delta_{n^{*}-2}\doteq T_{n^{*}-1} < T$, $T_{n^{*}-1}+\delta_{n^{*}-1}\geq T\doteq T_{n^{*}}$. Then we have intervals $[0, T_1], [T_1, T_2]$, $[T_2, T_3]$, $\cdots$, $[T_{n^{*}-2}, T_{n^{*}-1}]$, $[T_{n^{*}-1}, T]$, and solutions $x_i\in C^{1}(T_{i-1}, T ] \cap C[T_{i-1}, T]$ of the following initial value problem defined in the interval $[T_{i-1}, T]$

where $p_i = p(T_{i-1}, x_{i-1}(T_{i-1}))$ satisfying

and

$j = 1, 2, \ldots, i-1$, $i = 5, 6, \cdots, n^{*}$, $T_0 = 0, T_{n^{*}} = T$. For details, please refer to the analysis of problems (2.2), (2.8), (2.15) and (2.22).

From the arguments above, we obtain a function $x^{*}\in C[0, T]$ defined by

Now, we propose the new definition of approximate solution to the initial value problem (1.1), which is crucial in our work.

Definition 2.1. If there exists natural number $n^{*}\in N$ and intervals $[0, T_1], (T_1, T_2], \cdots$, $(T_{n^{*}-1}, T]$, and initial value problems (2.2), (2.8), (2.15), (2.22), (2.23) exist solutions $x_1\in C^{1}(0, T]\cap C[0, T]$, $x_2\in C^{1}(T_1, T]\cap C[T_1, T]$, $x_3\in C^{1}(T_2, T]\cap C[T_2, T]$, $x_4\in C^{1}(T_3, T]\cap C[T_3, T]$, $x_i\in C^{1}(T_{i-1}, T]\cap C[T_{i-1}, T]$ ($i = 5, \cdots, n^{*}$) respectively, then the function $x\in C[0, T]$ defined by

is called an approximate solution of the initial value problem (1.1).

Remark 2.3. If $x_1(t), x_2(t), \cdots, x_{n^{*}}(t)$ are both unique, then we say $x(t)$ defined by (2.27) is unique approximate solution of the initial value problem $(1.1)$.

Remark 2.4. Based on Remark 2.1, Definition 2.1 seems suitable.

Remark 2.5. From Definition 2.1, we notice that approximate solution $x(t)$ of the initial value problem (1.1) in interval $[T_2, T_3]$ is $x_3(t)$ which is a solution of the initial value problem (2.15). Obviously, the state of $x_3(t)$ is affected by the state of $x_1(t)$ and $x_2(t)$. That is, the state of $x(t)$ in interval $[T_2, T_3]$ is affected by the state of $x(t)$ in interval $[0, T_2]$. Hence, Definition 2.1 is suitable and reasonable according to Remark 2.2.

Remark 2.6. In our previous analysis, we chose functions (2.6), (2.13), etc, so that we obtain the initial value problems (2.8), (2.15), etc. Such a choice must meet the following three reasons at the same time. The first reason is operability, for instance, choosing function (2.13) enable us to calculate functions $\phi_{x_1}(t), \phi_{x_2}(t)$, and obtain the initial value problem (2.15) defined in $[T_2, T]$. The second reason is for fitting Remark 2.1 and Remark 2.5. If we take $x'(t) = x_1(T_1)$ for $0\leq t\leq T_1$ (here $x_1(t)$ is the solution of the initial value problem (2.2)), we may easily calculate function $\varphi_{x_1}(t) = \frac{x_1(T_1)(t^{1-p_1}-(t-T_1)^{1-p_1})}{\Gamma(2-p_1)}$, and then have the initial value problem (2.8) with such $\varphi_{x_1}(t)$. However, we see that the state of solution of problem (2.8) is only affected by $x_1(T_1)$, but not affected by the state of $x_1(t), 0\leq t\leq T_1$. The third reason is the rationality of the obtained equation. For instance, according to the definition the Caputo fractional derivative ${}^{C}\!D^{p_{2}}_{T_{1}+}x\left(t\right) = \frac{1}{\Gamma \left(1-p_{2}\right) } \int^{t}_{T_{1}} \left(t-s\right)^{-p_{2}} x^{\prime }\left(s\right) ds$, the solution of initial value problem (2.8) exists only if the term $f(t, \cdot)-\varphi_{x_1}(t)$ is absolutely continuous in the interval $[T_1, T]$, otherwise one can not obtain the existence of solution of the initial value problems (2.8). If we take $x(t) = x_1(t)$ for $0\leq t\leq T_1$ (here $x_1(t)$ is the solution of initial value problem (2.2)), we may easily get function $\varphi_{x_1}(t) = \int_0^{T_1}\frac{(t-s)^{-p_2}x'_1(s)ds}{\Gamma(1-p_2)}$, and then obtain initial value problem (2.8) with such $\varphi_{x_1}(t)$. However, we can't obtain the existence of solution of initial value problem (2.8) with this $\varphi_{x_1}(t)$.

Example 2.1. According to Definition 2.1, we consider the approximate solution of the following initial value problem

By the definition of the variable order Caputo fractional derivative, there is no way to obtain explicit expression of its solution, even hardly conventional methods to study the existence of its solution. Next, according to Definition 2.1, we try to seek its approximate solution.

Here $p(t, x(t)) = \frac 12+\frac {t}{1000(1+t^2)}+\frac{x(t)}{3(1+x^2(t))}$, then we take $p_1 = p(0, 0) = \frac 12$.

First, we consider initial value problem as following

By simple calculation, the solution of the problem (2.29) is given by

Since $x_1(t)$ is right continuous at point $0$, then for $\varepsilon = 0.002$, we take $\delta_{01} = \frac 12$ such that

Notice that $p(t, x_1(t))$ is right continuous at point $(0, x_1(0)) = (0, 0)$, then together with (2.31), for the above $\varepsilon = 0.002$, we take $\delta_{0} = \frac 12$, such that

We take point $T_1 = \delta_{0} = \frac 12$, and let

We denote

Next, we consider the following initial value problem

From the facts of constant order fractional calculus, the solution of the initial value problem (2.34) is

We notice that

Since $x_2(t)$ is right continuous at point $T_1 = \frac 12$, then for $\varepsilon = 0.002$, we take $\delta_{11} = \frac 12$ such that for $\frac{1}{2} < t\leq1$, we have

Because $p(t, x_2(t))$ is right continuous at point $(T_1, x_2(T_1)) = (\frac 12, x_2(\frac 12))$, then together with the estimation above, for $\varepsilon = 0.002$, we take $\delta_{1} = \frac 12$ such that for $\frac 12 < t\leq 1$,

From the arguments above, the function $x^{*}\in C[0, 1]$ defined by

is the continuous approximate solution of problem (2.28) according to Definition 2.1(see Figure 1).

3.   Definition of approximate solution for $p(t)$

In this section, we study the initial value problem (1.1) for $p(t, x(t))\equiv p(t)$. Since $p(t)$ is a function of one variable, we may propose another definition of approximate solution of the initial value problem (1.1) such that we can simplify the analysis process in the section 2.

The following result is crucial for us to propose another definition of approximate solution of the initial value problem (1.1) for the particular order $p(t)$.

Lemma 3.1. Assume that $p:[0, T]\rightarrow (0, 1)$ is a continuous function, then for arbitrary small $\varepsilon > 0$, there exists natural number $n^{*}$ and intervals $[0, T_1], (T_1, T_2], \cdots, (T_{n^{*}-1}, T]$ and a piecewise function $\alpha:[0, T]\rightarrow (0, 1)$ defined by

where $p_k = p(T_{k-1})(k = 1, 2, \cdots, n^{*}, T_{0} = 0)$, such that for arbitrary small $\varepsilon > 0$,

Proof. Since $p(t)$ is right continuous at point $0$, then for arbitrary small $\varepsilon > 0$, there exists $\delta_0 > 0$ such that

We take point $\delta_0\doteq T_1$ (if $T_1 < T$, we consider continuity of $p(t)$ at point $T_1$, otherwise, we end this procedure). Since $p(t)$ is right continuous at point $T_1$, then for the above $\varepsilon$, there exists $\delta_1 > 0$ such that

We take point $T_1+\delta_1\doteq T_2$ (if $T_2 < T$, we consider continuity of $p(t)$ at point $T_2$, otherwise, we end this procedure). Since $p(t)$ is right continuous at point $T_2$, then for the above $\varepsilon$, there exists $\delta_2 > 0$ such that

Since $[0, T]$ is a finite interval, we continue this analysis procedure and could finish it by finite steps. That is, there exists $\delta_{n^{*}-2} > 0$ and $\delta_{n^{*}-1} > 0$ ($n^{*}\in N)$ such that $T_{n^{*}-2}+\delta_{n^{*}-2}\doteq T_{n^{*}-1} < T$ and $T_{n^{*}-1}+\delta_{n^{*}-1}\geq T$. Then we have intervals $[0, T_1], [T_1, T_2]$, $\cdots$, $[T_{n^{*}-2}, T_{n^{*}-1}]$, $[T_{n^{*}-1}, T]$, such that for the above $\varepsilon$,

$i = 1, 2, \ldots, n^{*}$, $T_{0} = 0$ and $T_{n^{*}} = T$.

We denote

Thus, we define a piecewise function $\alpha:[0, T]\rightarrow (0, 1)$ as following

From (3.3)–(3.8), we obtain that for the arbitrary small $\varepsilon > 0$,

The proof is completed.

Similar to the analysis in the section 2, we consider approximate solution of the initial value problem (1.1) in the following sense: if $p(t)$ and $\alpha(t)$ satisfy (3.2), then solution $x(t)$ of the following initial value problem

is called the approximate solution of the initial value problem (1.1).

We start off by analyzing the problem (3.9), and then propose a new definition of continuous approximate solutions to the initial value problem (1.1) with $p(t, x(t)) = p(t)$.

For the initial value problem (3.9) in the interval $[0, T_1]$, by (3.1), we have the initial value problem

By similar analysis in section 2, we may consider the initial value problem defined in the interval $[T_1, T_2]$ as following

where $x_1(t)$ is the solution of the initial value problem (3.10) and

in which $x'(s) = \int_0^sx_1(\tau)d\tau$.

Using the same method, we may consider the initial value problem defined in the interval $[T_2, T_3]$ as following

where $x_1(t)$ is the solution of the initial value problem (3.10), $x_2(t)$ is the solution of the initial value problem (3.11) and $\phi_{x_i}(t)$ is the function defined by

in which $x'(s) = \int_{T_{i-1}}^sx_i(\tau)d\tau$, $i = 1, 2, T_0 = 0$.

Similarly, we may consider the initial value problem defined in the interval $[T_{i-1}, T_i]$ as following

where $x_1(t)$ is the solution of the initial value problem (3.10), $x_2(t)$ is the solution of the initial value problem (3.11), $x_3(t)$ is the solution of the initial value problem (3.12), $x_i(t)$ is the solution of the initial value problem (3.13) and $\psi_{x_j}$ is the function defined by

in which $x'(s) = \int_{T_{j-1}}^sx_j(\tau)d\tau$, $j = 1, 2, \cdots, i-1$, $i = 4, \cdots, n^{*}$, $T_0 = 0, T_{n^{*}} = T$.

Based on the arguments above, we propose the definition of solution to the initial value problem (3.9), which is crucial in our work.

Definition 3.1. If the initial value problems (3.10), (3.11), (3.12) and (3.13) exist solutions $x_1:[0, T_1]\rightarrow \mathbb{R}$, $x_2:[T_1, T_2]\rightarrow \mathbb{R}$, $x_3:[T_2, T_3]\rightarrow \mathbb{R}$, $x_i:[T_{i-1}, T_i]\rightarrow \mathbb{R}$($i = 4, \cdots, n^{*}$, $T_{n^{*}} = T$) respectively, then we call function $x:[0, T]\rightarrow \mathbb{R}$ defined by

is a solution of the initial value problem (3.9).

Remark 3.1. If $x_1(t), x_2(t), \cdots, x_{n^{*}}(t)$ are unique, then we say $x(t)$ defined by (3.14) is unique solution of the initial value problem (3.9).

The following is the definition of the approximate solution of the initial value problem (1.1) with $p(t, x(t))\equiv p(t)$.

Definition 3.2. We call the solution of initial value problem (3.9) defined by Definition 3.1 is an (unique) approximate solution of initial value problem (1.1) with $p(t, x(t)) = p(t)$ if $\alpha(t)$ is defined by Lemma 3.1.

Remark 3.2. For the initial value problem (1.1) with $p(t, x(t))\equiv p(t)$, its approximate solution defined by Definition 3.2 is consistent with by Definition 2.1.

Example 3.1. We consider the following initial value problem for linear equation

By the definition of the variable order Caputo fractional derivative, there is no way to obtain explicit expression of its solution, even hardly conventional methods to study the existence of its solution. Next, we try to seek its approximate solution in the sense of Definition 3.2.

Here $p(t) = \frac 13+\frac {t}{1000(1+t^2)}$. Obviously, $p(t)$ is continuous on $[0, 1]$ and $0 < p(t) < 1$.

By the right continuity of function $p(t)$ at point $0$, for given arbitrary small $\varepsilon = 0.00035$, taking $\delta_0 = \frac 13$, when $0\leq t\leq\delta_0 = \frac 13$, we have

Then, we get $T_1 = \delta_0 = \frac 13$. By the right continuity of function $p(t)$ at point $T_1$, for the above $\varepsilon$, taking $\delta_1 = \frac 13$, when $T_1 < t\leq T_1+\delta_1$($\frac{1}{3} < t\leq \frac{2}{3}$), by differential mean value theorem, we have

where $T_1 < \xi < t < \frac{2}{3}$.

We let $T_2 = T_1+\delta_1 = \frac 23$. By the right continuity of function $p(t)$ at point $T_2$, for $\varepsilon = 0.00035$, taking $\delta_2 = \frac 13$, when $T_2 < t\leq T_2+\delta_2$($\frac{2}{3} < t\leq 1$), we have

We see that $T_2+\delta_2 = 1$, hence, we obtain three intervals $[0, \frac 13]$, $(\frac 13, \frac 23]$, $(\frac 23, 1]$ and piecewise constant function $\alpha(t)$ defined by

which satisfies

Thus, according to Definition 3.2, we first consider the following initial value problem

By the facts of constant order fractional calculus, the unique solution of the initial value problem (3.17) is

Let

Next, we consider the following initial value problem

where $x_{1}$ and $\psi_{x_{1}}$ are given by (3.18) and (3.19) respectively. By simple calculation, we get the solution of the initial value problem (3.20) as following

Let

Finally, we consider the following initial value problem

where $x_1$ is given by (3.18), $x_2$ is presented by (3.21) and $\psi_{x_{i}}(i = 1, 2)$ is given by (3.22). Thus, the solution of the initial value problem (3.23) is obtained by

According to Definition 3.2, the approximate solution of initial value problem (3.15) is given by

Obviously, $x(t)$ defined by (3.25) is continuous.

Example 3.2. We consider the continuous approximate solution of initial value problem (3.15) in the sense of Definition 2.1.

According Definition 2.1, let

We first consider the following initial value problem

By the arguments above, we know that the solution of initial value problem (3.27) is

Since $p(t)$ is right continuous at point $0$, for $\varepsilon = 0.00035$, there exists $\delta_{0} = \frac{1}{3}$ such that

We take $T_{1} = \delta_{0} = \frac{1}{3}$. Let

By Definition 2.1, we next seek the solution of the initial value problem

where $x_1(t)$ is the function given by (3.28), and $\varphi_{x_1}(t)$ is the function defined by

Thus, the solution of the initial value problem (3.30) is

Since $p(t)$ is right continuous at point $T_{1}$, for $\varepsilon = 0.00035$, there exists $\delta_{1} = \frac{1}{3}$ such that

We take $T_{2} = T_{1}+\delta_{1} = \frac{2}{3}$. Let

According to Definition 2.1, then we seek the solution of the initial value problem

where $x_1(t)$ is given by (3.28), $x_2(t)$ is presented by (3.32) and $\psi_{x_i}(t)(i = 1, 2)$ is the function as followings

Thus, the solution of the initial value problem (3.35) is

where $\psi_{x_{i}}(i = 1, 2)$ is given by (3.36).

Hence, the approximate solution of initial value problem (3.15) in the sense of Definition 2.1 is given by

Obviously, $x(t)$ defined by (3.38) is continuous.

From the analysis above, for the same $\varepsilon$ and the same interval $[0, \frac{1}{3}]$, $[\frac{1}{3}, \frac{2}{3}]$, $[\frac{2}{3}, 1]$, continuous approximate solution of initial value problem (3.15) obtained by Definition 3.2 is consistent with which is obtained by Definition 2.1.

4.   Conclusions

This paper propose a new definition of continuous approximate solution to initial value problem for differential equations involving variable order Caputo fractional derivative on finite interval. It provide a new method to consider the existence of solution to more general variable order fractional differential equations. This method is different from common numerical methods and it also can be apply to variable order fractional boundary value problem.The new results generalize some existing results in the literature.

Acknowledgments

The authors would like to thank the referees for the helpful suggestions. The research are supported by the National Natural Science Foundation of China (No. 12071302) and the Fundamental Research Funds for the Central Universities (No. 2021YJSLX01).

Conflict of interest

The authors declare no conflict of interest.