A novel class of fractional adams method for solving uncertain fractional differential equation

TL;DR

This paper introduces a new fractional Adams numerical method for solving uncertain fractional differential equations, improving accuracy and efficiency over existing methods, with stability analysis and practical simulations demonstrating its effectiveness.

Contribution

A novel fractional Adams method derived from interval weighting extends predictor-corrector approaches for UFDE, with stability and error analysis and enhanced numerical performance.

Findings

Higher accuracy in numerical simulations

Extended predictor-corrector to higher order

Validated stability and error bounds

Abstract

Uncertain fractional differential equation (UFDE) is a kind of differential equation about uncertain process. As an significant mathematical tool to describe the evolution process of dynamic system, UFDE is better than the ordinary differential equation with integer derivatives because of its hereditability and memorability characteristics. However, in most instances, the precise analytical solutions of UFDE is difficult to obtain due to the complex form of the UFDE itself. Up to now, there is not plenty of researches about the numerical method of UFDE, as for the existing numerical algorithms, their accuracy is also not high. In this research, derive from the interval weighting method, a class of fractional adams method is innovatively proposed to solve UFDE. Meanwhile, such fractional adams method extends the traditional predictor-corrector method to higher order cases. The stability and truncation error limit of the improved algorithm are analyzed and deduced. As the application, several numerical simulations (including $\alpha$-path, extreme value and the first hitting time of the UFDE) are provided to manifest the higher accuracy and efficiency of the proposed numerical method.

Tables

Table 1. Table 1: Comparison of numerical approaches for uncertain (fractional) differential equations

Work	Author	Order	Algorithm	Solution
[11]	Kai Yao &Xiaowei Chen	$p=1$	Euler’s method	Analytical solution &Numerical solution
[12]	Xiangfeng Yang &Dan A Ralescu	$p=1$	Adams method	Numerical solution
[13]	Rong Gao	$p=1$	Miline methodd	Numerical solution
[14]	Xiao Wang &Yufu Ning	$p=1$	Adams-simpson method	Numerical solution
[15]	Yi Zhang etal.	$p=1$	Haming method	Analytical solution
[16]	GuoCheng Wu et al.	$p\in (0,1]$	Adams method	Numerical solution
[17]	Luo Cheng et al.	$p\in (0,1]$	Truncation method	Analytical solution &Numerical solution
[18]	Ziqiang Lu &Yuanguo Zhu	$p\in (0,1]$	Truncation method	Numerical solution

Equations

$$\mathcal{M}\left\{\prod _{k=1}^{\infty }{\Lambda }_k\right\}=\underset{k=1}{\overset{\infty }{\bigwedge }}{\mathcal{M}}_k\left\{{\Lambda }_k\right\}$$

$${\Phi }_t(x)={\left(1+\exp \left(-\frac{\pi x}{\sqrt{3}t}\right)\right)}^{-1}$$

$${\Phi }_t^{-1}(\alpha )=\frac{\sqrt{3}t}{\pi }\ln \frac{\alpha }{1-\alpha }.$$

$$\mathrm{d}{X}_t=f\left(t,X_t\right)\mathrm{d}t+g\left(t,X_t\right)\mathrm{d}{C}_t$$

$$\mathrm{d}{X}_t^{\alpha }=f\left(t,X_t^{\alpha }\right)\mathrm{d}t+\left|g\left(t,X_t^{\alpha }\right)\right|{\Phi }^{-1}(\alpha )\mathrm{d}t,0<\alpha <1$$

$${\Phi }^{-1}(\alpha )=\frac{\sqrt{3}}{\pi }\ln \frac{\alpha }{1-\alpha }.$$

$$\mathcal{M}\left\{X_t\le X_t^{\alpha },\forall t\right\}=\alpha ,$$

$$\mathcal{M}\left\{X_t>X_t^{\alpha },\forall t\right\}=1-\alpha .$$

$$I_a^{\nu }f(t)=\frac{1}{\Gamma (\nu )}{\int }_a^t(t-s{)}^{\nu -1}f(s,X_s)\mathrm{d}s,$$

$${}_a^C{D}_t^{\nu }f(t)=I_{a+}^{n-\nu }{f}^{(n)}(t)=\frac{1}{\Gamma (n-\nu )}{\int }_a^t(t-s{)}^{n-\nu -1}{f}^{(n)}(s,X_s)\mathrm{d}s.$$

$${}_a^C{D}_t^{\nu }{X}_t=f\left(t,X_t\right)+g\left(t,X_t\right)\frac{\mathrm{d}{C}_t}{\mathrm{d}t.}$$

$${}_a^C{D}_t^{\nu }{X}_t^{\alpha }=f\left(t,X_t^{\alpha }\right)+\left|g(t,X_t^{\alpha })\right|{\Phi }^{-1}(\alpha ).$$

$$|f(t,x)-f(t,y)|+|g(t,x)-g(t,y)|\le L|x-y|,\forall x,y\in {\mathbb{R}}^n,t\in [0,+\infty )$$

$$|f(t,x)|+|g(t,x)|\le L(1+|x|),\forall x\in {\mathbb{R}}^n,t\in [0,+\infty ).$$

$$\left\{\begin{array}[]{l}\displaystyle{}_{a}^{C}D_{t}^{\nu}X_{t}=f\left(t,X_{t}\right)+g(t,X_{t})\frac{\mathrm{d}C_{t}}{\mathrm{d}t}\\ \left.X_{t}^{(i)}\right|_{t=a}=x_{i},i=0,1,\ldots,n-1\end{array}\right.$$

$$X_t=\sum _{i=0}^{n-1}\frac{(t-a{)}^i{x}_i}{\Gamma (i+1)}+\frac{1}{\Gamma (\nu )}{\int }_a^t(t-s{)}^{\nu -1}f\left(s,X_s\right)\mathrm{d}s+\frac{1}{\Gamma (\nu )}{\int }_a^t(t-s{)}^{\nu -1}g\left(s,X_s\right)\mathrm{d}{C}_s,$$

$$\left\{\begin{array}[]{l}\displaystyle{}_{a}^{C}D_{t}^{\nu}X_{t}^{\alpha}=f\left(t,X_{t}^{\alpha}\right)+g(t,X_{t}^{\alpha})\frac{\mathrm{d}C_{t}}{\mathrm{d}t},~{}\alpha\in(0,1)\\ \left.[X_{t}^{\alpha}]^{(i)}\right|_{t=a}=x^{\alpha}_{i},i=0,1,\ldots,n-1,\end{array}\right.$$

$$X_t^{\alpha }=$$

$$+\frac{1}{\Gamma (v)}{\int }_a^t(t-s{)}^{v-1}\left|g\left(s,X_s^{\alpha }\right)\right|{\Phi }^{-1}(\alpha )\mathrm{d}s.$$

$$\hat{I}=\sum _{i=0}^n\sum _{j=0}^m{w}_{ij}f(t_{ij}),$$

$$I={\int }_a^{b}p(s)f(s)\mathrm{d}s$$

$$X(t_{n+1})=x_0+\frac{1}{\Gamma (\nu )}{\int }_{t_0}^{t_{n+1}}(t_{n+1}-s{)}^{\nu -1}f(s,X_s)\mathrm{d}s,$$

$${\int }_{t_0}^{t_{n+1}}(t_{n+1}-s{)}^{\nu -1}f(s,X_s)\mathrm{d}s\approx \sum _{j=0}^{n+1}{a}_{j,n+1}f(t_j,X_{t_j})$$

$${\mathcal{L}}_{j,n+1}(s)=\{\begin{array}{ll}\left(s-t_{j-1}\right)/\left(t_j-t_{j-1}\right)& ,\text{ if }{t}_{j-1}<s<t_j\\ \left(t_{j+1}-s\right)/\left(t_{j+1}-t_j\right)& ,\text{ if }{t}_j<s<t_{j+1}\\ 0& ,\text{ otherwise }\end{array}.$$

$${\int }_{t_i}^{t_{i+1}}(t_{n+1}-s{)}^{\nu -1}{L}_{ij}(s)\mathrm{d}s$$

$$M(\nu-1,k)=-\frac{{{s}^{k}}}{k+\nu}{{(t_{n+1}-s)}^{\nu}}\bigg{|}_{t_{i}}^{t_{i+1}}+\frac{kt_{n+1}}{k+\nu}M(\nu-1,k-1).$$

$$\int_{t_{i}}^{t_{i+1}}{{{(t_{n+1}-s)}^{\nu-1}}{{s}^{k}}\mathrm{d}s}=-\frac{{{s}^{k}}}{\nu}{{(t_{n+1}-s)}^{\nu}}\bigg{|}_{t_{i}}^{t_{i+1}}+\frac{k}{\nu}\int_{t_{i}}^{t_{i+1}}{{{(t_{n+1}-s)}^{\nu}}{{s}^{k-1}}\mathrm{d}s},$$

$${\int }_{t_i}^{t_{i+1}}{(t_{n+1}-s)}^{\nu }{s}^{k-1}\text{d}s=t{\int }_{t_i}^{t_{i+1}}{(t_{n+1}-s)}^{\nu -1}{s}^{k-1}\mathrm{d}s-{\int }_{t_i}^{t_{i+1}}{(t_{n+1}-s)}^{\nu -1}{s}^k\mathrm{d}s.$$

$$M(\nu-1,k)=-\frac{{{s}^{k}}}{\nu}{{(t_{n+1}-s)}^{\nu}}\bigg{|}_{t_{i}}^{t_{i+1}}+\frac{k}{\nu}M(\nu,k-1),$$

$$M(\nu ,k-1)=tM(\nu -1,k-1)-M(\nu -1,k).$$

$$M(\nu-1,k)=-\frac{{{s}^{k}}}{\nu}{{(t_{n+1}-s)}^{\nu}}\bigg{|}_{t_{i}}^{t_{i+1}}+\frac{k}{\nu}\left[tM(\nu-1,k-1)-M(\nu-1,k)\right].$$

Full text

Keywords: Uncertain fractional differential equation; Fractional adams method; Interval weighting; $\alpha$-path; Extreme value

1 Introduction

The most common mathematical tools used to describe uncertain phenomena are frequency-based probability theory and degree of belief based on uncertainty theory. In real life, the precondition of using probability theory to deal with random phenomena is that there is sufficient sample data so that the probability distribution can be inferred to approximate the actual situation. Then the probability theory can solve many problems and really have good effects. Owing to practical or technical reasons, sufficient sample data cannot be obtained in most cases, so the evaluation must be based on the trust of experts in the field. It is worth mentioning that the Nobel Prize winner in economics Kahneman and Tversky [1] pointed out that humans generally overestimate the likelihood of some events happening, which can make the results deviate from reality and even make decision makers make wrong decisions. To workaround the belief degree of experts mathematically, Liu [2, 3] established the uncertainty theory in 2007. In fact, the uncertainty theory just describes the case where the confidence level is usually much greater than the cumulative frequency for small or unavailable sample size. For more details about uncertainty theory, see reference [4].

After putting forward the uncertainty theory, Liu [2] defined the uncertain measure, uncertain space, uncertain variable together with its uncertain distribution, expectation and variance respectively. For more details, please refer to [5]. Liu [6] defined the uncertain process for the sake of describing the evolution of uncertain phenomenon over time in 2008. In order to study the uncertain calculus of uncertain process, Liu [5] proposed Liu process, which is a Lipschitz continuous uncertain process with normal uncertain variables as stationary independent increments. Moreover, he advanced the uncertain differential equations (UDEs) to describe the evolution of uncertainty over time. After that, researchers began to pay attention to the numerical method of solving UDE because UDE has many applications in uncertain optimal control [7] and uncertain finance [8, 9, 10]. In 2013, Yao and Chen [11] first putted forward the famous Yao-Chen formula.The relationship between ordinary differential equation and UDE has thus been established. Based on the Yao-Chen formula, a series of numerical methods have emerged: Among them, Yang and Ralescu [12] proposed the Adams method to solve the UDE, Milne method was designed by Gao[13] in 2016 for solving the UDEs. Wang and Ning [14] provided Adams-Simpson method, while Zhang and Gao [15] proposed Hamming method to solve the UDEs.

In some complex dynamic systems, ordinary differential equations usually cannot adequately describe the complex operating mechanisms and hereditability of the system. In this case, fractional differential equation (FDE) can be well used for describing the memorability and historical features. The fractional calculus method has been widely used in the financial field in recent years [19, 20], optimal control [21] and image encryption [22, 23]. For the latest researches on FDEs, please refer to [24, 25]. In 2013, Zhu [26] first combined the fractional theory with the uncertainty theory, defined two types of UFDEs, the Caputo one and the Riemann-Liouville one. Subsequently, based on linear growth condition and Lipschitz condition, Zhu [27] proposed the existence and uniqueness theorem of the UFDE’s solution. Using the $\alpha$-path in Yao-Chen formula, the relationship between the FDEs and the UFDEs is established, the solution of UFDEs can be represented by a cluster of solutions of FDEs, that is, $\alpha$-path is a numerical methods for solving UFDE. Lu [18] proposed a numerical approach to solve UFDE involving Caputo derivatives by using $\alpha$-path, and also provided a formula to calculate the expected value of the monotone function of the solution about UFDE. In many other fields, UFDE also has its own applications. Jin [28] studied the extreme value of a kind of Caputo type UFDE’s solution and subsequently applied it to the American option pricing model. In addition, Jin [29] also used the predictor-corrector method to give the uncertain distributions of their first hitting time of a kind of nonlinear Caputo type UFDE, and applied it to a novel uncertain risk index model. Wu et al. [16] studied the parameter estimation of UFDE based on fractional Adams method. Considering the generalization of UFDE, Luo et al. [17] studied the uniqueness and existence of the solution of the generalized fractional uncertain differential equation (GUFDE), and gives the extreme values and solutions of GUFDEs. It is worth noting that in these studies, the accuracy of the method used to calculate the uncertain distribution is not too high, so it is necessary to study a kind of numerical algorithm with higher accuracy and faster arithmetic speed.

As the main motivation, our work aims to study the improvement of fractional order Adams numerical algorithm. Kai Diethelm [30] first proposed the fractional Adams method. Li [31] further proposed the fractional Adams method based on Simpson method, which approximated the fractional integral and the fractional derivative with the use of higher order piecewise interpolating polynomials. This approach was designed to solve the analytical expression of the uncertain integral. Thus when there are too many nodes, the complexity of the solution will be very large. Based on the motivation of improving the computational accuracy of Adams method, our study proposes a novel fractional Adams method which can be extended to any node.

The frame of article is as follows: In Section 2, some concepts and properties of the UFDE, coupled with the product integration method and Adams method, are reviewed. In Section 3, the Adams method is extended to order $n$. Relevant numerical experiments can be found in Section 4, which calculate the extreme value, the inverse distribution and the first hitting time (FHT) of UFDE, respectively. Finally, a brief conclusion is given in Section 5.

2 Preliminary

In this section, some definitions and theorems about uncertainty theory are introduced. The brief introductions of the Adams method and UFDE are given.

2.1 Uncertainty theory and UFDE

Definition 2.1

(Liu[3]) Assume that $\Gamma$ is a nonempty set, $\mathcal{L}$ is a $\sigma$-algebra over $\Gamma$. Call each element $\Lambda\in\mathcal{L}$ an event. Call a set function $\mathcal{M}$ defined on the $\sigma$-algebra an uncertain measure, if the following four axioms are satisfied:

a. $\mathcal{M}\{\Gamma\}=1$.

b. $\mathcal{M}\{\Lambda\}+\mathcal{M}\left\{\Lambda^{c}\right\}=1$ for any event $\Lambda$.

c. $\mathcal{M}\left\{\cup_{i=1}^{\infty}\Lambda_{i}\right\}\leq\sum\limits_{i=1}^{\infty}\mathcal{M}\left\{\Lambda_{i}\right\}$ for events $\Lambda_{1},\Lambda_{2},\ldots$

d. Let $\left(\Gamma_{k},\mathcal{L}_{k},\mathcal{M}_{k}\right)~{}(k\in\mathbb{N}^{+})$ be uncertain spaces. The product uncertain measure $\mathcal{M}$ is an uncertain measure satisfied

[TABLE]

.

Definition 2.2

(Liu[5]) Call an uncertain process $C_{t}$ a canonical Liu process, if

a. $C_{0}=0$ and almost all sample paths are Lipschitz continuous.

b. $C_{t}$ has stationary and independent increments.

c. every increment $C_{s+t}-C_{s}$ is a normal uncertain variable with expected value [math] and variance $t^{2}$.

The uncertain distribution of $C_{t}$ is

[TABLE]

and inverse uncertain distribution is

[TABLE]

Definition 2.3

(Yao-Chen[11]) An UDE

[TABLE]

is said to have an $\alpha$-path $X_{t}^{\alpha}$ if it solves the differential equation

[TABLE]

where

[TABLE]

is the inverse standard normal uncertain distribution.

Theorem 2.1

(Yao-Chen[11]) Let $X_{t}$ be the solution, $X_{t}^{\alpha}$ be the $\alpha$-path of the UDE(2.3), respectively. Thus

[TABLE]

[TABLE]

Definition 2.4

([32]) For any $\nu>0,t>a$, the Riemann-Liouville type fractional integral is defined as

[TABLE]

where $\Gamma(\alpha)$ is Gamma Function.

Definition 2.5

([32]) Assume that $\nu>0,t>a$ and $n-1<\nu\leq n~{}(n\in\mathbb{N}^{+}$), Caputo type fractional differential is defined as

[TABLE]

Definition 2.6

(Li[31]) Suppose $C_{t}$ is a canonical Liu process, $\nu>0$, $f:[0,+\infty)\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ and $g:[0,+\infty)\times\mathbb{R}^{n}\to\mathbb{R}^{n+l}$ are two given functions. An UFDE

[TABLE]

The corresponding $\alpha$-path $X_{t}^{\alpha}$ of $X_{t}$ is a function of $t$ which solves the following FDE

[TABLE]

Theorem 2.2

(Zhu[27]) An UFDE has a unique solution $X_{t}$ in $[0,+\infty)$, if the coefficients $f(t,X_{t})$ and $g(t,X_{t})$ satisfy the Lipschitz condition

[TABLE]

and the linear growth condition

[TABLE]

Theorem 2.3

(Lu[33]) A Caputo type UFDE with initial value conditions

[TABLE]

has an integral form

[TABLE]

and if its $\alpha$-path $X_{t}^{\alpha}$ satisfies

[TABLE]

then

[TABLE]

3 Improved $n$-order fractional Adams method

In this section, the numerical method of a class of integral equations, namely the product integration method, and the fractional Adams method, will be introduced. On this basis, this paper will improve the Fractional Adams Method to apply it to any number of nodes. This paper proves the operability of this method and gives the truncation error.

3.1 Improved fractional Adams method

Product Integration Method was originally proposed to solve the integral equation. The numerical solution of fractional differential equation can be converted into the numerical method of the second kind of Volterra integral equation with a circular singular kernel. The problem of The fractional Adams method proposed by Kai Diethelm [31] is the application of two-point Lagrange interpolation in Product Integration Method.

We can take

[TABLE]

as an approximate estimate of integral

[TABLE]

where ${{w}_{ij}}=\int_{t_{i}}^{t_{i+1}}p(s)L_{ij}(s)\mathrm{d}t$ and $L_{ij}$ is base function of Lagrange interpolation.

Let us consider the integral equation with $0<\nu\leq 1$

[TABLE]

using fractional Adams method, we have

[TABLE]

where $a_{j,n+1}=\int^{t_{n+1}}_{t_{0}}(t_{n+1}-s)^{\nu-1}\mathcal{L}_{j,n+1}(s)\mathrm{d}s$ and

[TABLE]

In the premise of stability, this paper tries to explore a higher order fractional Adams method, which means increasing the number of nodes of Lagrange interpolation. Li [31] proposed the fractional Adams method based on Simpson method in 2011. However, solving the coefficient before the node involves solving the equation

[TABLE]

or the corresponding analytical expression of indefinite integral, with the number of nodes increases, the difficulty of solving is also increasing. Therefore, we need to improve the fractional Adams method to obtain a more accurate solution.

Back to the fractional Adams method, if we want to use more points to improve the calculation accuracy, we need to give a general formula to calculate the term $\displaystyle\int_{t_{i}}^{t_{i+1}}(t_{n+1}-s)^{\nu-1}s^{k}\mathrm{d}s$. Only when this term has an exact value or analytical formula in different cases of $k$ can it be applied to fractional integration. Now we give the calculation method of the exact value of this item.

Theorem 3.1

Let $M(\nu-1,k)=\displaystyle\int_{t_{i}}^{t_{i+1}}(t_{n+1}-s)^{\nu-1}s^{k}\mathrm{d}s$, then

[TABLE]

Proof: For $\displaystyle\int_{t_{i}}^{t_{i+1}}(t_{n+1}-s)^{\nu-1}s^{k}\mathrm{d}s$, we have

[TABLE]

[TABLE]

That is,

[TABLE]

[TABLE]

Simultaneous Eq. (6) and Eq. (7), we have

[TABLE]

Thus, the theorem is proved.

Remark 3.1

According to Theorem 3.1, when the form of the integral $\displaystyle\int_{t_{i}}^{t_{i+1}}({{t}_{n+1}}-s)^{\nu-1}\mathrm{d}s$ is known, we can easily compute the integral $\displaystyle\int_{t_{i}}^{t_{i+1}}(t_{n+1}-s)^{\nu-1}s^{k}\mathrm{d}s$ for any $k\geq 1$. The original complex integral calculation is transformed into solving the relationship between different $k\geq 1$, which greatly reduce the computation complexity.

In this way, we successfully find the relation about $\displaystyle\int_{t_{i}}^{t_{i+1}}(t_{n+1}-s)^{\nu-1}s^{k}\mathrm{d}s$ under different $k$. In addition, most node-related data generated during the calculation process can be reused to avoid redundancy.

Theorem 3.2

The product integral over the interval $[t_{i},t_{i+1}]$

[TABLE]

can be written as

[TABLE]

Then, rewrite the Lagrange interpolation for $m$ nodes into a form of polynomial

[TABLE]

Proof: Substituting the expression of $L_{ij}$ into $\displaystyle\int_{{{t}_{i}}}^{{{t}_{i+1}}}(t_{n+1}-s)^{\nu-1}{L}_{ij}(s)\mathrm{d}s$, we can obtain

[TABLE]

then substitute it into Eq. (9), the conclusion apparently be proofed.

In this way, combining the relation of the integral terms with Eq. (10), we can get the expression of the numerical algorithm.

Considering the Lagrange interpolation method for node $\{t_{i}\}_{0\leq i\leq n}$,

where the formula and polynomial form of the interpolation of the $j$ term are

[TABLE]

and

[TABLE]

repectively.

3.2 Predictor-Corrector method of improved fractional Adams method

Given the node $t_{k-m},t_{k-m+1},\cdots,t_{k-1}$ and the corresponding value $X_{t_{k-m}},X_{t_{k-m+1}},\cdots,X_{t_{k-1}}$, we estimate

[TABLE]

of the interval $[t_{k-1},t_{k}]$. Let the predictor term be

[TABLE]

where

[TABLE]

[TABLE]

Then the corrector term is

[TABLE]

Theorem 3.3

Considering $f(t)\in C^{n}[a,t]$, for the $n$-th order fractional Adams method on the subinterval $[t_{n-1},t_{n}]$, the truncation error limit is:

[TABLE]

where $\omega_{(}s)=(s-t_{0})(s-t_{1})\cdots(s-t_{n-1}),M_{n}=\max\limits_{a\leq s\leq t}f^{(n)}(x)$.

To solve the FDE as follow

[TABLE]

The following algorithm is given.

In the case of limiting the number of intervals to be divided, it is recommended to divide the intervals first intensively then incompactly, which is because in the case of only one initial value, before using $n$ order Adams algorithm, we need to use $m=1,2,\cdots,n-1$ order Adams algorithm to calculate the first $n$ points successively. Therefore, the error of the initial point will affect the subsequent calculation, so the intervals of the first few points is reduced to avoid its impact on the accuracy in the subsequent weighted calculation. Or you can use a more expensive method to calculate the value of the first few points with higher accuracy, and then use $n$ order Adams algorithm for calculation.

4 Numerical Simulations

This section introduces an example of using the modified $n$ order Adams algorithm to calculate the extreme value of UFDE. The differential equation used in this example has the analytical solution. This paper give the error analysis of the inverse distribution $X^{\alpha}_{t}$ of UFDE’s solution for $\alpha\in[0.01,0.99]$ and $t\in[0.01,1]$. The calculation results using the algorithm are given, and the magnitude of error for different $n$ are compared.

Based on two differential equations without analytical solutions, numerical methods are constructed. One of which is applied to the inverse distribution of UFDE’s solution, and the other is applied to calculate solve the first hitting time of the UFDE. Both of them have achieved good results.

4.1 Extreme value of UFDE’s solution

In this subsection, we calculate extreme values of UFDE’s solution with a numerical format, which is described in Algorithm 3. According to the description in [28], let $X_{t}$ is the unique solution for Eq. (1) and the $X_{t}^{\alpha}$ is $\alpha$-path. $\Psi_{t}$ is the uncertain distribution of $X_{t}$. The IUD of the infimum $\inf\limits_{0\leq t\leq s}J(X_{t})$ exists,

[TABLE]

Example 4.1

Assume the following linear Caputo type UFDE

[TABLE]

with initial value $X_{0}=0.5$ and $J(x)=x$.

We calcuate the absolute errors of different $\alpha$-path with parameter values $a=0.6,b=1,t=1,\upsilon=2$ at $n\in[2,3,4]$, see Figs. 1(a), 1(c) and 1(e). Moreover, the absolute error of uncertain distributions of $X^{\alpha}_{t}$ with different $t$ and $\alpha$ at $n\in[2,3,4]$ provided by Algorithm 2 is shown in Figs. 1(b), 1(d) and 1(f). The uncertain distribution of extreme value is shown in Fig. 3(a) and the absolute error of extreme value is shown in Fig. 3(b). In order to show the absolute error under different $n$ better, we perform a logarithmic on the error results.

As can be seen from Figs. 1(a), 1(c) and 1(e), when $n=2,3,4$, the order of error is $10^{-4},\,\,10^{-6}$ and $10^{-7}$. In Figs. 1(b), 1(d) and 1(f), when $n$ increases , the change rate of error about $t$ also increases accordingly with the same value $\alpha$. In fact, as the number of nodes increases, the stability of the algorithm will also deteriorate. In Fig. 1(e), when $n=4$, part of the function image about the error of $\alpha$ appears jagged.

When $n=5$, in Fig. 2, this situation worsens. This image of the absolute value of error about $\alpha$ shows that it is very sensitive to the change of the inverse distribution of diffusion term $bt^{\nu}\Phi^{-1}(\alpha)$. For this example, if $n$ is too large, its effect will not be quite as impressive in terms of sensitivity or accuracy. Considering that the essence of fractional Adams method is the application of Lagrange interpolation, this phenomenon may be caused by over-fitting caused by too many nodes.

Fig. 3(a) shows the extreme value of $\alpha$-path under the given parameters. Fig. 3(b) is the image that compares Figs. 1(a), 1(c) and 1(e) after logarithmization. As shown in Fig. 3(b) and Fig. 2, compared with $n\geq 5$, the absolute error of the results is acceptable when $n=3$ or $4$.

Then we analyze the error in the case of order $\nu\in[0.1,0.9]$ and parameter $\upsilon\in[1,3]$ respectively, and the results are as follows:

In Fig. 4(a), The $Y$-axis represents the change of MAE after logarithmizing the $alpha$-path, and the $X$-axis represents the change of order of UFDE. In Fig. 4(b), $X$-axis represents the change of parameter $\upsilon$.

Obviously, with the increase of order $\nu$, MAE is also decreasing, which shows that the order of fractional differential equation also has an impact on the error. From Fig. 4(a) shows a negative correlation between the two. For the parameter $\upsilon$, it is positively correlated with MAE. With the increase of $\upsilon$, undering the same $t$, the integer derivative value of the right term of the FDE also increases correspondingly. According to the Eq. (15), the error bound will become larger.

4.2 Inverse distribution of UFDE’s solution

In many cases, UFDE is unable to find an exact analytical solution. In this section, we present a numerical method for calculating the inverse uncertain distribution of its solution. The algorithm is given as follows.

Example 4.2

Assume the following Caputo type UFDE,

[TABLE]

with initial value $X_{0}=0$.

When $a=1.2,\mu=0.05,\sigma=0.04,\nu=0.8$, the numerical results of UFDE is shown in Fig. 5.

4.3 First hitting time of UFDE’s solution

When UFDE is nonlinear, it is generally difficult to find the exact solution. In this subsection, we use example to show how to apply fractional Adams method to calculate the FHT of nonlinear fractional equation. When $z>J(X_{0})$ and $J(x)$ is a nondecreasing function, according to [29], the distribution of the FHT is

[TABLE]

Example 4.3

Assume that a nonlinear UFDE of the Caputo type is

[TABLE]

with the initial value $X_{0}=3$ and $J(x)=x$, where $J(X_{0})<z$.

We assign values to the parameters $z=4,\nu=0.8$. According to Algorithm 2, the distribution of FHT is shown in Fig. 6.

In subsection 4.2 and 4.3, it is laborious to verify the accuracy of the results. We only show the rationality of this method through linear and nonlinear examples and calculation of different indicators of the model. According to the previous equation on accuracy (15) and the example in subsection 4.1, the accuracy of calculation is guaranteed. Since the calculation of fractional differential equations involves the weighting of the whole time domain, the stability of the algorithm is not given, but only shown from the stability of the numerical results. It can be seen from the numerical simulation that the stability cannot be guaranteed with the increase of $n$.

5 Conclusion

A novel numerical method has been developed to solve UFDE with initial value conditions. On the basis of the existing Adams method, according to the product integration method, we gave the polynomial form of lagrange basis function in the case of several values, which extended the traditional Adams method to the fractional Adams method with any node, and did not cause redundancy in computation time. Moreover, three numerical algorithms have been designed for calculating the extreme value, the inverse uncertain distribution and the first hitting time of the solution for UFDE to verify the effectiveness of the methods. Before using the $n$-order Adams method, we need to calculate the first $n$ points, so the accuracy of the initial points is very important. With the aim of more accurate results, we will further study and develop more efficient numerical methods in future research to calculate the estimated values of the first $n$ nodes.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No.12201304 and No.12071219) and supported by Academic Program Development of Jiangsu Higher Education Institutions (PAPD), Natural Science Foundation of Jiangsu Province (No.BK20210605), the General Research Projects of Philosophy and Social Sciences in Colleges and Universities (2022SJYB0140), the Jiangsu Province Student Innovation Training Program (202110298040Z and 202210298050Z).