The inverse uncertainty distribution of the solutions to a class of higher-order uncertain differential equations

1.   Introduction

Events with known frequencies of occurrence are classified as random, while those with unknown frequencies are termed uncertain ^[1]. With the rapid advancement of science and technology, a multitude of uncertain factors have emerged in real life, rendering the phenomena of uncertainty in the objective world undeniable. Consequently, scholars have begun to incorporate these uncertain factors into the establishment of mathematical models, leading to the research and development of uncertainty theory.

Integrating uncertain factors into differential equations results in the formation of UDEs, a type of differential equation established by Liu in 2007 ^[1], designed to describe the dynamics of uncertain phenomena. Yao and Chen provided an effective formula for calculating the inverse uncertainty distribution of the solutions to UDEs, known as the Yao-Chen formula ^[2].

The Yao-Chen formula yields a family of solutions to ordinary differential equations, denoted as $\alpha$-paths, and it has been indicated ^[2] that these $\alpha$-paths represent the inverse uncertainty distribution of the solutions. Therefore, to determine the inverse uncertainty distribution of the solution, we need to solve a family of ordinary differential equations to obtain the $\alpha$-paths. For first-order scalar UDE, the inverse uncertainty distribution of the solutions can be determined using the Yao-Chen formula. Building on this foundation, research has been conducted in many fields, including finance ^[3], optimal control ^[4], population growth ^[5], pharmacokinetics ^[6,7], epidemiology ^[8], and heat conduction ^[9,10]. However, in many practical contexts, first-order scalar UDEs may not fully capture the complexity of real-world scenarios. Often, more intricate cases emerge, such as higher-order, fractional-order, functional differential equations, etc. Thus, investigating theories concerning those UDEs is of significant importance. This paper primarily focuses on the inverse uncertainty distribution problem of a class of higher-order UDEs.

Zhang ^[11] has rigorously defined the concept of multiple integrals and higher-order derivatives of uncertain processes, establishing the framework for higher-order UDEs. These contributions have addressed previous theoretical deficiencies in higher-order UDEs and provided new analytical methods for solving more complex uncertain system problems. However, the definition of the $\alpha$-path, which is crucial for determining the inverse uncertainty distribution of UDEs' solutions, was not provided. Therefore, in this article, we define the $\alpha$-path for second-order UDEs and a class of higher-order UDEs, and determine the corresponding inverse uncertainty distributions for the solutions of these types of UDEs. This will enable a more comprehensive understanding and analysis of higher-order uncertain processes, offering new perspectives and tools for the theoretical development and practical application of uncertain calculus.

To determine the inverse uncertainty distribution of the solutions to UDEs, it is essential to identify the $\alpha$-paths. The formation of $\alpha$-paths must satisfy fundamental conditions, primarily concerning the monotonicity with respect to $\alpha$. That is, at any given time $t$, the value of $X_t^\alpha$ on an $\alpha$-path should monotonically increase with respect to $\alpha$, as illustrated in Figure 1(a). It is impermissible to encounter a scenario as depicted in Figure 1(c) and Figure 2(b). The main focus of this paper is to study under what conditions the $\alpha$-paths of UDEs behave like the one in Figure 1(a), rather than exhibiting the situations shown in (b) and (c).

The rest of this paper is organized as follows: In Section 2, we review related concepts and conclusions of uncertainty theory. In Section 3, we define the $\alpha$-paths for some second-order UDEs and a class of higher-order UDEs, deriving the inverse uncertainty distribution corresponding to the solutions. In Section 4, we provide positive and negative instances of the theorem in Section 3. In Section 5, we give a brief summary of this paper. Figure 2 illustrates the conceptual framework of this manuscript, encapsulating the core ideas and interconnections presented in the paper.

2.   Preliminary

In this section, we introduce some basic concepts and theorems about uncertain processes and uncertain calculus.

Definition 2.1 (Liu ^[12]) Let $\Gamma$ be a non-empty set, let $\Lambda$ be a $\sigma$-algebra over $\Gamma$, and let $\mathcal{M}$ be an uncertain measure. Then the triplet $(\Gamma_k, \mathcal{L}_k, \mathcal{M}_k)$ is called an uncertainty space.

Theorem 2.1. ^[13] (Measure Inversion Theorem) Let $\xi$ be an uncertain variable with uncertainty distribution $\Phi$. Then for any real number $x$, we have

Definition 2.2 (Liu ^[14]) An uncertain process $C_t$ is said to be a Liu process if

(i) $C_0 = 0$ and almost all sample paths are Lipschitz continuous,

(ii) $C_t$ has stationary and independent increments,

(iii) every increment $C_{s+t}$-$C_s$ is a normal uncertain variable with an expected value of $0$ and variance $t^2$.

Definition 2.3 (Yao-Chen^[2]) Let $\alpha$ be a number between $0$ and $1$. An uncertain differential equation

is said to have an $\alpha$-path $X_t^\alpha$ if it solves the corresponding ordinary differential equation

where $\Phi^{-1}(\alpha)$ is the inverse standard normal uncertainty distribution, i.e.,

Definition 2.4 (Liu^[12]) Suppose $f$ and $g$ are continuous functions. An uncertain differential equation

is said to satisfy the regular condition if

Theorem 2.2. (Liu^[12]) The uncertain differential equation

has a unique solution if the functions $f(t, x)$ and $g(t, x)$ satisfy the linear growth condition

and the Lipschitz condition

Without loss of generality, suppose $L$. Moreover, the solution is sample continuous.

Theorem 2.3. (Liu^[12]) Let $X_t^\alpha$ be the $\alpha$-path of the regular uncertain differential equation

If the linear growth, Lipschitz, and regular conditions hold, then $X_t^\alpha$ is a strictly increasing function with respect to $\alpha$ at each time $t > 0$.

Theorem 2.4. (Liu^[12]) Let $X_t^\alpha$ be the $\alpha$-path of the regular uncertain differential equation

Then $X_t^\alpha$ is a continuous function with respect to $\alpha$ at each time $t > 0$.

Theorem 2.5. (Yao-Chen formula ^[2]) Let $X_t$ and $X_t^\alpha$ be the solution and $\alpha$-path of the regular uncertain differential equation

respectively. Then

Definition 2.5. (Liu ^[14]; Chen and Ralescu^[15]; Ye^[16]) Let $C_t$ be a Liu process, and let $Z_t$ be an uncertain process. If there exist two sample-continuous uncertain processes $\mu_t$ and $\delta_t$ such that

for any $t\ge 0$, then $Z_t$ is called a general Liu process with drift $\mu_t$ and diffusion $\delta_t$. Furthermore, $Z_t$ has an uncertain differential

and a first-order derivative

where $\dot{C}_t$ is the formal derivative ${\rm d}C_t/{\rm d}t$.

Definition 2.6 (Zhang ^[11]) Let $C_t$ be a Liu process, and let $Z_t$ be an uncertain process. If there exist sample-continuous uncertain processes $\mu _t$ and $\delta_t$ such that

for any $t \ge 0$, then $Z_t$ is called a second-order Liu process and has a second-order derivative

where $\ddot{Z_t}$ is the formal second-order derivative ${\rm d}^2 Z_t/{\rm d}t^2.$

Theorem 2.6. (Zhang ^[11]) Let $f(t, s)$ be an uncertain field, and let $X_t$ and $Y_s$ be general Liu processes. For any partition of the closed region $[0, a] \times [0, b]$ with

the mesh is written as

Then the double Liu integral of $f(t, s)$ with respect to $X_t$ and $Y_s$ is defined as

provided that there is an uncertain variable to which the above sum converges almost surely as $\Delta \rightarrow 0$. In this case, the uncertain field $f(t, s)$ is said to be integrable with respect to $X_t$ and $Y_s$.

Theorem 2.7. Zhang ^[11] (Fubini's Theorem) Let $X_t$ and $Y_s$ be general Liu processes. Suppose $f(t, s)$ is an integrable uncertain field with respect to $X_t$ and $Y_s$. Then

(1) $\int_{0}^{a}f(t, s){\rm d}X_t$ and $\int_{0}^{b}f(t, s){\rm d}Y_s$ exist almost surely;

(2) $\int_{0}^{a}f(t, s){\rm d}X_t$ and $\int_{0}^{b}f(t, s){\rm d}Y_s$ are integrable with respect to $Y_s$ and $X_t$, respectively;

(3)

Definition 2.7. (Liu^[12]) Suppose $f$ and $g$ are continuous functions. An uncertain differential equation

is said to be regular if

Definition 2.8. (Zhang ^[11]) An uncertain process $X_t$ is called a solution of the higher-order uncertain integral equation if

where $C_t$ is a Liu process, and $f$ and $g$ are continuous functions. Equivalently, the above equation can be simply written as the differential form

that is called a higher-order uncertain diferential equation.

Theorem 2.8. (Liu ^[17]) A function $\Phi^{-1}:(0, 1) \rightarrow \Re$ is the inverse uncertainty distribution of an uncertain variable $\xi$ if and only if it is continuous and

3.   The inverse uncertainty distribution to some higher-order UDEs

Initially, for the significant research conclusions obtained in this paper, such as Theorem 3.6, a schematic diagram is employed for a more intuitive representation of the research process, as shown in Figure 3.

Next, we introduce Lemma 3.1, which serves as a foundation for the proof of Theorems 3.3, 3.4, 3.7, and 3.8.

Lemma 3.1. Lemma 3.1. Let $X_t$ satisfies the following higher-order UDE:

Then, $X_t$ also satisfies the following integral equation:

For simplicity and clarity, we denote $X_t(\gamma)$ as $X_t$. Henceforth, we shall use $X_t$ to represent the solution to the UDE and $X_t^\alpha$ to denote its corresponding $\alpha$-path.

Proof. Introduce two variables, $Z_t$ and $W_t$, which respectively satisfy the following two equations:

It can be easily derived that $X_t = Z_t+W_t$. For $Z_t$, we integrate both sides directly:

For $W_t$, we use intregration by parts repeatedly as follows:

Thus, adding Eqs (3.1) and (3.2), we derive the integral equation satisfied by $X_t$:

□

Definition 3.1. Let $\alpha$ be a number between $0$ and $1$. An uncertain differential equation

is said to have an $\alpha$-path $X_t^\alpha$ if it solves the corresponding ordinary differential equation

where $\Phi^{-1}(\alpha)$ is the inverse standard normal uncertainty distribution, i.e.,

Example 1. In the uncertain spring vibration equation ^[18] denoted as

The solution is given by

it has an $\alpha$-path

and the inverse uncertainty distribution is expressed as

Note 1.

(1) There are generally two approaches to obtaining the inverse uncertain distribution $\Psi_t^{-1}(\alpha)$ of the solution to a UDE. One approach involves directly solving the equation to obtain $\Psi_t^{-1}(\alpha)$^[18], and the other involves solving its $\alpha$-path, which entails solving a family of deterministic ordinary differential equations to determine $\Psi_t^{-1}(\alpha)$.

(2) The first method is straightforward, allowing for the direct computation of the solution $X_t$, and consequently, the inverse uncertainty distribution. Example 1 is linear and thus allows for the computation of $X_t$. However, for the majority of equations, analytical solutions cannot be derived, and currently, there are no numerical methods available for directly solving uncertain systems. Therefore, this method has significant limitations. The second method does not suffer from such limitations, as it involves solving a family of ordinary differential equations for the $\alpha$-path, which has a variety of solution methods. Consequently, the second method is more meaningful in practical applications and represents the direction of research in this paper.

$(3)$ Regarding $X_t^\alpha$ and $\Psi_t^{-1}(\alpha)$, if we consider them as bivariate functions, they represent the same function. If we view them as univariate functions with a single parameter, $X_t^\alpha$ is a function with $\alpha$ as the parameter and $t$ as the independent variable. $\Psi_t^{-1}(\alpha)$ is a function with $t$ as the parameter and $\alpha$ as the independent variable. Therefore, knowing $X_t^\alpha$ allows us to determine $\Psi_t^{-1}(\alpha)$ by simply interchanging the roles of the parameter and the independent variable.

Definition 3.2. Suppose $f$ and $g$ are continuous functions. An uncertain differential equation

is said to be regular if

Theorem 3.1. The uncertain differential equation is

where

It has a unique solution if the functions $f(t, \textbf{X})$ and $g(t, \textbf{X})$ satisfy the linear growth condition

and the Lipschitz condition

Without loss of generality, suppose $L \ge 2$. Moreover, the solution is sample continuous.

Proof. At first, Theorem 14.1 in ^[12] says that there exists an event $\Lambda$ with $\mathcal{M}\{\Lambda\} = 1$, such that $C_t(\gamma)$ is Lipschitz continuous with respect to $t$ for each $\gamma \in \Lambda$. Next, we prove the existence and continuity of the solution by a successive approximation method. Define

and

By integrating,

for each integer $n$. It follows from the linear growth condition that

for any $t \ge 0$, where $K_\gamma$ is the Lipschitz constant to the sample path $C_t(\gamma)$. Assume

It follows from the Lipschitz condition that

The above induction shows that

for each integer $n$. This means that, for each $\gamma \in \Lambda$, the sequence $X_{t}^{(n)}(\gamma)$ converges uniformly on any given time interval as $n \rightarrow \infty$. Furthermore, since $X_t^{(0)}$ is constant, obviously it is continuous. Suppose $X_t^{(k)}$ is continuous. Then, $X_t^{(k)}$ is also continuous with respect to $t$. So, by induction, $X_t^{(n)}$ is continuous for all $n$. Provided $X_t^{(n)}$ converges uniformly, that is $\lim\limits_{n \rightarrow \infty}X_t^{(n)}(\gamma) = X_t(\gamma)$, the limit function $X_t$ is also continuous.

The solution to the first equation below is $X_t(\gamma)$.

We then prove that the solution is unique. Assume that both $\textbf{X}_t$ and $\textbf{X}_t^{\ast}$ are solutions of the UDE. For each $\gamma \in \Lambda$, it follows from the Lipschitz condition that

By using the Gronwall inequality, we obtain

Hence $\textbf{X}_t = \textbf{X}_t^{\ast}$. The uniqueness is verified. The theorem is proved. □

Note 2. The conditions on $f$ and $g$ in the aforementioned theorem are stringent, leading to an ideal conclusion: the global existence of solutions. However, in practical problems, such a perfect conclusion is not necessary. Therefore, we can relax the condition, for instance, by discussing the properties of solutions within a bounded domain. In this case, the linear growth condition for $f$ and $g$ is not required. Instead, it is sufficient that $f$ and $g$ have an upper bound $M$ on the domain $G$. This also allows us to derive th einequality

From this inequality, we can infer the existence, continuity, and uniqueness of the solution on the domain. These are more practical conditions for applications, and the examples provided later in the paper all meet this criterion.

Theorem 3.2. Let $X_t^\alpha$ be the $\alpha$-path of the uncertain differential equation

If $f$ and $g$ satisfy the Lipschitz and regular conditions hold, then $X_t^\alpha$ is a continuous function with respect to $\alpha$ at each time $t > 0$.

Proof. Since $f$ and $g$ satisfy the Lipschitz conditions, it follows from Theorem 3.1 that the UDEs

have a unique solution $X_t^\alpha$, where $\Phi^{-1}(\alpha)$ is the inverse standard normal uncertainty distribution. For any numbers $\alpha$ and $\beta$ between 0 and 1, it follows from the Lipschitz condition that

By using the Gronwall inequality, we obtain

Thus

for each time $t > 0$. Hence $X_t^\alpha$ is continuous with respect to $\alpha$.

Note 3. The proof of Theorem 3.2 is pivotal as it establishes a necessary condition for $X_t^\alpha$ to serve as an inverse uncertainty distribution. This condition will be instrumental in the subsequent proofs of Theorems 3.6 and 3.10.

Theorem 3.3. Let $X_t^\alpha$ be the $\alpha$-path of the regular uncertain differential equation

If the conditions

are met and the linear growth, Lipschitz, and regular conditions hold, then $X_t^\alpha$ is a continuous and strictly increasing function with respect to $\alpha$ at each time $t > 0$.

Proof. Since $f$ and $g$ satisfy the linear growth and Lipschitz conditions, the $\alpha$-path $X_t^\alpha$ is continuous with respect to $t$. Let $\Phi^{-1}$ be the inverse standard normal uncertainty distribution, and let $\alpha$ and $\beta$ be numbers with $0 < \alpha < \beta < 1$. Write the second-order UDEs in the form of a system of equations:

Define $\mu$ and $\nu$ as:

We have

Since $g(0, X_0, Y_0) > 0$ (regular condition), we have

Due to the continuous nature of $f$ and $g$, and the continuity of $X_t^{\alpha}$ and $Y_t^{\alpha}$, we can infer that the composite functions $\mu$ and $\nu$ are also continuous with respect to $t$.

By the continuity of $\mu$ and $\nu$, there exists a small number $r > 0$ such that

Thus, by Lemma 3.1, we have:

for any time $T \in (0, r]$.

If for any $t > r$, $X_t^\alpha < X_t^\beta$, and the theorem holds. We will prove that by contradiction.

Suppose there exists a time $b > r$ at which $X_t^\alpha$ and $X_t^\beta$ first meet, i.e.,

The next phase of our proof will be to compare $Y_t^\alpha$ and $Y_t^\beta$ in a similar way. Write

Since $g(0, X_0, Y_0) > 0$, we have

By the continuity of $\overline{\mu}$ and $\overline{\nu}$, there exists a small number $\overline{r} > 0$ such that

Thus,

for any time $t \in (0, \overline{r}]$.

We will prove $Y_t^\alpha < Y_t^\beta$ for $\overline{r} < t < b$ by contradiction.

Suppose there exists a time $\overline{r} < \overline{b} < b$ at which $Y_t^\alpha$ and $Y_t^\beta$ first meet, i.e.,

Due to conditions (H), we have

Then

By the continuity of $\overline{\mu}$ and $\overline{\nu}$, there exists a time $\overline{a} \in (0, \overline{b})$ such that

Thus,

which is in contradiction with the assumption $Y_{\overline{b}}^\alpha = Y_{\overline{b}}^\beta.$ Therefore,

Integrate Eqs (3.3) and(3.4)

and we have

which is in contradiction with the assumption $X_{b}^\alpha = X_{b}^\beta.$ The theorem is proved. □

Note 4. Condition (H) is indispensable in the proof of the theorem. Should condition (H) not hold, it would be impossible to establish the relationship between $X_{\overline{b}}^\alpha$ and $X_{\overline{b}}^\beta$ when proving $\overline{\mu}(\overline{b}) < \overline{\nu}(\overline{b})$ and thus the proof would be untenable.

By Theorem 3.3, we have established that $X_t$ is a continuous and strictly increasing function with respect to $\alpha$. We wonder whether this collection of $\alpha$-paths can determine the inverse uncertainty distribution of the solutions to second-order UDEs. The following Theorem 3.4 provides a definitive answer.

Theorem 3.4. Let $X_t$ and $X_t^\alpha$ be the solution and $\alpha$-path of a regular uncertain differential equation

If the conditions (H) are met and the linear growth, Lipschitz, and regular conditions hold, then we have

Proof. Theorem 14.3 in ^[12] constructs an event $\Lambda_1$ with $\mathcal{M}\{\Lambda_1\} = \alpha,$ and shows that for each $\gamma \in \Lambda_1$, there exists a small number $\delta > 0$ such that

for any times $s$ and $t$ with $s > t$, where $\Phi^{-1}$ is the inverse standard normal uncertainty distribution. Since $f$ and $g$ satisfy the linear growth and Lipschitz conditions, $X_t^\alpha$ and $X_t(\gamma)$ are continuous with respect to $t$. Write the second-order UDE in the form of a system of equations:

Define $\lambda$, $\mu$, and $\nu$ as:

Due to the continuous nature of $f$ and $g$, and the continuity of $X_t(\gamma)$ and $Y_t(\gamma)$, we can infer that the composite functions $\lambda$, $\mu$, and $\nu$ are also continuous with respect to $t$.

Since $g(0, X_0, Y_0) > 0$ (regular condition), we have

By the continuity of $\mu$ and $\nu$, there exists a small number $r > 0$ such that

By inequality (3.5), for any time $t \in [0, r]$ and any time $s \in (t, \infty)$, we have

By Lemma 3.1, it follows that

for any time $T \in (0, r]$.

We will then prove $X_t(\gamma) < X_t^\alpha$ for all $t > r$ by contradiction.

Suppose there exists a time $b > r$ at which $X_t(\gamma)$ and $X_t^\alpha$ first meet, i.e.,

The next phase of our proof will be to compare $Y_t^\alpha$ and $Y_t(\gamma)$ in a similar way. Write

Since $g(0, X_0, Y_0) > 0$, we have

By the continuity of $\overline{\mu}$ and $\overline{\nu}$, there exists a small number $\overline{r} > 0$ such that

By inequality (3.5), for any time $t \in [0, \overline{r}]$ and any time $s \in (t, \infty)$, we have

Thus,

for any time $t \in (0, \overline{r}]$.

We will prove $Y_t(\gamma) < Y_t^\alpha$ for all $\overline{r} < t < b$ by contradiction.

Suppose there exists a time $\overline{r} < \overline{b} < b$ at which $Y_t(\gamma)$ and $Y_t^\alpha$ first meet, i.e.,

Due to conditions (H), we have

and then

By the continuity of $\overline{\mu}$ and $\overline{\nu}$, there exists a time $\overline{a} \in (0, \overline{b})$ such that

By inequality (3.5), for any time $t \in [\overline{a}, \overline{b}]$ and any time $s \in (t, \infty)$, we have

Thus,

which is in contradiction with $Y_{\overline{b}}(\gamma) = Y_{\overline{b}}^\alpha.$ Therefore,

Integrate Eq (3.6),

and we have

which is in contradiction with $X_{b}(\gamma) = X_{b}^\alpha.$ Therefore,

Since $\mathcal{M}\{\Lambda_1\} = \alpha,$ we have

Theorem 14.3 in ^[12] constructs an event $\Lambda_2$ with $\mathcal{M}\{\Lambda_2\} = 1-\alpha,$ and shows that for each $\gamma \in \Lambda_2$, there exists a small number $\delta > 0$ such that

for any times $s$ and $t$ with $s > t$. Write

Since $g(0, X_0, Y_0) > 0$ (regular condition), we have

By the continuity of $\mu$ and $\nu$, there exists a small number $r > 0$ such that

By inequality (3.8), for any time $t \in [0, r]$ and any time $s \in (t, \infty)$, we have

By Lemma 3.1, it follows that

for any time $T \in (0, r]$.

We will then prove $X_t(\gamma) < X_t^\alpha$ for all $t > r$ by contradiction.

Suppose there exists a time $b > r$ at which $X_t(\gamma)$ and $X_t^\alpha$ first meet, i.e.,

The next phase of our proof will be to compare $Y_t^\alpha$ and $Y_t(\gamma)$ in a similar way. Write

Since $g(0, X_0, Y_0) > 0$, we have

By the continuity of $\overline{\mu}$ and $\overline{\nu}$, there exists a small number $\overline{r} > 0$ such that

By inequality (3.8), for any time $t \in [0, \overline{r}]$ and any time $s \in (t, \infty)$, we have

Thus,

for any time $t \in (0, \overline{r}]$.

We will prove $Y_t(\gamma) > Y_t^\alpha$ for all $\overline{r} < t < b$ by contradiction.

Suppose there exists a time $\overline{r} < \overline{b} < b$ at which $Y_t(\gamma)$ and $Y_t^\alpha$ first meet, i.e.,

Due to conditions (H), we have

and then

By the continuity of $\overline{\mu}$ and $\overline{\nu}$, there exists a time $\overline{a} \in (0, \overline{b})$ such that

By inequality (3.8), for any time $t \in [\overline{a}, \overline{b}]$ and any time $s \in (t, \infty)$, we have

Thus,

which is in contradiction with $Y_{\overline{b}}(\gamma) = Y_{\overline{b}}^\alpha.$ Therefore,

Integrate Eq (3.6),

and we have

which is in contradiction with $X_b(\gamma) = X_b^\alpha$. Therefore,

Since $\mathcal{M}\{\Lambda_2\} = 1-\alpha,$ we have

It follows from (3.7), (3.9) and

that

The proof is now complete.□

Theorem 3.5. Let $X_t$ and $X_t^\alpha$ be the solution and $\alpha$-path of a regular uncertain differential equation

If the conditions (H) are met, the linear growth, Lipschitz and regular conditions hold, then at any time $t > 0$, we have

Proof. Note that $\{X_t\le X_t^\alpha\} \supset \{X_s\le X_s^\alpha, \forall s > 0\}$ holds for any time $t > 0$. By using the Theorems 3.3 and 3.4, we obtain

Similarly, we also have

Since $\{X_t\le X_t^\alpha\}$ and $\{X_t > X_t^\alpha\}$ are opposite events for each $t > 0$, the duality axiom makes

The theorem is proved. □

Theorem 3.6. Let $X_t$ and $X_t^\alpha$ be the solution and $\alpha$-path of the uncertain differential equation

respectively. If the conditions (H) are met, the linear growth, Lipschitz and regular conditions hold, then $X_t$ has an inverse uncertainty distribution

Proof. Theorem 3.2 says that $X_t^\alpha$ is a continuous function with respect to $\alpha$ at each time $t > 0$. Theorem 3.4 says that

for any $\alpha \in (0, 1)$. It follows from Theorem 2.8 that the inverse uncertainty distribution of $X_t$ is $\Psi^{-1}_t(\alpha) = X_t^\alpha.$ □

Determining the inverse uncertainty distribution of solutions to second-order UDEs is more complex than that for first-order UDEs. In our analysis, we employ Condition H, which is fundamentally derived from the comparison theorem. Readers can refer to publications on the comparison theorem for higher-order differential equations ^[19], partial order relations, and quasi-monotone functions ^[20].

We now turn to the discussion of higher-order UDEs. In Definition 2.7, we defined higher-order UDEs and presented them in the form of a system of Eq (3.10). The first component of the solution to system (3.10) is the solution to the equation as defined in Definition 2.7.

However, since $f$ is an $(n+1)$-ary function, applying the condition of inverse monotonicity becomes more complex and less straightforward to prove. Yet, if our UDE is formulated as in Eq (3.11), we will proceed to provide a proof for the theorem below.

Theorem 3.7. Let $X_t^\alpha$ be the $\alpha$-path of a regular uncertain differential equation

If the linear growth, Lipschitz and regular conditions hold, then $X_t^\alpha$ is a continuous and strictly increasing function with respect to $\alpha$ at each time $t > 0$.

Proof. Since $f$ and $g$ satisfy the linear growth and Lipschitz conditions, the $\alpha$-path $X_t^\alpha$ is continuous with respect to $t$. Let $\Phi^{-1}$ be the inverse standard normal uncertainty distribution, and let $\alpha$ and $\beta$ be numbers with $0 < \alpha < \beta < 1$. Write

Due to the continuous nature of $f$ and $g$, and the continuity of $X_t^{\alpha}$ and $Y_t^{\alpha}$, we can infer that the composite functions $\mu$, $\nu$ and $\mu_1$, $\nu_1$ are also continuous with respect to $t$.

Since $g(0, X_0) > 0$ (regular condition), we have

By the continuity of $\mu$ and $\nu$, there exists a small number $r > 0$ such that

By Lemma 3.1, it follows that

for any time $T \in (0, r]$.

We will prove $X_t^\alpha < X_t^\beta$ for any $t > r$ by contradiction.

Suppose there exists a time $b > r$ at which $X_t^\alpha$ and $X_t^\beta$ first meet, i.e.,

Since $f(b, X_b^\beta) = f(b, X_b^\alpha), g(b, X_b^\beta) = g(b, X_b^\alpha) > 0$, and we have

By the continuity of $\mu_1$ and $\nu_1$, there exists a time $a \in (0, b)$ such that

Then, we have

Choose $a_1$ and $b_1$ such that $a < a_1 < b_1 < b$. Thus, we have

Note that any continuous function defined on a closed interval can always reach its minimum. Without loss of generality, suppose that $\nu_1(t)$-$\mu_1(t)$ reaches its minimum on $[a_1, b_1]$ at $t^\ast \in [a_1, b_1]$. Then, we have the following inequality:

Thus,

which is in contradiction with $X_b(\gamma) = X_b^\alpha$. The theorem is proved.□

By Theorem 3.7, we have established that $X_t$ is a continuous and strictly increasing function with respect to $\alpha$. We wonder whether this collection of $\alpha$-paths can determine the inverse uncertainty distribution of the solutions to higher-order UDEs. The following Theorem 3.8 provides a definitive answer.

Theorem 3.8. Let $X_t$ and $X_t^\alpha$ be the solution and $\alpha$-path of the regular uncertain differential equation

If the linear growth, Lipschitz and regular conditions hold, then

Proof. Theorem 14.3 in ^[12] constructs an event $\Lambda_1$ with $\mathcal{M}\{\Lambda_1\} = \alpha,$ and shows that for each $\gamma \in \Lambda_1$, there exists a small number $\delta > 0$ such that

for any times $s$ and $t$ with $s > t$, where $\Phi^{-1}$ is the inverse standard normal uncertainty distribution. Since $f$ and $g$ satisfy the linear growth and Lipschitz conditions, it follows from Theorem 3.1 that $X_t^\alpha$ and $X_t(\gamma)$ exist and are continuous with respect to $t$. Write the higher-order UDE in the form of a system of equations:

Due to the continuous nature of $f$ and $g$, and the continuity of $X_t(\gamma)$, we can infer that the composite functions $\lambda$, $\mu$, $\nu$ and $\lambda_1$, $\mu_1$, $\nu_1$ are also continuous with respect to $t$.

Since $g(0, X_0) > 0$ (regular condition), we have

By the continuity of $\mu$ and $\nu$, there exists a small number $r > 0$ such that

By inequality (3.14), for any time $t \in [0, r]$ and any time $s \in (t, \infty)$, we have

By Lemma 3.1, it follows that

for any time $T \in (0, r]$.

We will prove $X_t(\gamma) < X_t^\alpha$ for all $t > r$ by contradiction.

Suppose there exists a time $b > r$ at which $X_t(\gamma)$ and $X_t^\alpha$ first meet, i.e.,

Since $f(b, X_b^\alpha) = f(b, X_b(\gamma)), g(b, X_b^\alpha) = g(b, X_b(\gamma)) > 0$, and we have

By the continuity of $\mu_1$ and $\nu_1$, there exists a time $a \in (0, b)$ such that

By inequality (3.14), for any time $t \in [a, b)$ and any time $s \in (t, \infty)$, we have

By a similar process as in Theorem 3.7, which indicates that the integral of a non-negative function with a positive interval is also positive, we have the following inequality:

which is in contradiction with $X_b(\gamma) = X_b^\alpha$. Therefore,

Since $\mathcal{M}\{\Lambda_1\} = \alpha,$ we have

Theorem 14.3 in ^[1] constructs an event $\Lambda_2$ with $\mathcal{M}\{\Lambda_2\} = 1-\alpha,$ and shows that for each $\gamma \in \Lambda_2$, there exists a small number $\delta > 0$ such that

for any times $s$ and $t$ with $s > t$. Write

Since $g(0, X_0) > 0$, we have

By the continuity of $\mu$ and $\nu$, there exists a small number $r > 0$ such that

By inequality (3.17), for any time $t \in [0, r]$ and any time $s \in (t, \infty)$, we have

Thus,

for any time $T \in (0, r].$

We will prove $X_t(\gamma) > X_t^\alpha$ for all $t > r$ by contradiction.

Suppose there exists a time $b > r$ at which $X_t(\gamma)$ and $X_t^\alpha$ first meet, i.e.,

Since $f(b, X_b^\alpha) = f(b, X_b(\gamma)), g(b, X_b^\alpha) = g(b, X_b(\gamma)) > 0$, we have

By the continuity of $\mu_1$ and $\nu_1$, there exists a time $a \in (0, b)$ such that

By inequality (3.17), for any time $t \in [a, b]$ and any time $s \in (t, \infty)$, we have

Thus,

which is in contradiction with $X_b(\gamma) = X_b^\alpha$. Therefore,

Since $\mathcal{M}\{\Lambda_2\} = 1-\alpha,$ we have

It follows from (3.16), (3.18) and

that

hold. □

Theorem 3.9. Let $X_t$ and $X_t^\alpha$ be the solution and $\alpha$-path of the regular uncertain differential equation

If the linear growth, Lipschitz and regular conditions hold, then at any time $t > 0$, we have

Proof. The proof follows a similar process to Theorem 3.5.□

Theorem 3.10. Let $X_t$ and $X_t^\alpha$ be the solution and $\alpha$-path of the uncertain differential equation

respectively. If the linear growth, Lipschitz and regular conditions hold, then $X_t$ has an inverse uncertainty distribution

Proof. The proof follows a similar process to Theorem 3.6.□

4.   Numerical examples

Moving forward, we will provide examples to illustrate the significance of Condition (H).

Example 2. For the uncertain Duffing system, the equation is given by

The initial values are $X_{t_0}(\gamma) = X_{t_0}^{\alpha} = 0.1.$ For convenience, the parameters involved are illustrated in Table 1.

The uncertain external excitation represents the influence of climate on the object within the system. Figure 4 is a schematic diagram of the Duffing system, where the spring is a hardening spring.

We have

It is straightforward to deduce that

Upon observing Figure 5, it can be seen that the $\alpha$-path for $\alpha = 0.1$ and $\alpha = 0.2$ intersect at time $t_1$. The set of solutions $X_t^{\alpha}$ derived from the family of ordinary differential equations cannot substitute for the inverse uncertainty distribution of the solution to the UDE.

For $\alpha < 0$, the spring is a softening spring, where $\beta < 0$, the uncertain Duffing system is for

The initial values are $X_t(\gamma)\Big| _{t = 0} = X_t^{\alpha}\Big| _{t = 0} = 0.1$, $\frac{{\rm d}X_t(\gamma)}{{\rm d}t}\Big| _{t = 0} = \frac{{\rm d}X_t^\alpha}{{\rm d}t}\Big| _{t = 0} = 0.1$. And we have

Verify condition (H),

We define the region $G\in [0,100]\times[0, 10]\times[0, 10]$. It is straightforward to determine that the maximum value of the function $f$ is denoted as $M = 1100$, which satisfies the conditions provided in the remarks. Furthermore, as depicted in Figure 6, there are no intersecting $\alpha$-path. Hence, the set of solutions $X_t^{\alpha}$ derived from the family of ordinary differential equations corresponding to the $\alpha$-path can be used to represent the inverse uncertainty distribution of the UDE.

Example 3. The function $g$ represents a commonly utilized activation function within neural networks. Its second-order uncertain differential equation is given by

The initial values are $X_t(\gamma)\Big| _{t = 0} = X_t^{\alpha}\Big| _{t = 0} = 0.1$, $\frac{{\rm d}X_t(\gamma)}{{\rm d}t}\Big| _{t = 0} = \frac{{\rm d}X_t^\alpha}{{\rm d}t}\Big| _{t = 0} = 0.1$, where

We define the region $G\in [0,100]\times[0, 10]\times[0, 10]$. It is easy to establish that the maximum value of the functions $f$ and $g$ is $M = 1100$, which meets the conditions specified in the notes. Additionally, as shown in Figure 7, there are no intersections among the $\alpha$-paths, whether in the short term (a) or the long term (b). So the set of solutions $X_t^{\alpha}$ derived from the $\alpha$-path for a family of ordinary differential equations can be used to represent the inverse uncertainty distribution of the solution to the UDE.

Example 4.

The initial values are

We define the region $G\in [0,100]\times[0, 10]$. It is easy to establish that the maximum value of the function $f$ is $M = 11$, which meets the conditions specified in the notes. The $\alpha$-path diagram is depicted in Figure 8, from which it is observable that the individual $\alpha$-path do not intersect at any point in time. Thus, the set of solutions $X_t^{\alpha}$ derived from the $\alpha$-path for a family of ordinary differential equations can be used to represent the inverse uncertainty distribution of the UDE.

5.   Conclusions

The research presented in this paper is of significant importance and has broad applicability in practical applications. If the correct inverse uncertainty distribution of the solutions to UDEs cannot be obtained, it would be impossible to delve into the study of the behavior of UDEs and their solutions. Consequently, the construction of such UDE models would be meaningless.

This paper presents for the first time the inverse uncertainty distribution for second-order and a class of higher-order UDEs. (1) For second-order UDEs, if the conditions (H) are met and the linear growth, Lipschitz and regular conditions hold, we can determine the inverse uncertainty distribution of the second-order UDEs. (2) For a class of higher-order UDEs, if the linear growth, Lipschitz and regular conditions hold, we can similarly ascertain the inverse uncertainty distribution of the second-order UDEs. Based on the theorems presented in this paper, it is possible to conduct uncertain dynamical analyses of these types of UDEs, which constitutes a foundational work in the field of uncertainty theory. It is undeniable that these conditions have certain limitations. Scholars can continue to explore the statistical properties and other applications of the solutions to UDEs to potentially relax these constraints.

Based on our research, there are three areas for potential future exploration: (1) It merits investigation whether the sufficient conditions for determining the inverse unncertainty distribution of solutions to second-order or higher-order UDEs can be expanded and whether there exist necessary and sufficient conditions. (2) While uncertainty theory and this paper discuss the transition from ordinary differential equations to UDEs, our research group is currently working on more complex equations, such as uncertain fractional-order differential equations and uncertain functional differential equations with constant delays. We have conceived a potential sufficient condition for the $\alpha$-paths of systems of UDEs to determine the inverse uncertainty distribution of their solutions. Additionally, uncertain partial differential equations also warrant investigation. (3) There is a multitude of differential equations that satisfy our conditions, with numerous examples provided in the final version under submission. On this basis, more in-depth issues can be explored, such as uncertain dynamics, uncertain bifurcation, and uncertain chaos.

Author contributions

Zeman Wang: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Writing–original draft, Writing–review & editing; Qiubao Wang: Conceptualization, Funding acquisition, Project administration, Supervision, Validation, Writing–review & editing; Zhong Liu: Data curation, Formal analysis; Zikun Han: Data curation, Investigation; Xiuying Guo: Investigation, Supervision. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

This work was supported by the Innovation and Entrepreneurship Curriculum Construction in Hebei Province: Ordinary Differential Equations; Key Project of Natural Science Foundation of Hebei Province (Basic Discipline Research) (A2023210064).

Conflict of interest

All authors declare no conflict of interest that could affect the publication of this paper.