Methods of uncertain partial differential equation with application to internet public opinion problem

Abstract

Uncertain partial differential equation is a type of partial differential equation driven by Liu processes. This paper first provides two methods for solving the first order linear uncertain partial differential equation and the inverse uncertainty distribution of solution is discussed. Moreover, on the basis of the study of uncertainty theory and the propagation process of internet public opinion, three types of internet public opinion (IPO) problem are proposed based on uncertain partial differential equation. Finally, some numerical examples are given.

Keywords

Uncertainty theory Uncertain process Partial differential equation Internet public opinion

1 Introduction

Uncertainty theory is a branch of mathematics for modeling belief degrees established by Liu [1], it develops following normality, duality, subadditivity and product axioms. In addition, Liu [2] investigated Liu process-a type of stationary independent increment process. After that, uncertain differential equation as a type of differential equation involving uncertain processes was first built by Liu [3]. Chen and Liu [4] provided the solution of linear uncertain differential equation under the condition of Lipschitz continuous and linear growth. Nowadays, uncertain differential equation has been widely used in many fields such as uncertain finance [5], string vibration [6], and optimal control [7]. What is more, since partial differential equation has been studied widely as an important role in mathematics, Buckley [25] proposed fuzzy partial differential equations, then scholars have investigated and developed it [26 –29]. Recently, Yang and Yao [8] introduced uncertain partial differential equation. This work relates partial differential equation and uncertainty theory, and it points out a new direction to partial differential equation.

The internet public opinion (IPO) problem as a hot issue was pioneered by Sudbury [9] in 1985. Following that, many scholars built models in different complex networks [10 –15]. Meanwhile, Fang [16] investigated the IPO problem by cellular automata, Wang [17] and Han [18] built models based on game theory. All of the above methods are based on differential equation and probability theory with setting parameters to study. However, internet public opinion often sudden outbreaks with no historical data, using probability theory to deal with those uncertain factors is not completely justified. For this reason, Su [19] first built the IPO models based on uncertainty theory which regarded the IPO problem as an uncertain process over time. But, it is clear that the flexibility of internet public opinion is different at different times, this paper will investigates the IPO problem under two parameters, time and flexibility, and builds three types of IPO models to explore the process of IPO problem. Besides, in order to deal with the proposed models, a solution of first order linear uncertain partial differential equation will be investigated first and the inverse uncertainty distribution of solution is discussed.

The rest of paper is organized as follows. In Section 2, some basic concepts and properties of uncertainty theory are introduced. Section 3 investigates two methods for solving the first order linear uncertain partial differential equation. Section 4 obtains the inverse uncertainty distribution of solution for uncertain partial differential equation. In Section 5, three types of IPO models are proposed. Section 6 gives some numerical examples. Section 7 contains a brief summary to this paper.

2 Preliminary

This section will introduce some basic concepts and properties in uncertainty theory which will be used in this paper.

Definition 1. (Liu [1]) Let Γ be a nonempty set, and ℒ a σ-algebra over Γ; the set function ℳ is called an uncertain measure if it satisfies the following axioms:

(Normality Axiom) ℳ {Γ} =1 for the universal set Γ;

(Duality Axiom) ℳ {Λ} + ℳ {Λ^c} =1 for any event Λ;

(Subadditivity Axiom) For every countable sequence of events Λ₁, Λ₂, ⋯, we have $ℳ {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} ℳ {Λ_{i}} .$

The triplet (Γ, ℒ, ℳ) is called an uncertainty space.

Liu [2] defined the uncertain measure on the product σ-algebra:

(Product Axiom) Let (Γ_k, ℒ_k, ℳ_k) be uncertainty spaces for k = 1, 2, ⋯. The product uncertain measure ℳ is an uncertain measure satisfying $ℳ {\prod_{i = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} ℳ_{k} {Λ_{k}},$ where Λ_k are arbitrarily chosen events from ℒ_k for k = 1, 2, ⋯, respectively.

The concept of uncertain variable is:

Definition 2. (Liu [1]) Suppose (Γ, ℒ, ℳ) is an uncertainty space. An uncertain variable is a measurable function ξ from (Γ, ℒ, ℳ) to the set of real numbers such that {ξ ∈ B} is an event for any Borel set B of real numbers.

Definition 3. (Liu [3]) Let (Γ, ℒ, ℳ) be an uncertainty space and let T be a totally ordered set. An uncertain process is a function X_t (γ) from T × (Γ, ℒ, ℳ) to the set of real numbers such that {X_t ∈ B} is an event for any Borel set B of real numbers at each time t.

Theorem 1. (Liu [23]) Let X_1t, X_2t, ⋯, X_nt be independent uncertain processes with regular uncertainty distributions Φ_1t, Φ_2t, ⋯, Φ_nt, respectively. If the function f (x₁, x₂, ⋯, x_n) is strictly increasing with respect to x₁, x₂, ⋯, x_m and strictly decreasing with respect to x_m+1, x_m+2, ⋯, x_n, then $X_{t} = f (X_{1 t}, X_{2 t}, \dots, X_{nt})$ has an inverse uncertainty distribution $\begin{matrix} Φ_{t}^{- 1} (α) & = & f (Φ_{1 t}^{- 1} (α), \dots, Φ_{mt}^{- 1} (α), \\ Φ_{m + 1, t}^{- 1} (1 - α), \dots, Φ_{nt}^{- 1} (1 - α)) \end{matrix}$

Definition 4. (Liu [2]) An uncertain process C_t is said to be a Liu process if

C₀ = 0 and almost all sample paths are Lipschitz continuous;

C_t has stationary and independent increments;

every increment C_s+t - C_s is a normal uncertain variable with expected value 0 and variance t².

It is clear that C_t is a type of stationary independent increment process which has a normal uncertainty distribution $Φ_{t} (x) = {(1 + \exp (- \frac{π x}{\sqrt{3} t}))}^{- 1}$ and a inverse uncertainty distribution $Φ_{t}^{- 1} (x) = \frac{\sqrt{3} t}{π} ln \frac{α}{1 - α} .$

Theorem 2. (Liu [20]) Let C_t be a Liu process. Then for each t, we have $\frac{t^{2}}{2} \leq E [C_{t}^{2}] \leq t^{2} .$

Theorem 3. (Iwamura and Kageyama [21]) Let C_t be a Liu process. Then for each t, we have $1.24 t^{4} < V [C_{t}^{2}] < 4.31 t^{4} .$

Definition 5. (Liu [2]) Let X_t be an uncertain process and let C_t be a Liu process. For any partition of closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_k+1 = b, the mesh is written as $Δ = max_{1 \leq i \leq k} | t_{i + 1} - t_{i} | .$

Then Liu integral of X_t with respect to C_t is defined as $\int_{a}^{b} X_{t} {dC}_{t} = lim_{Δ \to 0} \sum_{i = 1}^{k} X_{t} \cdot (C_{t_{i + 1}} - C_{t_{i}})$ provided that the limit exists almost surely and is finite. In this case, the uncertain process X_t is said to be integrable.

Definition 6. (Chen and Ralescu [22]) Let C_t be a Liu process and let Z_t be an uncertain process. If there exist uncertain processes μ_t and σ_t such that $Z_{t} = Z_{0} + \int_{0}^{t} μ_{s} d s + \int_{0}^{t} σ_{s} {dC}_{s}$ for any t ≥ 0, then Z_t is called a general Liu process with drift μ_t and diffusion σ_t. Furthermore, Z_t has an uncertain differential ${dZ}_{t} = μ_{t} d t + σ_{t} {dC}_{t} .$

Theorem 4. (Liu [2]) Let h (t, c) be a continuously differentiable function. Then Z_t = h (t, C_t) is a general Liu process and has an uncertain differential ${dZ}_{t} = \frac{\partial h}{\partial t} (t, C_{t}) d t + \frac{\partial h}{\partial c} (t, C_{t}) {dC}_{t} .$

Definition 7. (Liu [23]) Let (Γ, ℒ, ℳ) be an uncertainty space and let T be a partially ordered set. An uncertain field is a function X_t (γ) from T × (Γ, ℒ, ℳ) to the set of real numbers such that {X_t (γ) ∈ B} is an event for any Borel set B of real numbers at each t.

Definition 8. (Gao and Chen [24]) An uncertain field X_t is said to have an uncertainty distribution Φ_t (x) if for each t, the uncertain variable X_t has an uncertainty distribution Φ_t (x).

Definition 9. (Gao and Chen [24]) An uncertainty distribution Φ_t (x) is said to be regular if for any t, it is a continuous and strictly increasing function with respect to x such that 0 < Φ_t (x) <1, and $lim_{x \to - \infty} Φ_{t} (x) = 0, lim_{x \to + \infty} Φ_{t} (x) = 1 .$

Definition 10. (Yang and Yao [8]) Let C_t is a Liu process, and F is a function. Then

$\begin{matrix} F (t, x_{1}, \dots, x_{n}, u, \frac{\partial u}{\partial t}, \frac{\partial u}{\partial x_{1}}, \dots, \frac{\partial u}{\partial x_{n}}, \\ \frac{\partial^{2} u}{\partial x_{1}^{2}}, \dots, \frac{\partial^{2} u}{\partial x_{1} \partial x_{n}}, \dots, \dot{C_{t}}) = 0 \end{matrix}$ (1) is called an uncertain partial differential equation, where

$\dot{C_{t}} : = \frac{{dC}_{t}}{dt} .$

The solution is an uncertain field u (t, x₁, x₂, ⋯, x_n) that satisfies the Equation (1) identically. The order of uncertain partial differential equation is the order of the highest derivatives of Equation (1).

3 Two methods of uncertain partial differential equation

In this section, we will provide two methods for solving the first order linear uncertain partial differential equation

${\begin{matrix} a (t, x) \frac{\partial U}{\partial x} + b (t, x) \frac{\partial U}{\partial t} + c (t, x) U = f (t, x, \dot{C_{t}}) \\ U (0, x) = ω (x) . \end{matrix}$ (2)

Analytic method:

Theorem 5. Let a (t, x), b (t, x) and c (t, x) be functions of two variables with a (t, x) ≡0, b (t, x) ≠0, $f (t, x, \dot{C_{t}})$ is a function with $\dot{C_{t}} : = \frac{{dC}_{t}}{dt}$ , C_t is a Liu process. Then the uncertain partial differential Equation (2) which is equivalent to

$b (t, x) \frac{\partial U}{\partial t} + c (t, x) U = f (t, x, \dot{C_{t}})$ (3) has a solution

$\begin{matrix} U (t, x) \\ = \exp (- \int_{0}^{t} \frac{c (y, x)}{b (y, x)} d y) \\ \cdot [\int \exp (\int_{0}^{t} \frac{c (y, x)}{b (y, x)} d y) \frac{f (t, x, \dot{C_{t}})}{b (t, x)} d t + g (x)], \end{matrix}$ (4) where g (x) is an arbitrary function.

Proof. Since b (t, x) ≠0, the uncertain partial differential Equation (3) is that $\frac{\partial U}{\partial t} + \frac{c (t, x)}{b (t, x)} U = \frac{f (t, x, \dot{C_{t}})}{b (t, x)} .$ (5)

At first, both sides of the Equation (5) are multiplied by a factor $\exp (\int_{0}^{t} \frac{c (y, x)}{b (y, x)} d y)$ , we have

$\begin{matrix} \frac{\partial}{\partial t} (\exp (\int_{0}^{t} \frac{c (y, x)}{b (y, x)} d y) U) \\ = \exp (\int_{0}^{t} \frac{c (y, x)}{b (y, x)} d y) \frac{f (t, x, \dot{C_{t}})}{b (t, x)} . \end{matrix}$ (6)

Integrate t, then the partial differential equation (6) has a solution $\begin{matrix} U (t, x) \\ = \exp (- \int_{0}^{t} \frac{c (y, x)}{b (y, x)} d y) \\ \cdot [\int \exp (\int_{0}^{t} \frac{c (y, x)}{b (y, x)} d y) \frac{f (t, x, \dot{C_{t}})}{b (t, x)} d t + g (x)], \end{matrix}$ where g (x) is an arbitrary function.

Thus, the theorem is verified. □

Example 1. Consider the uncertain partial differential equation $\frac{\partial U}{\partial t} = x + t + \dot{C_{t}} .$

At first, we have b = 1, c = 0, and $f (t, x, \dot{C_{t}}) = x + t + \dot{C_{t}}$ .

It follows from Theorem 1 that the solution is $U (t, x) = \frac{1}{2} x^{2} + xt + C_{t} + g (x),$ where g (x) is an arbitrary function.

Numerical method:

Definition 11. Suppose t = φ (m, n) and x = ψ (m, n) are uncertain variables from (Γ₁, ℒ₁, ℳ₁) to the set of real numbers, and $U = f (t, x, \dot{C_{t}})$ is an uncertain field from T × (Γ₂, ℒ₂, ℳ₂) to the set of real numbers. Then the uncertain field $U = F (m, n, c) = f (φ (m, n), ψ (m, n), \dot{C_{φ}})$ (7) is composite, if

$\begin{matrix} {(t, x) ∣ t & = & φ (m, n), \\ x & = & ψ (m, n), (m, n) \in ℒ_{1}} \subset ℒ_{2} \end{matrix}$ (8) for each t. Where t, x is called intermediate uncertain variables of F, and m, n is called independent uncertain variables.

Theorem 6. (Chain Rule) Let t = φ (m, n) and x = ψ (m, n) be continuously differentiable functions, $U = f (t, x, \dot{C_{t}})$ is a general Liu process. Then $U = f (φ (m, n), ψ (m, n), \dot{C_{φ}})$ (9) is a general Liu process and has the following uncertain partial differential equations $\frac{\partial U}{\partial m} = \frac{\partial U}{\partial t} \frac{\partial φ}{\partial m} + \frac{\partial U}{\partial x} \frac{\partial ψ}{\partial m},$ (10) $\frac{\partial U}{\partial n} = \frac{\partial U}{\partial t} \frac{\partial φ}{\partial n} + \frac{\partial U}{\partial x} \frac{\partial ψ}{\partial n} .$ (11)

Proof. Write ΔC_t = C_t+Δt - C_t = C_Δt. Δt, Δx and ΔC_t are infinitesimals with the same order from Theorems 2 and 3.

It is clear that $Δ t = \frac{\partial φ}{\partial m} Δ m + \frac{\partial φ}{\partial n} Δ n + α_{1} Δ m + β_{1} Δ n,$ (12) $Δ x = \frac{\partial ψ}{\partial m} Δ m + \frac{\partial ψ}{\partial n} Δ n + α_{2} Δ m + β_{2} Δ n,$ (13) where α₁ → 0, α₂ → 0, β₁ → 0 and β₂ → 0, as Δm → 0 and Δn → 0.

It follows from Theorem 4 that $Δ U = \frac{\partial U}{\partial t} Δ t + \frac{\partial U}{\partial x} Δ x + \frac{\partial U}{\partial c} Δ C_{t} .$ (14)

Take Equations (12) (13) to Equation (14), we have $\begin{matrix} Δ U & = & (\frac{\partial U}{\partial t} \frac{\partial φ}{\partial m} + \frac{\partial U}{\partial x} \frac{\partial ψ}{\partial m}) Δ m \\ + (\frac{\partial U}{\partial t} \frac{\partial φ}{\partial n} + \frac{\partial U}{\partial x} \frac{\partial ψ}{\partial n}) Δ n \\ + \frac{\partial U}{\partial c} Δ C_{t} + (α_{1} + α_{2}) Δ m + (β_{1} + β_{2}) Δ n . \end{matrix}$

Since φ (m, n) and ψ (m, n) are continuously differentiable functions, we have Δt → 0, Δx → 0 and ΔC_t → 0 as Δm → 0 and Δn → 0. Then, the theorem is verified.□

Based on the Theorems 5 and 6, we can get a numerical method for the first order linear uncertain partial differential equation. Let a (t, x), b (t, x) and c (t, x) be functions of two variables with a (t, x) ≠0, b (t, x) ≠0, $f (t, x, \dot{C_{t}})$ is a function with $\dot{C_{t}} : = \frac{{dC}_{t}}{dt}$ , C_t is a Liu process. Then for the uncertain partial differential equation ${\begin{matrix} a (t, x) \frac{\partial U}{\partial x} + b (t, x) \frac{\partial U}{\partial t} + c (t, x) U = f (t, x, \dot{C_{t}}) \\ U (0, x) = ω (x), \end{matrix}$ (15) the method can be summarized as follows:

Step 1: Variable substitution ${\begin{matrix} ξ = φ (t, x) \\ η = ψ (t, x), \end{matrix}$ (16) where $J (φ, ψ) = \frac{\partial (φ, ψ)}{\partial (t, x)} = | \begin{matrix} \frac{\partial φ}{\partial t} & \frac{\partial φ}{\partial x} \\ \frac{\partial ψ}{\partial t} & \frac{\partial ψ}{\partial x} \end{matrix} | \neq 0 .$ (17)

Step 2: Turn the Equation (15) to the new equation

$\begin{matrix} (a \frac{\partial φ}{\partial t} + b \frac{\partial φ}{\partial x}) \frac{\partial U}{\partial ξ} \\ + (a \frac{\partial ψ}{\partial t} + b \frac{\partial ψ}{\partial x}) \frac{\partial U}{\partial η} + cU = f \end{matrix}$ (18) by using Theorem 6;

Step 3: Turn the Equation (18) to the new equation: $(a \frac{\partial ψ}{\partial t} + b \frac{\partial ψ}{\partial x}) \frac{\partial U}{\partial η} + cU = f$ (19) where ξ = φ (t, x) is the solution of first order homogeneous linear uncertain partial differential equation $a (t, x) \frac{\partial φ}{\partial t} + b (t, x) \frac{\partial φ}{\partial x} = 0;$ (20)

Step 4: By using Theorem 5, we get the solution U (ξ, η) of Equation (20);

Step 5: The solution U (t, x) is obtained by the initial condition U (0, x) = ω (x).

Example 2. Consider the uncertain partial differential equation ${\begin{matrix} \frac{\partial U}{\partial t} + \frac{\partial U}{\partial x} = x + t + \dot{C_{t}} \\ U (0, x) = 0 . \end{matrix}$ (21)

At first, we have a = 1, b = 1, c = 0, and $f (t, x, \dot{C_{t}}) = x + t + \dot{C_{t}}$ .

Variable substitution ${\begin{matrix} ξ = φ (t, x) = x - t \\ η = ψ (t, x) = t, \end{matrix}$

where

\begin{matrix} J (φ, ψ) & = & \frac{\partial (φ, ψ)}{\partial (t, x)} = | \begin{matrix} \frac{\partial φ}{\partial t} & \frac{\partial φ}{\partial x} \\ \frac{\partial ψ}{\partial t} & \frac{\partial ψ}{\partial x} \end{matrix} | \\ = & | \begin{matrix} - 1 & 1 \\ 1 & 0 \end{matrix} | = - 1 \neq 0 . \end{matrix}

Turn the Equation (21) to the new equation:

$\frac{\partial U}{\partial η} = ξ + 2 η + \dot{C_{η}} .$

By using Theorem 5, we have

$U (ξ, η) = ξ η + η^{2} + C_{η} + g (ξ) .$

That is,

$U (t, x) = xt + C_{t} + g (x - t) .$

Since U (0, x) =0, we have

$U (0, x) = g (x) = 0,$

Hence

$U (t, x) = xt + C_{t} .$

Example 3. Consider the uncertain partial differential equation

${\begin{matrix} \frac{\partial U}{\partial t} - 5 \frac{\partial U}{\partial x} + U = \dot{C_{t}} \\ U (0, x) = ω (x) . \end{matrix}$ (22)

At first, we have a = 1, b = -5, c = 1, and $f (t, x, \dot{C_{t}}) = \dot{C_{t}}$ .

Variable substitution

${\begin{matrix} ξ = φ (t, x) = x + 5 t \\ η = ψ (t, x) = t, \end{matrix}$ where

$\begin{matrix} J (φ, ψ) & = & \frac{\partial (φ, ψ)}{\partial (t, x)} = | \begin{matrix} \frac{\partial φ}{\partial t} & \frac{\partial φ}{\partial x} \\ \frac{\partial ψ}{\partial t} & \frac{\partial ψ}{\partial x} \end{matrix} | \\ = & | \begin{matrix} 5 & 1 \\ 1 & 0 \end{matrix} | = - 1 \neq 0 . \end{matrix}$

Turn the Equation (22) to the new equation:

$\frac{\partial U}{\partial η} = - U + \dot{C_{η}} .$

By using Theorem 5, we have $\begin{matrix} U (ξ, η) & = & \exp (- η) [\int \exp (η) d η + C_{η} + g (ξ)] \\ = & g (ξ) \exp (- η) + C_{η} . \end{matrix}$

That is, $U (t, x) = g (5 t + x) \exp (- t) + C_{t} .$

Since U (0, x) = ω (x), we have $U (0, x) = g (x) = ω (x) .$

Hence $U (t, x) = ω (x) \exp (- t) + C_{t} .$

4 Inverse uncertainty distribution of solution for uncertain partial differential equation

In this section, we will give the inverse uncertainty distribution of solution for the first order linear uncertain partial differential equation.

Theorem 7. Let α be a number with 0 < α < 1, U (t, x) = F (t, x, C_t) is the solution of the first order linear uncertain partial differential equations ${\begin{matrix} a (t, x) \frac{\partial U}{\partial x} + b (t, x) \frac{\partial U}{\partial t} + c (t, x) U = f (t, x, \dot{C_{t}}) \\ U (0, x) = ω (x) . \end{matrix}$ (23)

If F (t, x, c) is a strictly increasing function with respect to c, then the uncertain field U (t, x) has an inverse uncertainty distribution $Ψ_{t, x}^{- 1} (α) = F (t, x, Φ^{- 1} (α)),$ (24)where $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}$ (25)is the inverse uncertainty distribution of standard normal uncertain variable.

Proof. Since C_t is a normal uncertain variable $N (0, t)$ , it follows from Theorem 1 that F (t, x, C_t) has an inverse uncertainty distribution $Ψ_{t, x}^{- 1} (α) = F (t, x, Φ^{- 1} (α)),$ where $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}$ is the inverse uncertainty distribution of standard normal uncertain variable. □

Theorem 8. Let α be a number with 0 < α < 1, U (t, x) = F (t, x, C_t) is the solution of the first order linear uncertain partial differential equations ${\begin{matrix} a (t, x) \frac{\partial U}{\partial x} + b (t, x) \frac{\partial U}{\partial t} + c (t, x) U = f (t, x, \dot{C_{t}}) \\ U (0, x) = ω (x) . \end{matrix}$ (26)

If F (t, x, c) is a strictly decreasing function with respect to c, then the uncertain field U (t, x) has an inverse uncertainty distribution $Ψ_{t, x}^{- 1} (α) = F (t, x, Φ^{- 1} (1 - α))$ (27)where $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}$ (28)is the inverse uncertainty distribution of standard normal uncertain variable.

Proof. Since C_t is a normal uncertain variable $N (0, t)$ , it follows from Theorem 1 that F (t, x, C_t) has an inverse uncertainty distribution $Ψ_{t, x}^{- 1} (α) = F (t, x, Φ^{- 1} (1 - α)),$ where $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}$ is the inverse uncertainty distribution of standard normal uncertain variable. □

Example 4. Consider the uncertain partial differential equation ${\begin{matrix} \frac{\partial U}{\partial t} + \frac{\partial U}{\partial x} = x + t + \dot{C_{t}} \\ U (0, x) = 0 . \end{matrix}$

At first, it has a solution $U (t, x) = xt + C_{t} .$

It follows from Theorem 7 that $Ψ_{t, x}^{- 1} (α) = x + t + \frac{\sqrt{3} t}{π} ln \frac{α}{1 - α},$ which is shown in Fig. 1.

Fig.1

Inverse uncertainty distribution in Example 4.

Example 5. Consider the uncertain partial differential equation ${\begin{matrix} \frac{\partial U}{\partial t} + \frac{\partial U}{\partial x} = - x - t - \dot{C_{t}} \\ U (0, x) = 0 . \end{matrix}$

At first, it has a solution $U (t, x) = - xt - C_{t} .$

It follows from Theorem 8 that $Ψ_{t, x}^{- 1} (α) = - x - t - \frac{\sqrt{3} t}{π} ln \frac{α}{1 - α},$ which is shown in Fig. 2.

Fig.2

Inverse uncertainty distribution in Example 5.

5 IPO problem model in uncertain partial differential equation

Generally, uncertain partial differential equation is an effective tool for describing uncertainty dynamic systems. Besides, the uncertain factor influencing the internet public opinion is not single, it is clear that internet public opinion trends to be short spread and breakout fast, and the flexibility is not same in different times. That is, the longer the time, the higher the flexibility, then the more the propagation times. This paper will investigate the IPO problem by uncertain partial differential equation.

In the IPO problem, we introduce the following parameters and notations:

the days of propagation;

the flexibility of propagation;

U(0, 0)

the uncertain propagation times at the initial moment;

U(t, x)

the uncertain propagation times at time t and flexibility x;

the growth rate of propagation times in a unit;

the variance of growth rate, it is the volatility of propagation times;

C_t

a Liu process.

Due to different situations of IPO problem, we give the model construction in different ways. Here, we provide three types of IPO problem: soaring model, stationary model and recession model.

Soaring model: ${\begin{matrix} \frac{\partial U}{\partial t} + \frac{\partial U}{\partial x} = a (x + t) U + b \dot{C_{t}}, t > 0, x \in R \\ U (0, x) = 1, x \in R . \end{matrix}$ (29) where U (0, x) is that the opinion has just been put on the internet at initial moment.

This model is suitable for the beginning of IPO problem. When a public opinion has just been sent to the internet, people who knows the news is little, some of them are only spectators. As time goes on, the flexibility becomes higher, the number of propagation times increases rapidly in unit time, then it turns to a outbreak state.

Theorem 9. If C_t is a Liu process, then the uncertain field $U (t, x) = \exp (axt) + {bC}_{t}$ (30) is a solution of soaring model ${\begin{matrix} \frac{\partial U}{\partial t} + \frac{\partial U}{\partial x} = a (x + t) U + b \dot{C_{t}}, t > 0, x \in R \\ U (0, x) = 1, x \in R . \end{matrix}$ (31)

Proof. Variable substitution ${\begin{matrix} ξ = φ (t, x) = x - t \\ η = ψ (t, x) = t, \end{matrix}$

where $\begin{matrix} J (φ, ψ) & = & \frac{\partial (φ, ψ)}{\partial (t, x)} = | \begin{matrix} \frac{\partial φ}{\partial t} & \frac{\partial φ}{\partial x} \\ \frac{\partial ψ}{\partial t} & \frac{\partial ψ}{\partial x} \end{matrix} | \\ = & | \begin{matrix} - 1 & 1 \\ 1 & 0 \end{matrix} | = - 1 \neq 0 . \end{matrix}$

Turn the Equation (31) to the new equation: $\frac{\partial U}{\partial η} = a (ξ + 2 η) U + b \dot{C_{η}} .$ (32)

From Theorem 5, we have $\begin{matrix} U (ξ, η) & = & \exp (a (ξ η + η^{2})) \\ \cdot [\int b \exp (- a (ξ η + η^{2})) \dot{C_{η}} d η + g (ξ)] \\ = & \exp (a (ξ η + η^{2})) \\ \cdot [\int b \exp (- a (ξ η + η^{2})) {d C}_{η} + g (ξ)] \\ = & g (ξ) \exp (a (ξ η + η^{2})) + {bC}_{η} . \end{matrix}$

That is, $U (t, x) = g (x - t) \exp (axt) + {bC}_{t} .$

Since U (0, x) =1, we have $U (0, x) = g (x) = 1,$

Hence $U (t, x) = \exp (axt) + {bC}_{t} .$ □

Theorem 10. Let α be a number with 0 < α < 1, U (t, x) = exp(axt) + bC_t is the solution of soaring model ${\begin{matrix} \frac{\partial U}{\partial t} + \frac{\partial U}{\partial x} = a (x + t) U + b \dot{C_{t}}, t > 0, x \in R \\ U (0, x) = 0, x \in R . \end{matrix}$ (33)

Then the uncertain field U (t, x) has an inverse uncertainty distribution $Ψ_{t, x}^{- 1} (α) = \exp (axt) + b Φ^{- 1} (α),$ (34) where $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}$ (35)is the inverse uncertainty distribution of standard normal uncertain variable.

Proof. At first, f (t, x, c) is a strictly increasing function with respect to c.

It follows from Theorem 7 that $Ψ_{t, x}^{- 1} (α) = \exp (axt) + b Φ^{- 1} (α),$ where $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}$ is the inverse uncertainty distribution of standard normal uncertain variable.□

Stationary model: ${\begin{matrix} \frac{\partial U}{\partial t} + \frac{\partial U}{\partial x} = a (x + t) (1 - \frac{U}{c}) U + b \dot{C_{t}}, \\ t > 0, x \in R \\ U (0, x) = ω (x), x \in R . \end{matrix}$ (36) where c is a constant represents the maximum number of propagation times in unit time, ω (x) is the propagation times at initial moment of stationary phase.

This model is suitable for a stationary phase of IPO problem. When the number of propagation times reaches a certain height, it will increase slowly. The growth rate of propagation times will go down, and it is a decreasing function with respect to U, we set it as $(1 - \frac{U}{c})$ . That is, as time goes on, the number of propagation times turns to a stationary phase instead of rapid growth phase in unit time.

Recession model: ${\begin{matrix} \frac{\partial U}{\partial t} + \frac{\partial U}{\partial x} = - a (x + t) U + b \dot{C_{t}}, t > 0, x \in R \\ U (0, x) = ω (x), x \in R . \end{matrix}$ (37) where ω (x) is the propagation times of stationary phase in unit time.

This model is suitable for the end of IPO problem. It is clear that every opinion will die out. As time goes on, the flexibility of propagation will go down, and the number of propagation times will decrease until the opinion disappears from the internet.

Theorem 11. If C_t is a Liu process, then the uncertain field $U (t, x) = ω (x) \exp (- axt) + {bC}_{t}$ (38) is a solution of recession model ${\begin{matrix} \frac{\partial U}{\partial t} + \frac{\partial U}{\partial x} = - a (x + t) U + b \dot{C_{t}}, t > 0, x \in R \\ U (0, x) = ω (x), x \in R . \end{matrix}$ (39)

Proof. Variable substitution ${\begin{matrix} ξ = φ (t, x) = x - t \\ η = ψ (t, x) = t, \end{matrix}$ where $\begin{matrix} J (φ, ψ) & = & \frac{\partial (φ, ψ)}{\partial (t, x)} = | \begin{matrix} \frac{\partial φ}{\partial t} & \frac{\partial φ}{\partial x} \\ \frac{\partial ψ}{\partial t} & \frac{\partial ψ}{\partial x} \end{matrix} | \\ = & | \begin{matrix} - 1 & 1 \\ 1 & 0 \end{matrix} | = - 1 \neq 0 . \end{matrix}$

Turn the Equation (39) to the new equation: $\frac{\partial U}{\partial η} = - a (ξ + 2 η) U + b \dot{C_{η}} .$

From Theorem 5, we have $\begin{matrix} U (ξ, η) & = & \exp (- a (ξ η + η^{2})) \\ \cdot [\int b \exp (a (ξ η + η^{2})) \dot{C_{η}} d η + g (ξ)] \\ = & \exp (- a (ξ η + η^{2})) \\ \cdot [\int b \exp (a (ξ η + η^{2})) {d C}_{η} + g (ξ)] \\ = & g (ξ) \exp (- a (ξ η + η^{2})) + {bC}_{η} . \end{matrix}$

That is, $U (t, x) = g (x - t) \exp (- axt) + {bC}_{t} .$

Since U (0, x) = ω (x), we have $U (0, x) = g (x) = ω (x),$

Hence $U (t, x) = ω (x) \exp (- axt) + {bC}_{t} .$ □

Theorem 12. Let α be a number with 0 < α < 1, U (t, x) = ω (x) exp(- axt) + bC_t is the solution of soaring model ${\begin{matrix} \frac{\partial U}{\partial t} + \frac{\partial U}{\partial x} = - a (x + t) U + b \dot{C_{t}}, t > 0, x \in R \\ U (0, x) = ω (x), x \in R . \end{matrix}$ (40) then the uncertain field U (t, x) has an inverse uncertainty distribution $Ψ_{t, x}^{- 1} (α) = ω (x) \exp (- axt) + b Φ^{- 1} (α),$ (41) where $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}$ (42)is the inverse uncertainty distribution of standard normal uncertain variable.

Proof. At first, f (t, x, c) is a strictly increasing function with respect to c.

It follows from Theorem 7 that $Ψ_{t, x}^{- 1} (α) = ω (x) \exp (- axt) + b Φ^{- 1} (α),$ where $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}$ is the inverse uncertainty distribution of standard normal uncertain variable. □

6 Numerical examples

Example 6. Assume that an internet public opinion is just breakout, and it fits for the soaring model. Furthermore, the domain experts give the growth rate is a = 0.6, and the variance is b = 0.03, combined with their own professional knowledge and experience.

That is, ${\begin{matrix} \frac{\partial U}{\partial t} + \frac{\partial U}{\partial x} = 0.6 (x + t) U + 0.03 \dot{C_{t}}, t > 0, x \in R \\ U (0, x) = 1, x \in R . \end{matrix}$

Then the solution is $U (t, x) = \exp (0.6 xt) + 0.03 C_{t},$ where the inverse uncertainty distribution of solution is $Ψ_{t, x}^{- 1} (α) = \exp (0.6 xt) + 0.03 \frac{\sqrt{3}}{π} ln \frac{α}{1 - α},$ which is shown in Fig. 3.

Fig.3

An example of soaring model.

Example 7. Assume that an internet public opinion breaks out, and it has caused some influence on the internet. We now predict the trend of the event if it is controlled. Obviously, it fits the recession model.

The domain experts give the growth rate is a = 0.45, the variance is b = 0.02, combined with their own professional knowledge and experience. The uncertain propagation times at the initial moment is U (0, x) = ω (x) =17000.

That is, ${\begin{matrix} \frac{\partial U}{\partial t} + \frac{\partial U}{\partial x} = - 0.45 (x + t) U + 0.02 \dot{C_{t}}, \\ t > 0, x \in R \\ U (0, x) = 17000, x \in R . \end{matrix}$

Then the solution is $U (t, x) = 17000 \exp (- 0.45 xt) + 0.02 C_{t},$ where the inverse uncertainty distribution of solution is $Ψ_{t, x}^{- 1} (α) = 17000 \exp (- 0.45 xt) + 0.02 \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}$ which is shown in Fig. 4.

Fig.4

An example of recession model.

7 Conclusion

In this paper, two methods for solving the first order linear uncertain partial differential equations were first given. And then the inverse uncertainty distribution of solution was derived. In addition, this paper proposed three types of IPO models: soaring model, stationary model and recession model. Finally, some examples were given.

Footnotes

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (No. 2016MS65), and the National Natural Science Foundation of China (Nos. 71671064 and 11601150).

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