Analytic solutions of two types of nonlinear uncertain differential equations

Abstract

Uncertain differential equation is an essential tool in dealing with uncertain dynamic system, which is driven by canonical Liu process. This paper mainly studies two classes of nonlinear uncertain differential equations with exponential and power forms by two analytic methods. The corresponding solutions are obtained, and some examples are given to illustrate the effectiveness of these methods.

Keywords

Canonical Liu process Uncertain calculus Uncertain differential equation

1 Introduction

Uncertainty theory was founded by Liu [1] and refined by Liu [2], which is to deal with human’s belief degree based on normality, duality, subadditivity and product axioms. Uncertain variable is used to represent quantities with uncertainty. During the past few years, many scholars have contributed in the field. For example, Peng and Iwamura [3] proved a sufficient and necessary condition of uncertainty distribution for an uncertain variable. Liu and Ha [4] obtained some expressions of expected value of uncertain variable function. Chen and Dai [5] proposed the maximum entropy principle for uncertain variables.

In 2008, Liu introduced the concept of uncertain process and proposed a type of differential equations driven by canonical Liu process, called uncertain differential equation. So far, uncertain differential equation has become a main tool to deal with dynamic uncertain system in many fields such as finance,optimal control fields. Liu [7] established an uncertain stock model, and Chen and Gao [8] studied an uncertain interest rate model. Liu [9] proposed an uncertain insurance risk process, and Yao and Qin [10] modified the uncertain risk model and derived the uncertainty distribution. Recently, Li et al. [11] firstly applied uncertain differential equation to establish an uncertain SIS epidemic model and revealed the relationship of the uncertain and deterministic models.

In order to capture the properties of an uncertain differential equation, the solution plays an important role in theory and practice. Chen and Liu [12] obtained analytic solutions of linear uncertain differential equations. Liu [13] and Yao [14] provided some methods to get the solutions of some nonlinear uncertain differential equations, and Wang [15] extended their results. Yao and Chen [16] designed a numerical method to solve an uncertain differential equation.

In this paper, we will consider two types of nonlinear uncertain differential equations and provide the corresponding solutions, which extend the above results. The remainder of this paper is organized as follows. In Section 2, we review some basic concepts and results. Uncertain differential equations with exponential form are solved in Section 3. In Section 4, the solutions of uncertain differential equations with power form are obtained. Some discussions are made in Section 5.

2 Basic concepts and notation

In this section, we review some concepts and notation, including uncertain variable, uncertain process, uncertain calculus and uncertain differential equation. More details to see Liu [2] and Yao [10].

Let Γ be a non-empty set and $L$ be a σ-algebra over Γ. Each element $Λ \in L$ is called an event. A set function $M : L \to [0, 1]$ is called an uncertain measure if it satisfies four axioms:

Axiom 1. (Normality Axiom) For the universal set Γ, $M {Γ} = 1$ .

Axiom 2. (Duality Axiom) For any event Λ, $Λ ℳ {Λ} + ℳ {Λ^{c}} = 1$ .

Axiom 3. (Subadditivity Axiom) For every countable sequence of events Λ₁, Λ₂, …, $ℳ {\cup_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} ℳ {Λ_{i}} .$ The triplet $(Γ, L, M)$ is called an uncertainty space.

Axiom 4. (Product Axiom) Let $(Γ_{i}, L_{i}, M_{i})$ be uncertainty spaces and Λ_i be arbitrarily chosen events from $ℒ_{i}$ for i = 1, 2, …, then the product uncertain measure $M$ is an uncertain measure satisfying $ℳ {\prod_{i = 1}^{\infty} Λ_{i}} = Λ_{i = 1}^{\infty} ℳ_{i} {Λ_{i}} .$

An uncertain variable X is a measurable function from an uncertainty space $(Γ, L, M)$ to the set $ℝ$ of real numbers. The uncertainty distribution Φ of X is defined by $Φ (x) = M {X \leq x}$ for any real number x. Uncertain process X_t is a sequence of uncertain variables indexed by the time t. An uncertain process C_t is said to be a canonical 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 whose uncertainty distribution is $Φ_{t} (x) = (1 + \exp (- \frac{π x}{\sqrt{3} t}))^{- 1}, x \in ℝ .$

Let X_t be an uncertain process. For any partition of closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_k+1 = b, the mesh is written as $▵ = max_{1 \leq i \leq k} ∣ t_{i + 1} - t_{i} ∣ .$ Then the time integral of X_t with respect to t is $\int_{a}^{b} X_{t} dt = lim_{▵ \to 0} \sum_{i = 1}^{k} X_{t_{i}} \cdot (t_{i + 1} - t_{i}),$ provided that the limit exists almost surely and is finite, X_t is said to be time integrable. Let C_t be a canonical Liu process, Liu integral of X_t with respect to C_t is defined as $\int_{a}^{b} X_{t} {dC}_{t} = lim_{▵ \to 0} \sum_{i = 1}^{k} X_{t_{i}} \cdot (C_{t_{i + 1}} - C_{t_{i}}),$ provided that the limit exists almost surely and is finite, X_t is said to be integrable.

Suppose f and g are two measurable functions. Then, the equation ${dX}_{t} = f (t, X_{t}) dt + g (t, X_{t}) {dC}_{t}$ is called an uncertain differential equation (UDE). An uncertain process that satisfies the UDE identically at each time t is called a solution of the UDE.

Lemma 1. ([2]) Suppose X_t and Y_t are canonical Liu processes. Then $d (X_{t} Y_{t}) = Y_{t} {dX}_{t} + X_{t} {dY}_{t} .$ In the next sections, we provide some analytic methods to solve two classes of nonlinear uncertain differential equations, respectively.

3 Uncertain differential equation with exponential form

Let X_t be an uncertain process and C_t be a canonical Liu process. Suppose f and g are the functions of X_t and t. For a > 0, a ≠ 1 and p ≠ 0, we respectively consider analytic solutions of the following equations ${dX}_{t} = (u_{1 t} + u_{2 t} a^{{pX}_{t}}) dt + g (t, X_{t}) {dC}_{t},$ (1) ${dX}_{t} = f (t, X_{t}) dt + (σ_{1 t} + σ_{2 t} a^{{pX}_{t}}) {dC}_{t},$ (2) which are called uncertain differential equations (UDEs) with exponential form.

Theorem 1. Suppose u_1t, u_2t are two integrable uncertain processes. Then, the UDE (1) has an analytic solution $X_{t} = \frac{1}{p} [{log}_{a} Y_{t} - {log}_{a} (Z_{t} - Y_{t} G_{t})],$ where $Y_{t} = a^{p \int_{0}^{t} u_{1 s} ds}, G_{t} = p (ln a) Y_{t}^{- 1} \int_{0}^{t} u_{2 s} Y_{s} ds$ and Z_t satisfies an UDE ${dZ}_{t} = p (ln a) (Y_{t} G_{t} - Z_{t}) g (t, - \frac{1}{p} {log}_{a} (Y_{t}^{- 1} Z_{t} - G_{t})) {dC}_{t}$ with Z₀ = a^-pX₀.

Proof. Since da^-pX_t = - p (ln a) a^-pX_tdX_t, by the UDE (1), we have $\begin{matrix} {da}^{- {pX}_{t}} & = & - p (ln a) (u_{1 t} a^{- {pX}_{t}} + u_{2 t}) dt \\ - p (ln a) a^{- {pX}_{t}} g (t, X_{t}) {dC}_{t} . \end{matrix}$ Denote I_t = a^-pX_t, then $\begin{matrix} {dI}_{t} & = & - p (ln a) (u_{1 t} I_{t} + u_{2 t}) dt \\ - p (ln a) I_{t} g (t, - \frac{1}{p} {log}_{a} I_{t}) {dC}_{t} . \end{matrix}$ Let H_t = I_t + G_t. Note that G_t is the solution of the equation $G_{t}^{'} + p (ln a) u_{1 t} G_{t} - p (ln a) u_{2 t} = 0 .$ Thus, $\begin{matrix} {dH}_{t} & = & {dI}_{t} + {dG}_{t} \\ = & [- p (ln a) u_{1 t} I_{t} - p (ln a) u_{2 t} + G_{t}^{'}] dt \\ - p (ln a) I_{t} g (t, - \frac{1}{p} {log}_{a} I_{t}) {dC}_{t} \\ = & [- p (ln a) u_{1 t} H_{t} + G_{t}^{'} + p (ln a) u_{1 t} G_{t} \\ - p (ln a) u_{2 t}] dt - p (ln a) (H_{t} - G_{t}) \\ \times g (t, - \frac{1}{p} {log}_{a} (H_{t} - G_{t})) {dC}_{t} \\ = & - p (ln a) u_{1 t} H_{t} dt - p (ln a) (H_{t} - G_{t}) \\ \times g (t, - \frac{1}{p} {log}_{a} (H_{t} - G_{t})) {dC}_{t} \end{matrix}$ with initial value H₀ = I₀. From Lemma 1 and ${dY}_{t} = p (ln a) u_{1 t} Y_{t} dt,$ one has $\begin{matrix} d (H_{t} Y_{t}) & = & Y_{t} {dH}_{t} + H_{t} {dY}_{t} \\ = & - p (ln a) u_{1 t} Y_{t} H_{t} dt - p (ln a) Y_{t} (H_{t} - G_{t}) \\ \times g (t, - \frac{1}{p} {log}_{a} (H_{t} - G_{t})) {dC}_{t} + p (ln a) u_{1 t} H_{t} Y_{t} dt \\ = & - p (ln a) Y_{t} (H_{t} - G_{t}) g (t, - \frac{1}{p} {log}_{a} (H_{t} - G_{t})) {dC}_{t} . \end{matrix}$ Denote Z_t = H_tY_t. Thus, ${dZ}_{t} = p (ln a) (Y_{t} G_{t} - Z_{t}) g (t, - \frac{1}{p} {log}_{a} (Y_{t}^{- 1} Z_{t} - G_{t})) {dC}_{t}$ with Z₀ = a^-pX₀. Then, the UDE (1) has an analytic solution $X_{t} = \frac{1}{p} [{log}_{a} Y_{t} - {log}_{a} (Z_{t} - Y_{t} G_{t})] .$ □

The comparative results about some existing works and equation (1) are provided by Table 1.

Table 1

Comparative results of some UDEs and equation (1)

Uncertain differential equation	Solution	Source
dX_t = u_tdt + g (t, X_t) dC_t.	X_t = Y_t + Z_t, where $Y_{t} = \int_{0}^{t} u_{s} ds$	Yao [14]
	and dZ_t = g (t, Y_t + Z_t) dC_t, Z₀ = X₀.
dX_t = (u_tX_t + v_t) dt + g (t, X_t) dC_t.	$X_{t} = Y_{t}^{- 1} Z_{t} - G_{t}$ , where $Y_{t} = exp (- \int_{0}^{t} u_{s} ds), G_{t} = - Y_{t}^{- 1} \int_{0}^{t} v_{s} Y_{s} ds$	Y. Liu [13]
	and ${dZ}_{t} = Y_{t} g (t, (Y_{t}^{- 1} Z_{t} - G_{t}) {dC}_{t}, Z_{0} = X_{0} .$
dX_t = (u_1t + u_2ta^{pX _t}) dt + g (t, X_t) dC_t,	$X_{t} = \frac{1}{p}$ , where $Y_{t} = a^{p \int_{0}^{t} u_{1 s} ds}$ ,
a > 0, a ≠ 1 and p ≠ 0.	$G_{t} = p (ln a) Y_{t}^{- 1} \int_{0}^{t} u_{2 s} Y_{s} ds$ and	Equation (1)
	${dZ}_{t} = p (ln a) (Y_{t} G_{t} - Z_{t}) g (t, - \frac{1}{p} {log}_{a} (Y_{t}^{- 1} Z_{t} - G_{t})) {dC}_{t}, Z_{0} = a^{- {pX}_{0}} .$

Based on Theorem 1, it is easy to get the following corollary.

Corollary 1. Suppose u_1t, u_2t are two integrable uncertain processes. Then, the UDE ${dX}_{t} = (u_{1 t} + u_{2 t} exp ({pX}_{t})) dt + g (t, X_{t}) {dC}_{t}$ has an analytic solution $X_{t} = \frac{1}{p} [ln Y_{t} - ln (Z_{t} - Y_{t} G_{t})],$ where $Y_{t} = exp (p \int_{0}^{t} u_{1 s} ds), G_{t} = {pY}_{t}^{- 1} \int_{0}^{t} u_{2 s} Y_{s} ds$ and Z_t satisfies an UDE ${dZ}_{t} = p (Y_{t} G_{t} - Z_{t}) g (t, - \frac{1}{p} ln (Y_{t}^{- 1} Z_{t} - G_{t})) {dC}_{t}$ with Z₀ = exp(- pX₀).

Example 1. Let u, σ be two real numbers and u ≠ 0, σ ≠ 0. Consider an UDE ${dX}_{t} = udt + σ exp (X_{t}) {dC}_{t} .$ According to Corollary 1, we have $Y_{t} = exp (ut), G_{t} = 0,$ and Z_t satisfies ${dZ}_{t} = - σ exp (ut) {dC}_{t}$ with Z₀ = exp(- X₀). Further, $Z_{t} = exp (- X_{0}) - σ \int_{0}^{t} exp (us) {dC}_{s} .$ Then, the UDE has a solution $X_{t} = X_{0} + ut - ln (1 - σ exp (X_{0}) \int_{0}^{t} exp (us) {dC}_{s}) .$

Theorem 2. Suppose σ_1t, σ_2t are two integrable and differential uncertain processes with initial values σ₁₀ and σ₂₀.

(i) If σ_1t ≠ 0, then the UDE (2) has a solution $X_{t} = \frac{1}{p} [{log}_{a} Y_{t} - {log}_{a} (Z_{t} - σ_{1 t}^{- 1} σ_{2 t} Y_{t})],$ where $Y_{t} = a^{p \int_{0}^{t} σ_{1 s} {dC}_{s}}$ and Z_t satisfies $\begin{matrix} {dZ}_{t} & = & [(σ_{1 t}^{- 1} σ_{2 t})^{'} Y_{t} - p (ln a) (Z_{t} - σ_{1 t}^{- 1} σ_{2 t} Y_{t}) \\ \times f (t, - \frac{1}{p} {log}_{a} (Y_{t}^{- 1} Z_{t} - σ_{1 t}^{- 1} σ_{2 t}))] dt \end{matrix}$ with the initial value $Z_{0} = a^{- {pX}_{0}} + \frac{σ_{20}}{σ_{10}}$ .

(ii) If σ_1t = 0, then the UDE (2) has a solution $X_{t} = - \frac{1}{p} {log}_{a} (Y_{t} + Z_{t}),$ where $Y_{t} = - p (ln a) \int_{0}^{t} σ_{2 s} {dC}_{s}$ and Z_t satisfies ${dZ}_{t} = - p (ln a) (Y_{t} + Z_{t}) f (t, - \frac{1}{p} {log}_{a} (Y_{t} + Z_{t})) dt$ with Z₀ = a^-pX₀.

Proof. Let I_t = a^-pX_t. Similar to the proof of Theorem 1, one has $\begin{matrix} {dI}_{t} & = & - p (ln a) I_{t} f (t, - \frac{1}{p} {log}_{a} I_{t}) dt \\ - p (ln a) (σ_{1 t} I_{t} + σ_{2 t}) {dC}_{t} . \end{matrix}$

(i) If σ_1t ≠ 0, let $H_{t} = I_{t} + σ_{1 t}^{- 1} σ_{2 t}$ , then $\begin{matrix} {dH}_{t} & = & {dI}_{t} + d (σ_{1 t}^{- 1} σ_{2 t}) \\ = & - p (ln a) I_{t} f (t, - \frac{1}{p} {log}_{a} I_{t}) dt \\ - p (ln a) (σ_{1 t} I_{t} + σ_{2 t}) {dC}_{t} + (σ_{1 t}^{- 1} σ_{2 t})^{'} dt \\ = & [(σ_{1 t}^{- 1} σ_{2 t})^{'} - p (ln a) (H_{t} - σ_{1 t}^{- 1} σ_{2 t}) \\ \times f (t, - \frac{1}{p} {log}_{a} (H_{t} - σ_{1 t}^{- 1} σ_{2 t}))] dt \\ - p (ln a) σ_{1 t} H_{t} {dC}_{t} \end{matrix}$ with initial value $H_{0} = I_{0} + \frac{σ_{20}}{σ_{10}} .$ Based on Lemma 1 and ${dY}_{t} = p (ln a) σ_{1 t} Y_{t} {dC}_{t},$ we have $\begin{matrix} d (H_{t} Y_{t}) & = & Y_{t} {dH}_{t} + H_{t} {dY}_{t} \\ = & Y_{t} [(σ_{1 t}^{- 1} σ_{2 t})^{'} - p (ln a) (H_{t} - σ_{1 t}^{- 1} σ_{2 t}) \\ \times f (t, - \frac{1}{p} {log}_{a} (H_{t} - σ_{1 t}^{- 1} σ_{2 t}))] dt . \end{matrix}$ Denote Z_t = H_tY_t. By the above equation, Z_t satisfies $\begin{matrix} {dZ}_{t} & = & [(σ_{1 t}^{- 1} σ_{2 t})^{'} Y_{t} - p (ln a) (Z_{t} - σ_{1 t}^{- 1} σ_{2 t} Y_{t}) \\ \times f (t, - \frac{1}{p} {log}_{a} (Y_{t}^{- 1} Z_{t} - σ_{1 t}^{- 1} σ_{2 t}))] dt \end{matrix}$ with $Z_{0} = a^{- {pX}_{0}} + \frac{σ_{20}}{σ_{10}}$ . Therefore, the UDE (2) has an analytic solution $X_{t} = \frac{1}{p} [{log}_{a} Y_{t} - {log}_{a} (Z_{t} - σ_{1 t}^{- 1} σ_{2 t} Y_{t})] .$

(ii) If σ_1t = 0, then $\begin{matrix} {dI}_{t} = - p (ln a) I_{t} f (t, - \frac{1}{p} {log}_{a} I_{t}) dt - p (ln a) σ_{2 t} {dC}_{t} . \end{matrix}$ Since dY_t = - p (ln a) σ_2tdC_t, it is to get $\begin{matrix} d (I_{t} - Y_{t}) & = & {dI}_{t} - {dY}_{t} \\ = & - p (ln a) I_{t} f (t, - \frac{1}{p} {log}_{a} I_{t}) dt . \end{matrix}$ Denote Z_t = I_t - Y_t. Obviously, ${dZ}_{t} = - p (ln a) (Y_{t} + Z_{t}) f (t, - \frac{1}{p} {log}_{a} (Y_{t} + Z_{t})) dt$ with Z₀ = a^-pX₀. Thus, the UDE (2) has an analytic solution $X_{t} = - \frac{1}{p} {log}_{a} (Y_{t} + Z_{t}) .$ □

The comparative results about some existing works and equation (2) are provided by Table 2.

Based on Theorem 2, it is easy to get the following corollary.

Table 2

Comparative results of some UDEs and equation (2)

Uncertain differential equation	Solution	Source
dX_t = f (t, X_t) dt + σ_tdC_t.	X_t = Y_t + Z_t, where $Y_{t} = \int_{0}^{t} σ_{s} {dC}_{s}$	Yao [14]
	and dZ_t = f (t, Y_t + Z_t) dt, Z₀ = X₀.
dX_t = f (t, X_t) dt + (σ_1tX_t + σ_2t) dC_t.	$X_{t} = Y_{t}^{- 1} Z_{t} - \frac{σ_{2 t}}{σ_{1 t}}$ , where $Y_{t} = exp (- \int_{0}^{t} σ_{1 s} {dC}_{s})$	Y. Liu [13]
	and ${dZ}_{t} = Y_{t} (f (t, Y_{t}^{- 1} Z_{t} - \frac{σ_{2 t}}{σ_{1 t}}) + (\frac{σ_{2 t}}{σ_{1 t}})^{'}) dt, Z_{0} = X_{0} + \frac{σ_{20}}{σ_{10}} .$
dX_t = f (t, X_t) dt + (σ_1t + σ_2ta^{pX _t}) dC_t,	★ If $σ_{1 t} \neq 0, X_{t} = \frac{1}{p}$ ,
a > 0, a ≠ 1 and p ≠ 0.	where $Y_{t} = a^{p \int_{0}^{t} σ_{1 s} {dC}_{s}} and$	Equation (2)
	${dZ}_{t} = [(σ_{1 t}^{- 1} σ_{2 t})^{'} Y_{t} - p (ln a) (Z_{t} - σ_{1 t}^{- 1} σ_{2 t} Y_{t}) f (t, - \frac{1}{p} {log}_{a} (Y_{t}^{- 1} Z_{t}$
	$- σ_{1 t}^{- 1} σ_{2 t}))] dt, Z_{0} = a^{- {pX}_{0}} + \frac{σ_{20}}{σ_{10}} .$
	★ If $σ_{1 t} = 0, X_{t} = - \frac{1}{p} {log}_{a} (Y_{t} + Z_{t}),$ where $Y_{t} = - p (ln a) \int_{0}^{t} σ_{2 s} {dC}_{s}$
	and ${dZ}_{t} = - p (ln a) (Y_{t} + Z_{t}) f (t, - \frac{1}{p} {log}_{a} (Y_{t} + Z_{t})) dt, Z_{0} = a^{- {pX}_{0}} .$

Corollary 2. Suppose σ_1t, σ_2t are two integrable and differential uncertain processes with initial values σ₁₀ and σ₂₀. Consider the UDE $\begin{matrix} {dX}_{t} = f (t, X_{t}) dt + (σ_{1 t} + σ_{2 t} exp ({pX}_{t})) {dC}_{t} . \end{matrix}$

(i) If σ_1t ≠ 0, then the UDE has a solution $X_{t} = \frac{1}{p} [ln Y_{t} - ln (Z_{t} - σ_{1 t}^{- 1} σ_{2 t} Y_{t})],$ where $Y_{t} = exp (p \int_{0}^{t} σ_{1 s} {dC}_{s})$ and Z_t satisfies $\begin{matrix} {dZ}_{t} & = & [(σ_{1 t}^{- 1} σ_{2 t})^{'} Y_{t} - p (Z_{t} - σ_{1 t}^{- 1} σ_{2 t} Y_{t}) \\ \times f (t, - \frac{1}{p} ln (Y_{t}^{- 1} Z_{t} - σ_{1 t}^{- 1} σ_{2 t}))] dt \end{matrix}$ with $Z_{0} = exp (- {pX}_{0}) + \frac{σ_{20}}{σ_{10}} .$

(ii) If σ_1t = 0, then the UDE has a solution $X_{t} = - \frac{1}{p} ln (Y_{t} + Z_{t}),$ where $Y_{t} = - p \int_{0}^{t} σ_{2 s} {dC}_{s}$ and Z_t satisfies ${dZ}_{t} = - p (Y_{t} + Z_{t}) f (t, - \frac{1}{p} ln (Y_{t} + Z_{t})) dt$ with Z₀ = exp(- pX₀).

Example 2. Consider an UDE $\begin{matrix} {dX}_{t} = t exp (X_{t}) dt + t^{2} exp (t + X_{t}) {dC}_{t} . \end{matrix}$ From Corollary 2, we have $Y_{t} = - \int_{0}^{t} s^{2} exp (s) {dC}_{s}$ and dZ_t = - tdt with Z₀ = exp(- X₀). Further, $Z_{t} = exp (- X_{0}) - \frac{1}{2} t^{2} .$ Therefore, the UDE has a solution, $X_{t} = X_{0} - ln (1 - exp (X_{0}) (\frac{1}{2} t^{2} + \int_{0}^{t} s^{2} exp (s) {dC}_{s})) .$

4 Uncertain differential equation with power form

Chen and Liu [12] provided solution of a linear UDE

dX_t = (u_1tX_t + u_2t) dt + (v_1tX_t + v_2t) dC_t.

Liu [13] solved two non-linear UDEs $\begin{matrix} {dX}_{t} & = & u_{t} X_{t} dt + g (t, X_{t}) {dC}_{t}, \\ {dX}_{t} & = & f (t, X_{t}) dt + σ_{t} X_{t} {dC}_{t} . \end{matrix}$ Wang [15] extended the results of Liu [13] to the general type of the UDEs $\begin{matrix} {dX}_{t} & = & u_{t} X_{t}^{p} dt + g (t, X_{t}) {dC}_{t}, \\ {dX}_{t} & = & f (t, X_{t}) dt + σ_{t} X_{t}^{p} {dC}_{t} \end{matrix}$ for p ≠ 1. In this section, we combine the UDEs of Chen and Liu [12], and Wang [15] to study solutions of the following UDEs ${dX}_{t} = (u_{1 t} X_{t} + u_{2 t} X_{t}^{p}) dt + g (t, X_{t}) {dC}_{t},$ (3) ${dX}_{t} = f (t, X_{t}) dt + (σ_{1 t} X_{t} + σ_{2 t} X_{t}^{p}) {dC}_{t},$ (4) which are called UDEs with power form. Especially, if p = 1, the solutions of Liu [13] is those of the UDEs (3) and (4); if p ≠ 1 and u_1t = 0 or σ_1t = 0, the solutions of Wang [15] is those of the UDEs (3) or (4). Thus, in this section, we only discuss the cases p ≠ 1, u_1t ≠ 0 and σ_1t ≠ 0.

Theorem 3. Suppose u_1t, u_2t are two integrable uncertain processes. Then, the UDE (3) has a solution $X_{t} = (Y_{t}^{- 1} Z_{t} - G_{t})^{\frac{1}{1 - p}},$ where $\begin{matrix} Y_{t} = exp ((p - 1) \int_{0}^{t} u_{1 s} ds), \\ G_{t} = (p - 1) Y_{t}^{- 1} \int_{0}^{t} u_{2 s} Y_{s} ds, \end{matrix}$ and Z_t satisfies ${dZ}_{t} = (1 - p) Y_{t} (Y_{t}^{- 1} Z_{t} - G_{t})^{\frac{p}{p - 1}} g (t, (Y_{t}^{- 1} Z_{t} - G_{t})^{\frac{1}{1 - p}}) {dC}_{t}$ with $Z_{0} = X_{0}^{1 - p} .$

Proof. Let $I_{t} = X_{t}^{1 - p}$ , then ${dI}_{t} = (1 - p) (u_{1 t} I_{t} + u_{2 t}) dt + (1 - p) I_{t}^{\frac{p}{p - 1}} g (t, I_{t}^{\frac{1}{1 - p}}) {dC}_{t} .$ Denote H_t = I_t + G_t. Since $G_{t}^{'} + (p - 1) u_{1 t} G_{t} + (1 - p) u_{2 t} = 0,$ we have $\begin{matrix} {dH}_{t} & = & {dI}_{t} + {dG}_{t} \\ = & [(1 - p) (u_{1 t} I_{t} + u_{2 t}) + G_{t}^{'}] dt \\ + (1 - p) I_{t}^{\frac{p}{p - 1}} g (t, I_{t}^{\frac{1}{1 - p}}) {dC}_{t} \\ = & [(1 - p) u_{1 t} H_{t} + G_{t}^{'} + (p - 1) u_{1 t} G_{t} + (1 - p) u_{2 t}] dt \\ + (1 - p) (H_{t} - G_{t})^{\frac{p}{p - 1}} g (t, (H_{t} - G_{t})^{\frac{1}{1 - p}}) {dC}_{t} \\ = & (1 - p) u_{1 t} H_{t} dt + (1 - p) (H_{t} - G_{t})^{\frac{p}{p - 1}} \\ \times g (t, (H_{t} - G_{t})^{\frac{1}{1 - p}}) {dC}_{t} \end{matrix}$ with initial value H₀ = I₀. By Lemma 1 and ${dY}_{t} = (p - 1) u_{1 t} Y_{t} dt,$ it is to get $\begin{matrix} d (H_{t} Y_{t}) & = & Y_{t} {dH}_{t} + H_{t} {dY}_{t} \\ = & (1 - p) u_{1 t} Y_{t} H_{t} dt + (1 - p) Y_{t} (H_{t} - G_{t})^{\frac{p}{p - 1}} \\ \times g (t, (H_{t} - G_{t})^{\frac{1}{1 - p}}) {dC}_{t} + (p - 1) u_{1 t} Y_{t} H_{t} dt \\ = & (1 - p) Y_{t} (H_{t} - G_{t})^{\frac{p}{p - 1}} g (t, (H_{t} - G_{t})^{\frac{1}{1 - p}}) {dC}_{t} . \end{matrix}$ Let Z_t = H_tY_t. Thus, Z_t satisfies

${dZ}_{t} = (1 - p) Y_{t} (Y_{t}^{- 1} Z_{t} - G_{t})^{\frac{p}{p - 1}} g (t, (Y_{t}^{- 1} Z_{t} - G_{t})^{\frac{1}{1 - p}}) {dC}_{t}$ with $Z_{0} = X_{0}^{1 - p} .$ Then, the UDE (3) has an analytic solution $X_{t} = (Y_{t}^{- 1} Z_{t} - G_{t})^{\frac{1}{1 - p}} .$ □

The comparative results about some existing works and equation (3) are provided by Table 3.

Table 3

Comparative results of some UDEs and equation (3)

Uncertain differential equation	Solution	Source
dX_t = (u_1t X_t + u_2t) dt + (v_1t X_t + v_2t) dC_t.	$X_{t} = U_{t} (X_{0} + \int_{0}^{t} \frac{u_{2 s}}{U_{s}}) ds + \int_{0}^{t} \frac{v_{2 s}}{U_{s}} {dC}_{s}$	Chen, Liu [12]
	where $U_{t} = exp (\int_{0}^{t} u_{1 s} ds + \int_{0}^{t} v_{1 s}) {dC}_{s}$ .
dX_t = α_tX_tdt + g (t, X_t) dC_t.	$X_{t} = Y_{t}^{- 1} Z_{t}$ , where $Y_{t} = exp (- \int_{0}^{t} α_{s} ds)$	Y.H. Liu [13]
	and ${dZ}_{t} = Y_{t} g (t, Y_{t}^{- 1} Z_{t}) {dC}_{t}, Z_{0} = X_{0}$ .
${dX}_{t} = u_{t} X_{t}^{p} dt + g (t, X_{t}) {dC}_{t}, p \neq 1 .$	$X_{t} = (Y_{t} + Z_{t})^{\frac{1}{1 - p}}$ , where $Y_{t} = (1 - p) \int_{0}^{t} u_{s} ds$ and	Wang [15]
	${dZ}_{t} = (1 - p) (Y_{t} + Z_{t})^{- \frac{p}{(1 - p)}} g (t, (Y_{t} + Z_{t})^{\frac{1}{1 - p}}) {dC}_{s}, Z_{0} = X_{0}^{1 - p} .$
${dX}_{t} = (u_{1 t} X_{t} + u_{2 t} X_{t}^{p}) dt + g (t, X_{t}) {dC}_{t},$	$X_{t} = (Y_{t}^{- 1} Z_{t} - G_{t})^{\frac{1}{1 - p}}$ , where $Y_{t} = exp ((p - 1) \int_{0}^{t} u_{1 s} ds)$ ,	Equation (3)
p ≠ 1 and u_1t ≠ 0.	$G_{t} = (p - 1) Y_{t}^{- 1} \int_{0}^{t} u_{2 s} Y_{s} ds$ and
	${dZ}_{t} = (1 - p) Y_{t} (Y_{t}^{- 1} Z_{t} - G_{t})^{\frac{p}{p - 1}} g (t, (Y_{t}^{- 1} Z_{t} - G_{t})^{\frac{1}{1 - p}}) {dC}_{t}, Z_{0} = X_{0}^{1 - p} .$

Example 3. Let σ be a real number and σ ≠ 0. Consider an UDE ${dX}_{t} = (X_{t} + 1) dt + σ t (X_{t} - exp (t) + 1) {dC}_{t} .$ From Theorem 3, $Y_{t} = exp (- t), G_{t} = 1 - exp (t)$ and Z_t satisfies the equation dZ_t = σtZ_tdC_t with Z₀ = X₀. Further, $Z_{t} = X_{0} exp (σ \int_{0}^{t} {sdC}_{s}) .$ Then, the UDE has a solution $X_{t} = (X_{0} exp (σ \int_{0}^{t} {sdC}_{s}) + 1) exp (t) - 1 .$

Theorem 4. Suppose σ_1t, σ_2t are two integrable and differential uncertain processes with initial values σ₁₀ and σ₂₀. Then, the UDE (4) has a solution $X_{t} = (Y_{t}^{- 1} Z_{t} - σ_{1 t}^{- 1} σ_{2 t})^{\frac{1}{1 - p}},$ where $Y_{t} = exp ((p - 1) \int_{0}^{t} σ_{1 s} {dC}_{s})$ and Z_t satisfies $\begin{matrix} {dZ}_{t} & = & Y_{t} dt \end{matrix}$ with the initial value $Z_{0} = X_{0}^{1 - p} + \frac{σ_{20}}{σ_{10}}$ .

Proof. Let $I_{t} = X_{t}^{1 - p}$ , then $\begin{matrix} {dI}_{t} = (1 - p) I_{t}^{\frac{p}{p - 1}} f (t, I_{t}^{\frac{1}{1 - p}}) dt + (1 - p) (σ_{1 t} I_{t} + σ_{2 t}) {dC}_{t} . \end{matrix}$ Let $H_{t} = I_{t} + σ_{1 t}^{- 1} σ_{2 t}$ , we have $\begin{matrix} {dH}_{t} & = & {dI}_{t} + d (σ_{1 t}^{- 1} σ_{2 t}) \\ = & (1 - p) I_{t}^{\frac{p}{p - 1}} f (t, I_{t}^{\frac{1}{1 - p}}) dt \\ + (1 - p) (σ_{1 t} I_{t} + σ_{2 t}) {dC}_{t} + (σ_{1 t}^{- 1} σ_{2 t})^{'} dt \\ = & [(1 - p) I_{t}^{\frac{p}{p - 1}} f (t, I_{t}^{\frac{1}{1 - p}}) + (σ_{1 t}^{- 1} σ_{2 t})^{'}] dt \\ + (1 - p) (σ_{1 t} I_{t} + σ_{2 t}) {dC}_{t} \\ = & [(1 - p) (H_{t} - σ_{1 t}^{- 1} σ_{2 t}^{\frac{p}{p - 1}} f (t, (H_{t} - σ_{1 t}^{- 1} σ_{2 t})^{\frac{1}{1 - p}}) \\ + (σ_{1 t}^{- 1} σ_{2 t})^{'}] dt + (1 - p) σ_{1 t} H_{t} {dC}_{t} \end{matrix}$ with initial value $H_{0} = I_{0} + \frac{σ_{20}}{σ_{10}}$ . Since ${dY}_{t} = (p - 1) σ_{1 t} Y_{t} {dC}_{t},$ it is to get $\begin{matrix} d (H_{t} Y_{t}) & = & Y_{t} {dH}_{t} + H_{t} {dY}_{t} \\ = & Y_{t} {[(1 - p) (H_{t} - σ_{1 t}^{- 1} σ_{2 t}^{\frac{p}{p - 1}} \\ \times f (t, (H_{t} - σ_{1 t}^{- 1} σ_{2 t})^{\frac{1}{1 - p}}) + (σ_{1 t}^{- 1} σ_{2 t})^{'}] dt \\ + (1 - p) σ_{1 t} H_{t} {dC}_{t}} + (p - 1) σ_{1 t} Y_{t} H_{t} {dC}_{t} \\ = & Y_{t} [(1 - p) (H_{t} - σ_{1 t}^{- 1} σ_{2 t})^{\frac{p}{p - 1}} \\ \times f (t, (H_{t} - σ_{1 t}^{- 1} σ_{2 t})^{\frac{1}{1 - p}}) + (σ_{1 t}^{- 1} σ_{2 t})^{'}] dt \end{matrix}$ Denote Z_t = H_tY_t. Thus, Z_t satisfies $\begin{matrix} {dZ}_{t} & = & Y_{t} [(1 - p) (Y_{t}^{- 1} Z_{t} - σ_{1 t}^{- 1} σ_{2 t})^{\frac{p}{p - 1}} \\ \times f (t, (Y_{t}^{- 1} Z_{t} - σ_{1 t}^{- 1} σ_{2 t})^{\frac{1}{1 - p}}) + (σ_{1 t}^{- 1} σ_{2 t})^{'}] dt \end{matrix}$ with $Z_{0} = X_{0}^{1 - p} + \frac{σ_{20}}{σ_{10}}$ . Then, UDE (4) has an analytic solution $X_{t} = (Y_{t}^{- 1} Z_{t} - σ_{1 t}^{- 1} σ_{2 t})^{\frac{1}{1 - p}} .$ □

The comparative results about some existing works and equation (4) are provided by Table 4.

Table 4

Comparative results of some UDEs and equation (4)

Uncertain differential equation	Solution	Source
dX_t = u_tX_tdt + v_tX_tdC_t.	$X_{t} = X_{0} exp (\int_{0}^{t} u_{s} ds + \int_{0}^{t} v_{s} {dC}_{s}) .$	Liu [1]
dX_t = f (t, X_t) dt + σ_tX_tdC_t.	$X_{t} = Y_{t}^{- 1} Z_{t}$ , where $Y_{t} = exp (- \int_{0}^{t} σ_{s} {dC}_{s})$	Y.H. Liu [13]
	and ${dZ}_{t} = Y_{t} f (t, Y_{t}^{- 1} Z_{t}) dt, Z_{0} = X_{0}$ .
${dX}_{t} = f (t, X_{t}) dt + σ_{t} X_{t}^{p} {dC}_{t}, p \neq 1 .$	$X_{t} = (Y_{t} + Z_{t})^{\frac{1}{1 - p}}$ , where $Y_{t} = (1 - p) \int_{0}^{t} σ_{s} {dC}_{s}$	Wang [15]
	and ${dZ}_{t} = (1 - p) (Y_{t} + Z_{t})^{- \frac{p}{(1 - p)}} f (t, (Y_{t} + Z_{t})^{\frac{1}{1 - p}}) dt, Z_{0} = X_{0}^{1 - p} .$
${dX}_{t} = f (t, X_{t}) dt + (σ_{1 t} X_{t} + σ_{2 t} X_{t}^{p}) {dC}_{t},$	$X_{t} = (Y_{t}^{- 1} Z_{t} - σ_{1 t}^{- 1} σ_{2 t})^{\frac{1}{1 - p}}$ , where $Y_{t} = exp ((p - 1) \int_{0}^{t} σ_{1 s} {dC}_{s})$ and	Equation (4)
p ≠ 1 and σ_1t ≠ 0.	${dZ}_{t} = Y_{t} [(1 - p) (Y_{t}^{- 1} Z_{t} - σ_{1 t}^{- 1} σ_{2 t})^{\frac{p}{p - 1}} f (t, (Y_{t}^{- 1} Z_{t} - σ_{1 t}^{- 1} σ_{2 t})^{\frac{1}{1 - p}})$
	$+ (σ_{1 t}^{- 1} σ_{2 t})^{'}] dt, Z_{0} = X_{0}^{1 - p} + \frac{σ_{20}}{σ_{10}} .$

Example 4. Let u, σ be real numbers with u ≠ 0 and σ ≠ 0. Consider an UDE $\begin{matrix} {dX}_{t} = ut exp (t) X_{t}^{- 1} dt + σ (X_{t} + X_{t}^{- 1}) {dC}_{t} . \end{matrix}$ Obviously, Y_t = exp(-2σC_t) and Z_t satisfies the equation dZ_t = 2ut exp(t - 2σC_t) dt with the initial value $Z_{0} = X_{0}^{2} + 1$ . Thus, $Z_{t} = X_{0}^{2} + 2 u \int_{0}^{t} s exp (s - 2 σ C_{s}) ds + 1 .$ From Theorem 4, the UDE has a solution $X_{t} = [exp (2 σ C_{t}) (X_{0}^{2} + 2 u \int_{0}^{t} s exp (s - 2 σ C_{s}) ds + 1) - 1]^{\frac{1}{2}} .$

5 Discussions

This paper mainly provides analytic solutions of some nonlinear uncertain differential equations. Theorems 1 and 2 obtain the solutions of the UDEs with exponential form. Theorems 3 and 4 show the solutions of the UDEs with power form, which are the supplements for the results of Chen and Liu [12], and Wang [15]. Tables 1, 2, 3 and 4 provide the comparative results about existing works.

In this paper, we use two analytic methods for solving some nonlinear uncertain differential equations: the first and second analytic methods. The first analytic method solves some UDEs such as (1) and (2). The UDEs (3) and (4) are solved by the second analytic method. Some examples are illustrated the effectiveness of the proposed method. These methods can be applied to solve other differential equation like fuzzy differential equations, see Qiu et al. [18, 19], Qiu and Zhang [20]. By one of both methods, solutions of more complex differential equations may be obtained in future work.

Footnotes

Acknowledgements

This research is funded by the National Natural Science Foundation of China (Grant No. 11661076).

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