Applications of three-dimensional uncertain heat equations

Abstract

Uncertain heat equation is a partial differential equation describing temperature changes with time, while the strength of heat source is affected by the interference of noise. This paper applies the three-dimensional uncertain heat equation in some practical problems, including average temperature, maximum temperature, minimum temperature and minimum time of reaching a given temperature by using mathematical tools of time integral, space integral, extreme value and first hitting time.

1 Introduction

Real life is usually full of indeterminacy. For example, tomorrow’s stock price is unknown. Many things are undetermined, especially in the future. Due to this, real decisions are usually made under indeterminacy. Although indeterminacy is elusive, there are two mathematical systems used to deal with indeterminacy: One is probability theory, and the other is uncertainty theory. Probability theory is used to deal with frequency generated by historical data, while uncertainty theory found by Liu (2007) and perfected by Liu (2009) is used to deal with belief degree evaluated by experts.

Uncertain process as a sequence of uncertain variables indexed by totally ordered set was proposed by Liu (2008) to describe the evolution of uncertain phenomena based on uncertainty theory. As a special uncertain process, Liu process proposed by Liu (2009) is a Lipschitz continuous and stationary independent increment process whose increments are normal uncertain variables. Based on Liu process, Liu (2008) presented uncertain differential equation. Then, Chen and Liu (2010) proved the existence and uniqueness theorem for uncertain differential equation. For the linear uncertain differential equation, Chen and Liu (2010) successfully provided an analytic solution. However, since the general uncertain differential equation is complex and of many kinds, analytical solutions are usually unavailable. Fortunately, Yao and Chen (2013) constructed a bridge between the solution of an uncertain differential equation and a spectrum of ordinary differential equations. Due to the wonderful and vital work, the inverse uncertainty distribution of the solution can be obtained. Furthermore, Yao and Chen (2013) gave the expected value of the solution, and Yao (2013) calculated the time integral, extreme value and first hitting time of the solution.

Assume an uncertain process follows an uncertain differential equation and some realizations of this process are observed. In order to make a connection between uncertain differential equation and observed data, Liu and Liu (2022b) proposed the concept of residual. With the help of residuals, Ye and Liu (2022b) suggested an uncertain hypothesis test (Ye and Liu 2022a) to determine whether or not an uncertain differential equation fits the observed data. In practice, it is important to estimate unknown parameters in an uncertain differential equation to fit observed data as much as possible. For that purpose, the moment estimation (Yao and Liu 2020), minimum cover estimation (Yang et al. 2020), least squares estimation (Sheng et al. 2020) and maximum likelihood estimation (Liu and Liu 2022a) were proposed based on difference scheme. However, the above parameter estimation methods are not suitable for the case where the interval times between observations are not short enough. In order to overcome this shortage, the method of moments was revised by Liu and Liu (2022b) with the help of residuals. Up to now, uncertain differential equation has been widely applied in many fields such as chemical reaction (Tang and Yang 2021), electric circuit (Liu 2021), pharmacokinetics (Liu and Yang 2021), software reliability (Liu and Kang 2022), COVID-19 (Lio and Liu 2021) and Alibaba stock (Liu and Liu 2022b).

Uncertain partial differential equation driven by Liu process was first proposed by Yang and Yao (2017) when they studied one-dimensional uncertain heat equation describing temperature changes with time, while the strength of heat source is affected by the interference of noise which is described by Liu process. Then, Yang and Ni (2017) presented the existence and uniqueness theorem, and Yang (2018) discussed numerical methods of solving uncertain heat equations via numerical methods. However, the real world is a three-dimensional space. Thus, it is significant for Ye and Yang (2022) to study the three-dimensional uncertain heat equation as the extension of the one-dimensional uncertain heat equation. They gave a solution of a three-dimensional uncertain heat equation and its inverse uncertainty distribution. Based on their work, there are many applications since three-dimensional uncertain heat equation has certain practical significance. For example, when heating an iron rod with a heat source, we are sometimes interested in the average temperature and the maximum temperature of this iron rod or when the iron rod will be burned out. This paper is aimed to propose some formulas to calculate the time integral, space integral, expected value, extreme value and first hitting time of the solution of a three-dimensional uncertain heat equation.

The rest of the paper is organized as follows. Section 2 shows some basic concepts and theorems in uncertainty theory. Section 3 introduces three-dimensional uncertain heat equation. Section 4 gives some applications of the solution, including time integral, space integral, expected value, extreme value and first hitting time of the solution of a three-dimensional uncertain heat equation. Section 5 shows some examples of these applications. At last, some conclusions are made in Sect. 6.

2 Preliminaries

In this section, some basic concepts and theorems in uncertainty theory are introduced.

Definition 1

(Liu 2007) Let $\text{ L }$ be a $\sigma$-algebra on a nonempty set $\Gamma$. A real-valued set function $\text{ M }$ on $\text{ L }$ is called an uncertain measure if it satisfies the following three axioms:

Axiom 1. (Normality Axiom) $\text{ M }\{\Gamma \}=1$ for the universal set $\Gamma$.

Axiom 2. (Duality Axiom) $\text{ M }\{\Lambda \}+\text{ M }\{\Lambda ^{c}\}=1$ for any event $\Lambda$.

Axiom 3. (Subadditivity Axiom) For every countable sequence of events $\Lambda _1, \Lambda _2, \!\ldots ,$ we have

$$\begin{aligned}\displaystyle \text{ M }\left\{ \bigcup _{i=1}^{\infty }\Lambda _i\right\} \le \sum _{i=1}^{\infty }\text{ M }\{\Lambda _i\}.\end{aligned}$$

Then, the triplet $(\Gamma ,\text{ L },\text{ M})$ is called an uncertainty space. Another axiom about product uncertain measure was added by Liu (2009) as follows:

Axiom 4. (Product Axiom) Let $(\Gamma _k,\text{ L}_k,\text{ M}_k)$ be uncertainty spaces for $k=1, 2, \ldots$The product uncertain measure $\text{ M }$ is an uncertain measure satisfying

$$\begin{aligned} \displaystyle \text{ M }\left\{ \prod _{k=1}^\infty \Lambda _k\right\} =\bigwedge _{k=1}^\infty \text{ M}_k\{\Lambda _k\}\end{aligned}$$

where $\Lambda _k$ are arbitrarily chosen events from $\text{ L}_k$ for $k=1, 2, \ldots$,respectively.

An uncertain variable $\xi$ is a measurable function from an uncertainty space $(\Gamma ,\text{ L },\text{ M})$ to the set of real numbers such that the set $\{ \xi \in B \} = \{ \gamma \in \Gamma \ | \ \xi (\gamma ) \in B \}$ is an event for any Borel set B of real numbers. The uncertainty distribution $\Phi$ of an uncertain variable $\xi$ is defined as $\Phi (x)=\text{ M }\{ \xi \le x\}$ for any real number x. An uncertainty distribution $\Phi (x)$ is said to be regular (Liu 2010) if it is a continuous and strictly increasing function with respect to x at which $0<\Phi (x)<1$ and satisfies

$$\begin{aligned}\displaystyle \lim _{x \rightarrow -\infty } \Phi (x) = 0, \qquad \lim _{x \rightarrow \infty } \Phi (x) = 1. \end{aligned}$$

Then, the inverse function $\Phi ^{-1}(\alpha )$ of $\Phi (x)$ is said to be the inverse uncertainty distribution of $\xi$.

An uncertain variable $\xi$ is called normal if its uncertainty distribution is

$$\begin{aligned} \Phi (x)=\left( 1+\exp \left( \frac{\pi (e-x)}{\sqrt{3} \sigma } \right) \right) ^{-1}, x \in \Re \end{aligned}$$

denoted by ${\mathcal {N}}(e,\sigma )$ where e and $\sigma$ are real numbers satisfying $\sigma >0$, and the inverse uncertainty distribution of $\xi$ is

$$\begin{aligned} \Phi ^{-1}(\alpha )=e+\frac{\sigma \sqrt{3}}{\pi } \ln \frac{\alpha }{1-\alpha }. \end{aligned}$$

Definition 2

(Liu 2009) The uncertain variables $\xi _1$, $\xi _2$, $\ldots$, $\xi _n$ are said to be independent if

$$\begin{aligned} \text{ M }\left\{ \bigcap _{i=1}^{n} (\xi _{i} \in B_{i}) \right\} = \bigwedge _{i=1}^{n} \text{ M }\left\{ \xi _{i} \in B_{i} \right\} \end{aligned}$$

for any Borel sets $B_{1}$, $B_{2}$, $\ldots$, $B_{n}$ of real numbers.

Theorem 1

(Liu 2009) Suppose that $\xi _{1},\xi _{2},$ $\ldots ,\xi _{n}$ are independent uncertain variables with regular uncertainty distributions $\Phi _{1},\Phi _{2},$ $\ldots ,\Phi _{n}$, respectively, and $f(x_{1},x_{2},$ $\ldots ,x_{n})$ is a strictly increasing function with respect to $x_{1},x_{2},\ldots ,x_{m}$ and a strictly decreasing function with respect to $x_{m+1},x_{m+2},\ldots ,x_{n}$. Then, the inverse uncertainty distribution of $\xi =f(\xi _{1},\xi _{2},\ldots ,\xi _{n})$ is

$$\begin{aligned} \begin{aligned}&\Psi ^{-1}(\alpha )=\\&f(\Phi _{1}^{-1}(\alpha ),\ldots ,\Phi _{m}^{-1}(\alpha ),\Phi _{m+1}^{-1}(1-\alpha ),\ldots ,\Phi _{n}^{-1}(1-\alpha )). \end{aligned} \end{aligned}$$

Definition 3

(Liu 2007) Let $\xi$ be an uncertain variable. Then, the expected value of $\xi$ is defined as

$$\begin{aligned} E[\xi ] = \int _{0}^{+\infty } \text{ M }\{\xi \ge x \}\textrm{d} x - \int _{-\infty }^{0} \text{ M }\{\xi \le x \}\textrm{d} x \end{aligned}$$

provided that at least one of the two integrals is finite.

If the inverse uncertainty distribution $\Phi ^{-1}(\alpha )$ of $\xi$ exists, then

$$\begin{aligned} E[\xi ] = \int _{0}^{1} \Phi ^{-1}(\alpha ) \textrm{d}\alpha . \end{aligned}$$

(1)

An uncertain process is a sequence of uncertain variables indexed by time for modeling the evolution of uncertain phenomena.

Definition 4

(Liu 2008) Let $(\Gamma , \text{ L },\text{ M})$ be an uncertainty space, and let T be a totally ordered set. An uncertain process is a function $X_t(\gamma )$ from $T\times (\Gamma , \text{ L },\text{ M})$ to the set of real numbers such that $\{ X_t \in B\}$ is an event for any Borel set B of real numbers at each time t.

We call an uncertain process $X_t$ independent increment process if $X_{t_1},X_{t_2}-X_{t_1},X_{t_3}-X_{t_2},\ldots ,X_{t_k}-X_{t_{k-1}}$ are independent uncertain variables where $t_1$, $t_2,\ldots ,t_k$ are any times with $t_1<t_2<\ldots <t_k$. An uncertain process $X_t$ is said to have stationary increments if, for any given $t>0$, the increments $X_{t+s}-X_t$ are identically distributed uncertain variables for all $s>0$.

Definition 5

(Liu 2009) 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 expected value 0 and variance $t^2$.

The uncertainty distribution of $C_t$ is

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

and inverse uncertainty distribution is

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

Theorem 2

(Liu 2015) Let $C_t$ be a Liu process. Then for each time $t>0$, the ratio $C_t/t$ is a normal uncertain variable with expected value 0 and variance 1. That is,

$$\begin{aligned} \frac{C_t}{t}\sim {\mathcal {N}}(0,1) \end{aligned}$$

for any $t>0$.

Definition 6

(Liu 2009) Let $X_t$ be an uncertain process, and let $C_t$ be a Liu process. For any partition of closed interval [a, b] with $a=t_1<t_2<\ldots <t_{k+1}=b$, the mesh is written as

$$\begin{aligned} \Delta =\max _{1\le i\le k} |t_{i+1}-t_i|. \end{aligned}$$

Then, Liu integral of $X_t$ with respect to $C_t$ is defined as

$$\begin{aligned} \int _a^b X_t \textrm{d}C_t=\lim _{\Delta \rightarrow 0} \sum _{i=1}^k X_{t_i}\cdot (C_{t_{i+1}}-C_{t_{i}}) \end{aligned}$$

provided that the limit exists almost surely and is finite. In this case, the uncertain process $X_t$ is said to be integrable.

Definition 7

(Chen and Ralescu 2013; Ye 2021) 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 $\sigma _t$ such that

$$\begin{aligned} Z_t=Z_0+\int _0^t\mu _s\textrm{d}s+\int _0^t\sigma _s\textrm{d}C_s \end{aligned}$$

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

$$\begin{aligned} \textrm{d}Z_t=\mu _t\textrm{d}t+\sigma _t\textrm{d}C_t. \end{aligned}$$

Definition 8

(Liu 2008) Suppose f and g are continuous functions, and $C_t$ is a Liu process. Then,

$$\begin{aligned} \textrm{d}X_t=f(t,X_t)\textrm{d}t+g(t,C_t)\textrm{d}C_t \end{aligned}$$

(2)

is called an uncertain differential equation. A solution is an uncertain process $X_t$ that satisfies (2) identically in t.

3 Uncertain heat equation

Yang and Yao (2017) first presented a one-dimensional uncertain heat equation driven by Liu process. Following that, Ye and Yang (2022) studied a three-dimensional uncertain heat equation as follows,

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial u}{\partial t}-a^2 \left( \frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}+\frac{\partial ^2 u}{\partial z^2}\right) =q(t,x,y,z,{\dot{C}}_t)\\&u(0,x,y,z)=\varphi (x,y,z), \quad \quad t>0,x,y,z \in \Re \end{aligned} \right. \end{aligned}$$

(3)

where $a^2$ is a constant thermal diffusivity with $a>0$, ${\dot{C}}_t=\textrm{d}C_t/\textrm{d}t$ denotes the time white noise, $C_t$ is a Liu process defined in Definition 5, $q(t,x,y,z,{\dot{C}}_t)$ is a heat source, and $\varphi (x,y,z)$ is a given initial temperature at time $t=0$. Then, Ye and Yang (2022) gave the solution of the three-dimensional uncertain heat equation (3)

$$\begin{aligned} \begin{aligned}&u(t,x,y,z)\\&\quad =\iiint _{\Re ^3} K(t,x-\xi ,y-\eta ,z-\tau )\varphi (\xi ,\eta ,\tau ) \textrm{d}\xi \textrm{d}\eta \textrm{d}\tau \\&\quad \quad +\int _0^t \iiint _{\Re ^3} K(t-s,x-\xi ,y-\eta ,z-\tau )\\&\quad \quad q(s,\xi ,\eta ,\tau ,{\dot{C}}_s)\textrm{d}\xi \textrm{d}\eta \textrm{d}\tau \textrm{d}s \end{aligned} \end{aligned}$$

(4)

where

$$\begin{aligned} K(t,x,y,z)=\frac{1}{(4\pi a^2t)^{3/2}} \exp {\left( -\frac{x^2+y^2+z^2}{4a^2 t} \right) }. \end{aligned}$$

Furthermore, if q(t, x, y, z, c) in (3) is a strictly increasing for c, then the solution u(t, x, y, z) in (4) has an inverse uncertainty distribution

$$\begin{aligned} \begin{aligned}&\Psi _{t,x,y,z}^{-1}(\alpha )\\&\quad =\iiint _{\Re ^3} K(t,x-\xi ,y-\eta ,z-\tau )\varphi (\xi ,\eta ,\tau ) \textrm{d}\xi \textrm{d}\eta \textrm{d}\tau \\&\qquad +\int _0^t \iiint _{\Re ^3} K(t-s,x-\xi ,y-\eta ,z-\tau )\\&\quad \quad q(s,\xi ,\eta ,\tau ,\Phi ^{-1}(\alpha ))\textrm{d}\xi \textrm{d}\eta \textrm{d}\tau \textrm{d}s, \end{aligned} \end{aligned}$$

(5)

and if q(t, x, y, z, c) is a strictly decreasing for c, then the solution u(t, x, y, z) has an inverse uncertainty distribution

$$\begin{aligned} \begin{aligned}&\Psi _{t,x,y,z}^{-1}(\alpha )\\&\quad =\iiint _{\Re ^3} K(t,x-\xi ,y-\eta ,z-\tau )\varphi (\xi ,\eta ,\tau ) \textrm{d}\xi \textrm{d}\eta \textrm{d}\tau \\&\qquad +\int _0^t \iiint _{\Re ^3} K(t-s,x-\xi ,y-\eta ,z-\tau )\\&\quad \quad q(s,\xi ,\eta ,\tau ,\Phi ^{-1}(1-\alpha ))\textrm{d}\xi \textrm{d}\eta \textrm{d}\tau \textrm{d}s \end{aligned} \end{aligned}$$

(6)

where $\Phi ^{-1}(\alpha )$ is the inverse uncertainty distribution of ${\mathcal {N}}(0,1)$, i.e.,

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

Example 1

The three-dimensional uncertain heat equation

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial u}{\partial t}- \left( \frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}+\frac{\partial ^2 u}{\partial z^2}\right) ={\dot{C}}_t\\&u(0,x,y,z)=0,\quad \quad t>0,x,y,z \in \Re \end{aligned} \right. \end{aligned}$$

has a solution

$$\begin{aligned} u(t,x,y,z)=C_t \end{aligned}$$

whose inverse uncertainty distribution is

$$\begin{aligned} \Psi _{t,x,y,z}^{-1}(\alpha )={} & {} \frac{\sqrt{3}t}{\pi } \ln \frac{\alpha }{1-\alpha }. \end{aligned}$$

Example 2

The three-dimensional uncertain heat equation

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial u}{\partial t}-\left( \frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}+\frac{\partial ^2 u}{\partial z^2}\right) =-e^{-t}{\dot{C}}_t\\&u(0,x,y,z)=0,\quad \quad t>0,x,y,z \in \Re \end{aligned} \right. \end{aligned}$$

has a solution

$$\begin{aligned} u(t,x,y,z)=-\int _0^t e^{-s} \textrm{d}C_s \end{aligned}$$

whose inverse uncertainty distribution is

$$\begin{aligned} \Psi _{t,x,y,z}^{-1}(\alpha )=\frac{\sqrt{3}}{\pi }\left( 1-e^{-t}\right) \ln \frac{\alpha }{1-\alpha }. \end{aligned}$$

Example 3

The three-dimensional uncertain heat equation

$$\begin{aligned} \left\{ \begin{aligned}&\!\!\frac{\partial u}{\partial t}\!-\!\!\left( \!\!\frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}+\frac{\partial ^2 u}{\partial z^2}\!\!\right) \!\!=\!\!(\sin x+\sin y+\sin z+3\!){\dot{C}}_t\\&u(0,x,y,z)=0,\quad \quad t>0,x,y,z \in \Re \end{aligned} \right. \end{aligned}$$

has a solution

$$\begin{aligned} u(t,x,y,z)=e^{-t}(\sin x+\sin y+\sin z)\int _0^t e^s \textrm{d}C_s+3C_t \end{aligned}$$

whose inverse uncertainty distribution is

$$\begin{aligned} \begin{aligned}&\Psi _{t,x,y,z}^{-1}(\alpha )\\&\quad =\frac{\sqrt{3}}{\pi }\left[ (\sin x+\sin y+\sin z)\left( 1-e^{-t}\right) +3t\right] \ln \frac{\alpha }{1-\alpha }. \end{aligned} \end{aligned}$$

4 Applications of the solution

In this section, some applications of the solution are introduced. The first is the expected value in Sect. 4.1. The second is the average temperature including time integral in Sect. 4.2 and space time in Sect. 4.3. The third is the maximum temperature and the minimum temperature in Sect. 4.4. The last one is the first hitting time in Sect. 4.5.

At first, we introduce an important theorem as follows.

Theorem 3

Assume u(t, x, y, z) is the solution of the uncertain heat equation (3). If q(t, x, y, z, c) is strictly monotone (increasing or decreasing) for c, then

$$\begin{aligned}{} & {} \text{ M }\left\{ u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} =\alpha , \\{} & {} \text{ M }\left\{ u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} =1-\alpha , \end{aligned}$$

where $\Psi _{t,x,y,z}^{-1}(\alpha )$ is the inverse uncertainty distribution of the solution u(t, x, y, z).

Proof

Assume q(t, x, y, z, c) is strictly increasing for c. It follows from equation (5) that

$$\begin{aligned} \begin{aligned}&\Psi _{t,x,y,z}^{-1}(\alpha )\\&\quad =\iiint _{\Re ^3} K(t,x-\xi ,y-\eta ,z-\tau )\varphi (\xi ,\eta ,\tau ) \textrm{d}\xi \textrm{d}\eta \textrm{d}\tau \\&\quad \quad +\int _0^t \iiint _{\Re ^3} K(t-s,x-\xi ,y-\eta ,z-\tau )\\&\quad \quad q(s,\xi ,\eta ,\tau ,\Phi ^{-1}(\alpha ))\textrm{d}\xi \textrm{d}\eta \textrm{d}\tau \textrm{d}s \end{aligned} \end{aligned}$$

where

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

For each $\alpha \in (0,1)$, on the one hand, we get

$$\begin{aligned} \begin{aligned}&\text{ M }\left\{ u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} \\&\quad \ge \text{ M }\left\{ q(s,\xi ,\eta ,\tau ,{\dot{C}}_s) \right. \nonumber \\&\quad \left. \le q(s,\xi ,\eta ,\tau ,\Phi ^{-1}(\alpha )), \forall s, \xi ,\eta ,\tau \right\} \\&\quad =\text{ M }\left\{ {\dot{C}}_s \le \Phi ^{-1}(\alpha ), \forall s \right\} =\alpha . \end{aligned} \end{aligned}$$

On the other hand,

$$\begin{aligned} \begin{aligned}&\text{ M }\left\{ u(t,x,y,z)> \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} \\&\quad \ge \text{ M }\left\{ q(s,\xi ,\eta ,\tau ,{\dot{C}}_s)\right. \nonumber \\&\qquad \quad \left.> q(s,\xi ,\eta ,\tau ,\Phi ^{-1}(\alpha )), \forall s, \xi ,\eta ,\tau \right\} \\&\quad =\text{ M }\left\{ {\dot{C}}_s > \Phi ^{-1}(\alpha ), \forall s \right\} =1-\alpha . \end{aligned} \end{aligned}$$

Note that $\left\{ u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\}$ and $\left\{ u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha ), \exists t, \exists x,\exists y,\exists z \right\}$ are opposite

with each other. By using the duality axiom, we have

$$\begin{aligned} \begin{aligned}&\text{ M }\left\{ u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} \\&\quad +\text{ M }\left\{ u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha ), \exists t, \exists x,\exists y,\exists z \right\} =1. \end{aligned} \end{aligned}$$

It follows from

$$\begin{aligned} \begin{aligned}&\left\{ u(t,x,y,z)> \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} \\&\quad \subset \left\{ u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha ), \exists t, \exists x,\exists y,\exists z \right\} , \end{aligned} \end{aligned}$$

and the monotonicity theorem that

$$\begin{aligned} \begin{aligned}&\text{ M }\left\{ u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} \\&\quad +\text{ M }\left\{ u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} \le 1. \end{aligned} \end{aligned}$$

Therefore, we have

$$\begin{aligned}{} & {} \text{ M }\left\{ u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} =\alpha ,\\{} & {} \text{ M }\left\{ u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} =1-\alpha . \end{aligned}$$

Assume q(t, x, y, z, c) is strictly decreasing for c. It follows from equation (6) that

$$\begin{aligned} \begin{aligned}&\Psi _{t,x,y,z}^{-1}(\alpha )\\&\quad =\iiint _{\Re ^3} K(t,x-\xi ,y-\eta ,z-\tau )\varphi (\xi ,\eta ,\tau ) \textrm{d}\xi \textrm{d}\eta \textrm{d}\tau \\&\quad \quad +\int _0^t \iiint _{\Re ^3} K(t-s,x-\xi ,y-\eta ,z-\tau )\\&\quad \quad q(s,\xi ,\eta ,\tau ,\Phi ^{-1}(1-\alpha ))\textrm{d}\xi \textrm{d}\eta \textrm{d}\tau \textrm{d}s \end{aligned} \end{aligned}$$

where

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

For each $\alpha \in (0,1)$, on the one hand, we get

$$\begin{aligned} \begin{aligned}&\text{ M }\left\{ u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} \\&\quad \ge \text{ M }\left\{ q(s,\xi ,\eta ,\tau ,{\dot{C}}_s)\right. \nonumber \\&\quad \left. \le q(s,\xi ,\eta ,\tau ,\Phi ^{-1}(\alpha )), \forall s, \xi ,\eta ,\tau \right\} \\&\quad =\text{ M }\left\{ {\dot{C}}_s \ge \Phi ^{-1}(1-\alpha ), \forall s \right\} =1-(1-\alpha )=\alpha . \end{aligned} \end{aligned}$$

On the other hand,

$$\begin{aligned} \begin{aligned}&\text{ M }\left\{ u(t,x,y,z)> \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} \\&\quad \ge \text{ M }\!\left\{ q(s,\xi ,\eta ,\tau ,{\dot{C}}_s)\! \right. \nonumber \\&\qquad \quad \left. >\! q(s,\xi ,\eta ,\tau ,\Phi ^{-1}(1-\alpha )), \forall s, \xi ,\eta ,\tau \right\} \\&\quad =\text{ M }\left\{ {\dot{C}}_s < \Phi ^{-1}(1-\alpha ), \forall s \right\} =1-\alpha . \end{aligned} \end{aligned}$$

Note that $\left\{ u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\}$ and $\left\{ u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha ), \exists t, \exists x,\exists y,\exists z \right\}$ are opposite

with each other. By using the duality axiom, we have

$$\begin{aligned} \begin{aligned}&\text{ M }\left\{ u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} \\&\quad +\text{ M }\left\{ u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha ), \exists t, \exists x,\exists y,\exists z \right\} =1. \end{aligned} \end{aligned}$$

It follows from

$$\begin{aligned} \begin{aligned}&\left\{ u(t,x,y,z)> \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} \\&\quad \subset \left\{ u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha ), \exists t, \exists x,\exists y,\exists z \right\} , \end{aligned} \end{aligned}$$

and the monotonicity theorem that

$$\begin{aligned} \begin{aligned}&\text{ M }\left\{ u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} \\&\quad +\text{ M }\left\{ u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} \le 1. \end{aligned} \end{aligned}$$

Therefore, we have

$$\begin{aligned} \text{ M }\left\{ u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} =\alpha , \\ \text{ M }\left\{ u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha ), \forall t, \forall x,\forall y,\forall z \right\} =1-\alpha . \end{aligned}$$

The theorem is proved. $\square$

4.1 Expected value of the solution

Theorem 4

Assume u(t, x, y, z) is the solution of the uncertain heat equation (3). If q(t, x, y, z, c) is strictly monotone (increasing or decreasing) for c, then

$$\begin{aligned} E[u(t,x,y,z)]=\int _0^1 \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}\alpha , \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&\Psi _{t,x,y,z}^{-1}(\alpha )\\&\quad =\iiint _{\Re ^3} K(t,x-\xi ,y-\eta ,z-\tau )\varphi (\xi ,\eta ,\tau ) \textrm{d}\xi \textrm{d}\eta \textrm{d}\tau \\&\quad \quad +\int _0^t \iiint _{\Re ^3} K(t-s,x-\xi ,y-\eta ,z-\tau )\\&\quad \quad q(s,\xi ,\eta ,\tau ,\Phi ^{-1}(\alpha ))\textrm{d}\xi \textrm{d}\eta \textrm{d}\tau \textrm{d}s \end{aligned} \end{aligned}$$

and

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

Proof

On the one hand, if q(t, x, y, z, c) is strictly increasing for c, then we have

$$\begin{aligned} E[u(t,x,y,z)]=\int _0^1 \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}\alpha , \end{aligned}$$

since $\Psi _{t,x,y,z}^{-1}(\alpha )$ is the inverse uncertainty distribution of the solution u(t, x, y, z).

On the other hand, if q(t, x, y, z, c) is strictly decreasing for c, then we have

$$\begin{aligned} \begin{aligned} E[u(t,x,y,z)]&=\int _0^1 \Psi _{t,x,y,z}^{-1}(1-\alpha ) \textrm{d}\alpha \\&=\int _0^1 \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}\alpha \end{aligned} \end{aligned}$$

since $\Psi _{t,x,y,z}^{-1}(1-\alpha )$ is the inverse uncertainty distribution of the solution u(t, x, y, z). The theorem is proved. $\square$

4.2 Time integral of the solution

Time integral of the solution represents the average temperature over a certain period of time at a given position when the heat source provides or absorbs heat.

Theorem 5

Assume u(t, x, y, z) is the solution of the uncertain heat equation (3). If q(t, x, y, z, c) is strictly monotone (increasing or decreasing) for c, then for any time $T>0$, the average temperature over time [0, T]

$$\begin{aligned} \frac{1}{T}\int _0^T u(t,x,y,z) \textrm{d}t \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \Upsilon _{x,y,z}^{-1}(\alpha )=\frac{1}{T} \int _0^T \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}t \end{aligned}$$

where $\Psi _{t,x,y,z}^{-1}(\alpha )$ is the inverse uncertainty distribution of the solution u(t, x, y, z).

Proof

For any given time $T>0$ and any position $(x, y, z)\in \Re ^3$, it follows from the basic property of time integral that

$$\begin{aligned} \begin{aligned}&\left\{ \frac{1}{T}\int _0^T u(t,x,y,z) \textrm{d}t \le \frac{1}{T} \int _0^T \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}t\right\} \\&\quad \supset \{u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ),\forall t, \forall x,\forall y,\forall z \}. \end{aligned} \end{aligned}$$

By using Theorem 3, we obtain

$$\begin{aligned} \begin{aligned}&\text{ M }\left\{ \frac{1}{T}\int _0^T u(t,x,y,z) \textrm{d}t \le \frac{1}{T} \int _0^T \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}t\right\} \\&\quad \ge \!\text{ M }\{u(t,x,y,z)\! \le \! \Psi _{t,x,y,z}^{-1}(\alpha ),\!\forall t, \!\forall x,\!\forall y,\!\forall z \}=\alpha . \end{aligned} \end{aligned}$$

Similarly, we have

$$\begin{aligned} \begin{aligned}&\text{ M }\left\{ \frac{1}{T}\int _0^T u(t,x,y,z) \textrm{d}t> \frac{1}{T} \int _0^T \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}t\right\} \\&\quad \ge \text{ M }\{u(t,x,y,z) \!>\! \Psi _{t,x,y,z}^{-1}(\alpha ),\!\forall t, \!\forall x,\!\forall y,\!\forall z \}\\&\quad =1-\alpha . \end{aligned} \end{aligned}$$

It follows from the duality axiom that

$$\begin{aligned} \text{ M }\left\{ \frac{1}{T}\int _0^T u(t,x,y,z) \textrm{d}t \le \frac{1}{T} \int _0^T \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}t\right\} =\alpha . \end{aligned}$$

The theorem is proved. $\square$

4.3 Space integral of the solution

Space integral of the solution represents the average temperature of the object at a given time when the heat source provides or absorbs heat.

Theorem 6

Assume u(t, x, y, z) is the solution of the uncertain heat equation (3). If q(t, x, y, z, c) is strictly monotone (increasing or decreasing) for c, then the average temperature of the bounded region $D\subset \Re ^3$ at time $t>0$

$$\begin{aligned} \frac{1}{|D|}\iiint _D u(t,x,y,z) \textrm{d}x \textrm{d}y \textrm{d}z \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \Upsilon _{t}^{-1}(\alpha )=\frac{1}{|D|}\iiint _D \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}x \textrm{d}y \textrm{d}z \end{aligned}$$

where $\Psi _{t,x,y,z}^{-1}(\alpha )$ is the inverse uncertainty distribution of the solution u(t, x, y, z), and |D| represents the volume of D.

Proof

For any given time $T>0$, it follows from the basic property of time integral that

$$\begin{aligned} \begin{aligned}&\bigg \{ \frac{1}{|D|}\iiint _D u(t,x,y,z) \textrm{d}x \textrm{d}y \textrm{d}z \\&\quad \le \frac{1}{|D|}\iiint _D \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}x \textrm{d}y \textrm{d}z\bigg \}\\&\quad \supset \{u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ),\forall t, \forall x,\forall y,\forall z \}. \end{aligned} \end{aligned}$$

By using Theorem 3, we obtain

$$\begin{aligned} \begin{aligned}&\text{ M }\bigg \{\frac{1}{|D|}\iiint _D u(t,x,y,z) \textrm{d}x \textrm{d}y \textrm{d}z \\&\quad \le \frac{1}{|D|}\iiint _D \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}x \textrm{d}y \textrm{d}z\bigg \}\\&\quad \ge \text{ M }\{u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ),\forall t, \forall x,\forall y,\forall z \}=\alpha . \end{aligned} \end{aligned}$$

Similarly, we have

$$\begin{aligned} \begin{aligned}&\text{ M }\bigg \{\frac{1}{|D|}\iiint _D u(t,x,y,z) \textrm{d}x \textrm{d}y \textrm{d}z \\&\quad> \frac{1}{|D|}\iiint _D \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}x \textrm{d}y \textrm{d}z\bigg \}\\&\quad \ge \text{ M }\{u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha ),\forall t, \forall x,\forall y,\forall z \}\\&\quad =1-\alpha . \end{aligned} \end{aligned}$$

It follows from the duality axiom that

$$\begin{aligned} \begin{aligned}&\text{ M }\bigg \{\frac{1}{|D|}\iiint _D u(t,x,y,z) \textrm{d}x \textrm{d}y \textrm{d}z \\&\quad \le \frac{1}{|D|}\iiint _D \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}x \textrm{d}y \textrm{d}z\bigg \}=\alpha . \end{aligned} \end{aligned}$$

The theorem is proved. $\square$

4.4 Supremum and infimum of the solution

Supremum of the solution represents the maximum temperature over a certain period of time when the heat source provides or absorbs heat, and infimum of the solution represents the minimum temperature.

Theorem 7

Assume u(t, x, y, z) is the solution of the uncertain heat equation (3). If q(t, x, y, z, c) is strictly monotone (increasing or decreasing) for c, then the maximum temperature over time [0, T]

$$\begin{aligned} \sup _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z) \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \Upsilon ^{-1}(\alpha )=\sup _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha ) \end{aligned}$$

and the minimum temperature over time [0, T]

$$\begin{aligned} \inf _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z) \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \Upsilon ^{-1}(\alpha )=\inf _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha ) \end{aligned}$$

where $\Psi _{t,x,y,z}^{-1}(\alpha )$ is the inverse uncertainty distribution of the solution u(t, x, y, z).

Proof

For any given time $t>0$, it follows from the basic property of time integral that

$$\begin{aligned} \begin{aligned}&\left\{ \sup _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z) \le \sup _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha )\right\} \\&\quad \supset \{u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ),\forall t, \forall x,\forall y,\forall z \}. \end{aligned} \end{aligned}$$

By using Theorem 3, we obtain

$$\begin{aligned} \begin{aligned}&\text{ M }\left\{ \sup _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z) \le \sup _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha )\right\} \\&\quad \ge \text{ M }\{u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ),\forall t, \forall x,\forall y,\forall z \}=\alpha . \end{aligned} \end{aligned}$$

Similarly, we have

$$\begin{aligned} \begin{aligned}&\text{ M }\left\{ \sup _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z)> \sup _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha )\right\} \\&\quad \ge \!\text{ M }\{u(t,x,y,z)\! >\! \Psi _{t,x,y,z}^{-1}(\alpha ),\!\forall t, \!\forall x,\!\forall y,\!\forall z \}\!=\!1-\alpha . \end{aligned} \end{aligned}$$

It follows from the duality axiom that

$$\begin{aligned} \text{ M }\left\{ \sup _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z) \le \sup _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha )\right\} =\alpha . \end{aligned}$$

Next, it follows from the basic property of time integral that

$$\begin{aligned} \begin{aligned}&\left\{ \inf _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z) \le \inf _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha )\right\} \\&\quad \supset \{u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ),\forall t, \forall x,\forall y,\forall z \}. \end{aligned} \end{aligned}$$

By using Theorem 3, we obtain

$$\begin{aligned} \begin{aligned}&\text{ M }\left\{ \inf _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z) \le \inf _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha )\right\} \\&\quad \ge \text{ M }\{u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha ),\forall t, \forall x,\forall y,\forall z \}=\alpha . \end{aligned} \end{aligned}$$

Similarly, we have

$$\begin{aligned} \begin{aligned}&\text{ M }\left\{ \inf _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z)> \inf _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha )\right\} \\&\quad \ge \text{ M }\{u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha ),\forall t, \forall x,\forall y,\forall z \}\\ {}&\quad =1-\alpha . \end{aligned} \end{aligned}$$

It follows from the duality axiom that

$$\begin{aligned} \text{ M }\left\{ \inf _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z) \le \inf _{\begin{array}{c} 0\le t \le T\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha )\right\} =\alpha . \end{aligned}$$

The theorem is proved. $\square$

4.5 First hitting time of the solution

First hitting time of the solution represents how long the object takes to reach a given temperature for the first time when the heat source provides or absorbs heat.

Theorem 8

Assume u(t, x, y, z) is the solution of the uncertain heat equation (3). If q(t, x, y, z, c) is strictly monotone (increasing or decreasing) for c, then for any given temperature L, the first hitting time

$$\begin{aligned} \tau _L=\inf \left\{ t \ge 0 | \sup _{(x,y,z)\in \Re ^3} u(t,x,y,z)=L\right\} \end{aligned}$$

has an uncertainty distribution

$$\begin{aligned} \Upsilon (s)={\left\{ \begin{array}{ll} \displaystyle 1-\inf \left\{ \alpha | \sup _{0\le t \le s}\sup _{(x,y,z)\in \Re ^3} \Psi _{t,x,y,z}^{-1}(\alpha ) \ge L\right\} ,\\ \displaystyle \text {if } L>\sup _{(x,y,z)\in \Re ^3}\varphi (x,y,z)\\ \displaystyle \sup \left\{ \alpha | \inf _{0\le t \le s}\sup _{(x,y,z)\in \Re ^3} \Psi _{t,x,y,z}^{-1}(\alpha ) \le L\right\} , \\ \displaystyle \text {if } L<\sup _{(x,y,z)\in \Re ^3}\varphi (x,y,z) \end{array}\right. } \end{aligned}$$

where $\Psi _{t,x,y,z}^{-1}(\alpha )$ is the inverse uncertainty distribution of the solution u(t, x, y, z).

Proof

At first, suppose

$$\begin{aligned} L>\sup _{(x,y,z)\in \Re ^3}\varphi (x,y,z), \end{aligned}$$

and write

$$\begin{aligned} \alpha _0=\inf \left\{ \alpha | \sup _{0\le t \le s}\sup _{(x,y,z)\in \Re ^3} \Psi _{t,x,y,z}^{-1}(\alpha ) \ge L\right\} . \end{aligned}$$

Then,

$$\begin{aligned}{} & {} \sup _{0\le t \le s} \sup _{(x,y,z)\in \Re ^3} \Psi _{t,x,y,z}^{-1}(\alpha _0) = L,\\{} & {} \begin{aligned} \{\tau _L\le s\}&=\left\{ \sup _{0\le t \le s}\sup _{(x,y,z)\in \Re ^3} u(t,x,y,z) \ge L \right\} \\&\supset \{u(t,x,y,z) \ge \Psi _{t,x,y,z}^{-1}(\alpha _0),\forall t, \forall x,\forall y,\forall z \}, \end{aligned}\\{} & {} \begin{aligned} \{\tau _L> s\}&=\left\{ \sup _{0\le t \le s}\sup _{(x,y,z)\in \Re ^3} u(t,x,y,z)< L \right\} \\&\supset \{u(t,x,y,z) < \Psi _{t,x,y,z}^{-1}(\alpha _0),\forall t, \forall x,\forall y,\forall z \}. \end{aligned} \end{aligned}$$

By using Theorem 3, we obtain

$$\begin{aligned} \begin{aligned}&\text{ M }\{\tau _L\le s\} \\&\quad \ge \text{ M }\{u(t,x,y,z) \ge \Psi _{t,x,y,z}^{-1}(\alpha _0),\forall t, \forall x,\forall y,\forall z \}\\&\quad =1-\alpha _0, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&\text{ M }\{\tau _L> s\}\\&\quad \ge \text{ M }\{u(t,x,y,z) < \Psi _{t,x,y,z}^{-1}(\alpha _0),\forall t, \forall x,\forall y,\forall z \}\\&\quad =\alpha _0. \end{aligned} \end{aligned}$$

It follows from $\text{ M }\{\tau _L\le s\}+\text{ M }\{\tau _L> s\}=1$ that $\text{ M }\{\tau _L\le s\}=1-\alpha _0$. Hence, the first hitting time $\tau _L$ has an uncertainty distribution

$$\begin{aligned} \begin{aligned} \Upsilon _{x,y,z}(s)&=\text{ M }\{\tau _L\le s\}=1-\alpha _0\\&=1-\inf \left\{ \alpha | \sup _{0\le t \le s} \sup _{(x,y,z)\in \Re ^3}\Psi _{t,x,y,z}^{-1}(\alpha ) \ge L\right\} \!. \end{aligned} \end{aligned}$$

Similarly, suppose $L<u(0,x,y,z)$ and write

$$\begin{aligned} \alpha _0=\sup \left\{ \alpha |\inf _{0\le t \le s} \sup _{(x,y,z)\in \Re ^3} \Psi _{t,x,y,z}^{-1}(\alpha ) \le L\right\} . \end{aligned}$$

Then,

$$\begin{aligned}{} & {} \inf _{0\le t \le s}\sup _{(x,y,z)\in \Re ^3} \Psi _{t,x,y,z}^{-1}(\alpha _0) = L, \\{} & {} \begin{aligned}&\{\tau _L\le s\}=\left\{ \inf _{0\le t \le s}\sup _{(x,y,z)\in \Re ^3} u(t,x,y,z) \le L \right\} \\&\quad \supset \{u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha _0),\forall t, \forall x,\forall y,\forall z \}, \end{aligned} \\{} & {} \begin{aligned}&\{\tau _L> s\}=\left\{ \inf _{0\le t \le s}\sup _{(x,y,z)\in \Re ^3} u(t,x,y,z)> L \right\} \\&\quad \supset \{u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha _0),\forall t, \forall x,\forall y,\forall z \}. \end{aligned} \end{aligned}$$

By using Theorem 3, we obtain

$$\begin{aligned} \begin{aligned}&\text{ M }\{\tau _L\le s\} \\&\quad \ge \text{ M }\{u(t,x,y,z) \le \Psi _{t,x,y,z}^{-1}(\alpha _0),\forall t, \forall x,\forall y,\forall z \}\\&\quad =1-\alpha _0, \end{aligned} \\ \begin{aligned}&\text{ M }\{\tau _L> s\}\\&\quad \ge \text{ M }\{u(t,x,y,z) > \Psi _{t,x,y,z}^{-1}(\alpha _0),\forall t, \forall x,\forall y,\forall z \}\\&\quad =\alpha _0. \end{aligned} \end{aligned}$$

It follows from $\text{ M }\{\tau _L\le s\}+\text{ M }\{\tau _L> s\}=1$ that $\text{ M }\{\tau _L\le s\}=\alpha _0$. Hence, the first hitting time $\tau _L$ has an uncertainty distribution

$$\begin{aligned} \begin{aligned} \Upsilon (s)&=\text{ M }\{\tau _L\le s\}=\alpha _0\\&=\sup \left\{ \alpha | \inf _{0\le t \le s} \sup _{(x,y,z)\in \Re ^3}\Psi _{t,x,y,z}^{-1}(\alpha ) \le L\right\} . \end{aligned} \end{aligned}$$

The theorem is proved. $\square$

4.6 Further discussion

Uncertain heat equation is a tool to describe temperature changes of dynamic systems with time, while the strength of heat source is not certain but is affected by noise interference due to changes in the environment and different materials. In industry, there are many heating or cooling processes, such as heating processes of rubber or steel forgings, that are modeled by uncertain heat equations for further heat conduction analysis. With the help of uncertain heat equations, uncertainty distributions of future temperature over time at various locations can be predicted. Based on these uncertainty distributions, we can calculate some important process indicators of temperature which are shown as follows.

(1) Average temperature. This indicator represents the average temperature over a period of time or over an area when the heat source provides or absorbs heat. A lot of heating and cooling processes need to be concerned with this indicator. The tools of the time integral and the space integral introduced, respectively, in Sect. 4.2 and Sect. 4.3 can be used to model the average temperature of a heating or cooling process and predict the average temperature in a period of time in the future to provide a reasonable basis for dynamic adjustment of heat source strength.

(2) Extreme temperature. This indicator represents the extreme temperature including maximum temperature and minimum temperature when the heat source provides or absorbs heat. This is an important indicator to ensure the safety of the heating or cooling processes. The traditional method is to monitor the temperature of a location in real time to ensure that it does not exceed the safe range. However, it is almost impossible to measure the temperature in an area in real time. In this case, we can use the tool of extreme temperature introduced in Sect. 4.4 to simulate the maximum or minimum temperature of the system for the safety of the system and predict the maximum or minimum temperature of the system to control the heat source in advance for avoiding temperature exceeding the safe range.

(3) First hitting time. This indicator represents how long the object takes to reach a given temperature for the first time when the heat source provides or absorbs heat. The tools of the first hitting time introduced in Sect. 4.5 can be used to predict when the system will reach a specified or critical temperature for a given heat source to make full use of the heat source or ensure safety.

5 Examples

Example 4

It follows from Example 1 that the uncertain heat equation

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial u}{\partial t}- \left( \frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}+\frac{\partial ^2 u}{\partial z^2}\right) ={\dot{C}}_t\\&u(0,x,y,z)=0,\quad \quad t>0,x,y,z \in \Re \end{aligned} \right. \end{aligned}$$

has a solution

$$\begin{aligned} u(t,x,y,z)=C_t, \end{aligned}$$

with inverse uncertainty distribution

$$\begin{aligned} \Psi _{t,x,y,z}^{-1}(\alpha )=\frac{\sqrt{3}t}{\pi } \ln \frac{\alpha }{1-\alpha }. \end{aligned}$$

By using Theorem 4, the expected value of the solution is

$$\begin{aligned} E[u(t,x,y,z)]=\int _0^1 \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}\alpha =0. \end{aligned}$$

It follows from Theorem 5 that the average temperature over time [0, 1]

$$\begin{aligned} \int _0^1 u(t,x,y,z) \textrm{d}t \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \Upsilon _{x,y,z}^{-1}(\alpha )= \int _0^1 \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}t=\frac{\sqrt{3}}{2\pi } \ln \frac{\alpha }{1-\alpha }, \end{aligned}$$

which is shown in Fig. 1.

Fig. 1

Inverse uncertainty distribution of the average temperature over time [0, 1] in Example 4

From Theorem 6, the average temperature of the region $(0,1)\times (0,1)\times (0,1)$ at time $t>0$

$$\begin{aligned} \int _0^1\int _0^1\int _0^1 u(t,x,y,z) \textrm{d}x \textrm{d}y \textrm{d}z \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \Upsilon _{t}^{-1}(\alpha )=\int _0^1\int _0^1\int _0^1 \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}x \textrm{d}y \textrm{d}z\!=\!\frac{\sqrt{3}t}{\pi } \ln \frac{\alpha }{1-\alpha }, \end{aligned}$$

which is shown in Fig. 2.

Fig. 2

Inverse uncertainty distribution of the average temperature of the region $(0,1)\times (0,1)\times (0,1)$ in Example 4

It follows from Theorem 7 that the maximum temperature over time [0, 1]

$$\begin{aligned} \sup _{\begin{array}{c} 0\le t \le 1\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z) \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \begin{aligned} \Upsilon _{\text {max}}^{-1}(\alpha )&=\sup _{\begin{array}{c} 0\le t \le 1\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha )\\&=\left\{ \begin{array}{cl} 0,&{}\text {if} ~0<\alpha \le 0.5\\ \displaystyle \frac{\sqrt{3}}{\pi } \ln \frac{\alpha }{1-\alpha },&{}\text {if} ~0.5<\alpha <1 \end{array}\right. \end{aligned} \end{aligned}$$

which is shown in Fig. 3, and the minimum temperature over time [0, 1]

$$\begin{aligned} \inf _{\begin{array}{c} 0\le t \le 1 \\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z) \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \begin{aligned} \Upsilon _{\text {min}}^{-1}(\alpha )&=\inf _{\begin{array}{c} 0\le t \le 1\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha )\\&=\left\{ \begin{array}{cl} \displaystyle \frac{\sqrt{3}}{\pi } \ln \frac{\alpha }{1-\alpha },&{}\text {if} ~0<\alpha \le 0.5\\ 0,&{}\text {if} ~0.5<\alpha <1. \end{array}\right. \end{aligned} \end{aligned}$$

which is shown in Fig. 4.

Fig. 3

Inverse uncertainty distribution of the maximum temperature over time [0, 1] in Example 4

Fig. 4

Inverse uncertainty distribution of the minimum temperature over time [0, 1] in Example 4

By using Theorem 8, the time when the iron rod was first burned out, i.e., the temperature first reached 1,538 $^{\circ }$C (i.e., 2,800 $^{\circ }$F),

$$\begin{aligned} \tau _L=\inf \left\{ t \ge 0 | \sup _{(x,y,z)\in \Re ^3} u(t,x,y,z)=1538\right\} , \end{aligned}$$

has an uncertainty distribution

$$\begin{aligned} \Upsilon (s)=\left( 1+\exp {\left( \frac{1538\pi }{\sqrt{3}s} \right) } \right) ^{-1}, \end{aligned}$$

which is shown in Fig. 5. Note that

$$\begin{aligned} \Upsilon (s)<(1+1)^{-1}=0.5. \end{aligned}$$

That means the iron rod is no more than half likely to break no matter how long it is heated.

Fig. 5

Uncertainty distribution of the first hitting time in Example 4

Example 5

It follows from Example 2 that the uncertain heat equation

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial u}{\partial t}- \left( \frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}+\frac{\partial ^2 u}{\partial z^2}\right) =-e^{-t}{\dot{C}}_t\\&u(0,x,y,z)=0,\quad \quad t>0,x,y,z \in \Re \end{aligned} \right. \end{aligned}$$

has a solution

$$\begin{aligned} u(t,x,y,z)=-\int _0^t e^{-s} \textrm{d}C_s \end{aligned}$$

with inverse uncertainty distribution

$$\begin{aligned} \Psi _{t,x,y,z}^{-1}(\alpha )=\frac{\sqrt{3}}{\pi }\left( 1-e^{-t}\right) \ln \frac{\alpha }{1-\alpha }. \end{aligned}$$

By using Theorem 4, the expected value of the solution is

$$\begin{aligned} E[u(t,x,y,z)]=\int _0^1 \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}\alpha =0. \end{aligned}$$

It follows from Theorem 5 that the average temperature over time [0, 1]

$$\begin{aligned} \int _0^1 u(t,x,y,z) \textrm{d}t \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \Upsilon _{x,y,z}^{-1}(\alpha )= \int _0^1 \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}t=\frac{\sqrt{3}}{\pi }(1+e^{-1}) \ln \frac{\alpha }{1-\alpha }, \end{aligned}$$

which is shown in Fig. 6.

Fig. 6

Inverse uncertainty distribution of the average temperature over time [0, 1] in Example 5

From Theorem 6, the average temperature of the region $(0,1)\times (0,1)\times (0,1)$ at time $t>0$

$$\begin{aligned} \int _0^1\int _0^1\int _0^1 u(t,x,y,z) \textrm{d}x \textrm{d}y \textrm{d}z \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \begin{aligned} \Upsilon _{t}^{-1}(\alpha )&=\int _0^1\int _0^1\int _0^1 \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}x \textrm{d}y \textrm{d}z\\&=\frac{\sqrt{3}}{\pi }\left( 1-e^{-t}\right) \ln \frac{\alpha }{1-\alpha }, \end{aligned} \end{aligned}$$

which is shown in Fig. 7.

Fig. 7

Inverse uncertainty distribution of the average temperature of the region $(0,1)\times (0,1)\times (0,1)$ in Example 5

It follows from Theorem 7 that the maximum temperature over time [0, 1]

$$\begin{aligned} \sup _{\begin{array}{c} 0\le t \le 1\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z) \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \begin{aligned} \Upsilon _{\text {max}}^{-1}(\alpha )&=\sup _{\begin{array}{c} 0\le t \le 1\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha )\\&=\left\{ \begin{array}{cl} 0,&{}\text {if} ~0<\alpha \le 0.5\\ \displaystyle \frac{\sqrt{3}}{\pi }(1-e^{-1}) \ln \frac{\alpha }{1-\alpha },&{}\text {if} ~0.5<\alpha <1 \end{array}\right. \end{aligned} \end{aligned}$$

which is shown in Fig. 8, and the minimum temperature over time [0, 1]

$$\begin{aligned} \inf _{\begin{array}{c} 0\le t \le 1\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z) \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \begin{aligned} \Upsilon _{\text {min}}^{-1}(\alpha )&=\inf _{\begin{array}{c} 0\le t \le 1\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha )\\&=\left\{ \begin{array}{cl} \displaystyle \frac{\sqrt{3}}{\pi }(1-e^{-1}) \ln \frac{\alpha }{1-\alpha },&{}\text {if} ~0<\alpha \le 0.5\\ 0,&{}\text {if} ~0.5<\alpha <1, \end{array}\right. \end{aligned} \end{aligned}$$

which is shown in Fig. 9.

Fig. 8

Inverse uncertainty distribution of the maximum temperature over time [0, 1] in Example 5

Fig. 9

Inverse uncertainty distribution of the minimum temperature over time [0, 1] in Example 5

By using Theorem 8, the time when the iron rod was first burned out, i.e., the temperature first reached 1,538 $^{\circ }$C,

$$\begin{aligned} \tau _L=\inf \left\{ t \ge 0 | \sup _{(x,y,z)\in \Re ^3} u(t,x,y,z)=1538\right\} , \end{aligned}$$

has an uncertainty distribution

$$\begin{aligned} \Upsilon (s)=\left( 1+\exp {\left( \frac{1538\pi }{\sqrt{3}(1-\exp {(-s)})} \right) } \right) ^{-1}. \end{aligned}$$

Note that

$$\begin{aligned} \Upsilon (s)\le \left( 1+\exp {\left( \frac{1538\pi }{\sqrt{3}(1-0)} \right) } \right) ^{-1}<10^{-100}. \end{aligned}$$

That means the iron rod is almost impossible to burn out. Since the strength of heat source is

$$\begin{aligned} -e^{-t}{\dot{C}}_t \end{aligned}$$

whose uncertainty distribution is $\displaystyle {\mathcal {N}}\left( 0,e^{-t}\right)$, it is almost impossible for the heat source to burn the iron rod to 1538 $^{\circ }$C. Thus, the uncertainty distribution of the first hitting time $\tau _L$ is in line with expectations.

Example 6

It follows from Example 3 that the uncertain heat equation

$$\begin{aligned} \left\{ \begin{aligned}&\!\!\frac{\partial u}{\partial t}\!-\!\!\left( \!\!\frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}+\frac{\partial ^2 u}{\partial z^2}\!\!\right) \!\!=\!\!(\sin x+\sin y+\sin z+\!3){\dot{C}}_t\\&u(0,x,y,z)=0,\quad \quad t>0,x,y,z \in \Re \end{aligned} \right. \end{aligned}$$

has a solution

$$\begin{aligned} u(t,x,y,z)=e^{-t}(\sin x+\sin y+\sin z)\int _0^t e^s \textrm{d}C_s+3C_t, \end{aligned}$$

with inverse uncertainty distribution

$$\begin{aligned} \begin{aligned}&\Psi _{t,x,y,z}^{-1}(\alpha )\\&=\frac{\sqrt{3}}{\pi }\left[ (\sin x+\sin y+\sin z)\left( 1-e^{-t}\right) +3t\right] \ln \frac{\alpha }{1-\alpha }. \end{aligned} \end{aligned}$$

By using Theorem 4, the expected value of the solution is

$$\begin{aligned} E[u(t,x,y,z)]=\int _0^1 \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}\alpha =0. \end{aligned}$$

It follows from Theorem 5 that the average temperature over time [0, 1]

$$\begin{aligned} \int _0^1 u(t,x,y,z) \textrm{d}t \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \begin{aligned}&\Upsilon _{x,y,z}^{-1}(\alpha )= \int _0^1 \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}t\\&=\frac{\sqrt{3}}{2\pi }[2(\sin x+\sin y+\sin z)(1+e^{-1})+3] \ln \frac{\alpha }{1-\alpha }, \end{aligned} \end{aligned}$$

which is shown in Fig. 10.

Fig. 10

Inverse uncertainty distribution of the average temperature over time [0, 1] in Example 6

From Theorem 6, the average temperature of the region $(0,1)\times (0,1)\times (0,1)$ at time $t>0$

$$\begin{aligned} \int _0^1\int _0^1\int _0^1 u(t,x,y,z) \textrm{d}x \textrm{d}y \textrm{d}z \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \begin{aligned} \Upsilon _{t}^{-1}(\alpha )&=\int _0^1\int _0^1\int _0^1 \Psi _{t,x,y,z}^{-1}(\alpha ) \textrm{d}x \textrm{d}y \textrm{d}z\\&=\frac{3\sqrt{3}}{\pi }[(1-e^{-t})(1-\cos 1)+t] \ln \frac{\alpha }{1-\alpha }, \end{aligned} \end{aligned}$$

which is shown in Fig. 11.

Fig. 11

Inverse uncertainty distribution of the average temperature of the region $(0,1)\times (0,1)\times (0,1)$ in Example 6

It follows from Theorem 7 that the maximum temperature over time [0, 1]

$$\begin{aligned} \sup _{\begin{array}{c} 0\le t \le 1\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z) \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \begin{aligned} \Upsilon _{\text {max}}^{-1}(\alpha )&=\sup _{\begin{array}{c} 0\le t \le 1\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha )\\&=\left\{ \begin{array}{cl} 0,&{}\text {if} ~0<\alpha \le 0.5\\ \displaystyle \frac{3\sqrt{3}}{\pi }(2-e^{-1}) \ln \frac{\alpha }{1-\alpha },&{}\text {if} ~0.5<\alpha <1 \end{array}\right. \end{aligned} \end{aligned}$$

which is shown in Fig. 12, and the minimum temperature over time [0, 1]

$$\begin{aligned} \inf _{\begin{array}{c} 0\le t \le 1\\ (x,y,z)\in \Re ^3 \end{array}} u(t,x,y,z) \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \begin{aligned} \Upsilon _{\text {min}}^{-1}(\alpha )&=\inf _{\begin{array}{c} 0\le t \le 1\\ (x,y,z)\in \Re ^3 \end{array}} \Psi _{t,x,y,z}^{-1}(\alpha )\\&=\left\{ \begin{array}{cl} \displaystyle \frac{3\sqrt{3}}{\pi }(2-e^{-1}) \ln \frac{\alpha }{1-\alpha },&{}\text {if} ~0<\alpha \le 0.5\\ 0,&{}\text {if} ~0.5<\alpha <1 \end{array}\right. \end{aligned} \end{aligned}$$

which is shown in Fig. 13.

Fig. 12

Inverse uncertainty distribution of the maximum temperature over time [0, 1] in Example 6

Fig. 13

Inverse uncertainty distribution of the minimum temperature over time [0, 1] in Example 6

By using Theorem 8, the time when the iron rod was first burned out, i.e., the temperature first reached 1,538 $^{\circ }$C,

$$\begin{aligned} \tau _L=\inf \left\{ t \ge 0 | \sup _{(x,y,z)\in \Re ^3} u(t,x,y,z)=1538\right\} , \end{aligned}$$

has an uncertainty distribution

$$\begin{aligned} \Upsilon (s)=\left( 1+\exp {\left( \frac{1538\pi }{3\sqrt{3}(1+s-\exp {(-s)})}\right) } \right) ^{-1} \end{aligned}$$

which is shown in Fig. 14. Note that

$$\begin{aligned} \Upsilon (s)<(1+1)^{-1}=0.5. \end{aligned}$$

That means the iron rod is no more than half likely to break no matter how long it is heated.

Fig. 14

Uncertainty distribution of the first hitting time in Example 6

6 Conclusion

Three-dimensional uncertain heat equation is a kind of partial differential equation describing heat transfer in real life, while heat source is affected by the uncertain interference. This paper applied the solution of a three-dimensional uncertain heat equation in some issues, such as average temperature, maximum temperature, minimum temperature and minimum time of reaching a given temperature, with the help of some mathematical concepts like time integral, space integral, extreme value and first hitting time. Additionally, parameter estimation in three-dimensional uncertain heat equations can be further studied to obtain a three-dimensional uncertain heat equation that fits the observed data as much as possible.