近年来,分数阶微分方程理论已为研究非线性问题的热点,在弹性力学,非牛顿流体力学、振荡、扩散和输运理论、量子力学等领域中都有广泛应用^[1-7].例如,钙离子在细胞内扩散的时空分数阶反常扩散模型,带有热流边界条件的抛物型和双曲型生物传热问题,固体表面的超短激光脉冲加热问题,带有体积热源的时间分数阶Cattaneo热波模型等等.本文考虑带有分数阶热流条件的广义非线性扰动热波方程初始-边值问题.

关于非线性问题,近年来,很多近似方法被优化^[8-12].作者等也使用了各种渐近方法讨论了一类非线性方程大气物理、尘埃、等离子、孤波等问题^[13-32].在上述的背景下,本文进一步利用奇摄动伸长变量方法和泛函分析不动点理论来讨论一类分数阶扩散方程广义非线性扰动热波问题,并得到相关模型的渐近解析解.弥补了单纯用数值模拟方法得到的数值模拟解的不足.

由分数阶Cattaneo传热方程理论^[1]及相应的能量平衡方程可得如下分数阶广义非线性扰动热波方程

$ \kappa\frac{\partial^{2}u(t, x)}{\partial x^{2}} = \tau^{\alpha}_{1}\rho c \frac{\partial^{\alpha+1}u(t, x)}{\partial t^{\alpha+1}} +\rho c\frac{\partial u(t, x)}{\partial t}+f(u(t, x), \varepsilon), $

(2.1) $ \begin{align} 0<x<L, \ \ t>0, \ \ 0<\alpha\leq 1, \end{align} $

其中$ u(t, x) = T(t, x)-T_{0}, \ T $为介质的温度函数, $ T_{0} $为初始温度, $ f $为非线性传热扰动项,它在所考虑的变量变化范围内的充分光滑的有界函数,常数$ \kappa $为具有小热波导热速率的介质系数, $ \rho $为密度, $ c $为比热, $ \tau_{1} $为热松弛时间,它近似为$ \tau_{1} = \alpha_{0}/v^{2}_{0} $,其中$ \alpha_{0} $为导热系数, $ v_{0} $为热波的传播速率, $ \alpha $为分数阶的阶数,且$ u(t, x) $关于$ t $的$ \alpha $阶分数阶导数定义为

$ \frac{\partial^{1+\alpha} u(t, x)}{\partial t^{1+\alpha}} = \left\{\begin{array}{ll} { } \frac{1}{\Gamma(1-\alpha)}\int^{t}_{0} \frac{u(s, x)}{(t-s)^{\alpha}}{\rm d}s, & 0<\alpha<1, \\ { } \frac{\partial u(t, x)}{\partial t}, & \alpha = 1. \end{array} \right. $

现考虑在分数阶广义非线性扰动热波方程(2.1)下,热物质为常数的介质.假设介质在介质表面$ (x = 0) $瞬间作用一个瞬时热流满足零初值条件,并设介质在$ x = l $处边界保持绝热状态下,讨论方程(2.1)及满足初始-边值条件

(2.2) $ \begin{align} -\kappa\frac{\partial T(t, 0)}{\partial x} = T(t, 0) +\tau^{\alpha}_{1}\frac{\partial^{\alpha}}{\partial t^{\alpha}}T(t, 0), \ \ \frac{\partial T(t, l)}{\partial x} = 0, \end{align} $

(2.3) $ \begin{align} T(0, x) = \frac{\partial T(0, x)}{\partial t} = 0 \end{align} $

的分数阶广义非线性扰动热波方程初始-边值问题.

为方便书写,由方程(2.1)–(2.3),分数阶广义非线性扰动热波模型的无量纲形式的模型

(2.4) $ \begin{align} \varepsilon^{2}\frac{\partial^{2}T(t, x)}{\partial x^{2}} = D\frac{\partial^{\alpha+1}T(t, x)}{\partial t^{\alpha+1}} +\frac{\partial T(t, x)}{\partial t}+f(T(t, x), \varepsilon), \ \ 0<x<l, \ t>0, \end{align} $

(2.5) $ \begin{align} \varepsilon\frac{\partial T(t, 0)}{\partial x} = -g(t)-D\frac{\partial^{\alpha}}{\partial t^{\alpha}}g(t), \ \ \frac{\partial T(t, l)}{\partial x} = 0, \end{align} $

(2.6) $ \begin{align} T|_{t = 0} = \frac{\partial T}{\partial t}|_{t = 0} = 0, \end{align} $

其中$ \varepsilon $为小的正参数, $ f(T, x), \ g(t), \ (f(T, 0) = g(0) = 0) $为关于其变量为充分光滑的有界函数.

模型(2.4)–(2.6)的退化问题为

(3.1) $ \begin{align} D\frac{{\rm d}^{\alpha+1}T}{{\rm d}t^{\alpha+1}} +\frac{{\rm d}T}{{\rm d}t} = 0, \end{align} $

(3.2) $ \begin{align} T|_{t = 0} = \frac{\partial T}{\partial t}|_{t = 0} = 0, \end{align} $

初值问题(3.1)–(3.2),可转化为如下Fredholm型线性微分-积分方程初值问题

(3.3) $ \begin{align} \int^{t}_{0}\frac{T(s)}{(t-s)^{\alpha}}{\rm d}s+\frac{\Gamma(1-\alpha)}{D}\frac{\rm d}{{\rm d}t}T(t) = 0, \ \ T(0) = \frac{{\rm d}T(0)}{{\rm d}t} = 0, \end{align} $

由Fredholm型线性微分-积分方程初值问题(3.3),可得它的解$ \overline{T}_{0} $.再设分数阶广义非线性扰动热波问题(2.4)–(2.6)的外部解$ \overline{T}(t, x) $为

(3.4) $ \begin{align} \overline{T}(t, x) = \sum^{\infty}_{i = 0}\overline{T}_{i}(t, x)\varepsilon^{i}. \end{align} $

将(3.4)式代入(2.4)和(2.6)式,按$ \varepsilon $展开非线性项,合并$ \varepsilon^{i} $的同次幂项,并令各次幂项的系数为零.由$ \varepsilon^{0} $的系数为零,就是退化问题(3.1), (3.2),其解就是$ \overline{T}_{0} $.由$ \varepsilon^{i}\ (i = 1, 2, \cdots) $的系数为零,依次得

(3.5) $ \begin{align} D\frac{\partial^{\alpha+1}\overline{T}_{i}(t, x)}{\partial t^{\alpha+1}} +\frac{\partial \overline{T}_{i}(t, x)}{\partial t} = -\frac{\partial^{2}\overline{T}_{i-1}(t, x)}{\partial x^{2}}-F_{i}, \ \ i = 1, 2, \cdots, \end{align} $

(3.6) $ \begin{align} \overline{T}_{i}(0, x) = \frac{\partial \overline{T}_{i}(0, x)}{\partial t} = 0, \ \ i = 1, 2, \cdots, \end{align} $

其中

$ F_{i} = \frac{1}{(i-1)!}\frac{\partial^{i-1}}{\partial\varepsilon^{i-1}} \bigg[f\bigg(\sum^{\infty}_{i = 0}\overline{T}_{i}(t, x)\varepsilon^{i}, \varepsilon)\bigg)\bigg]_{\varepsilon = 0}. $

由分数阶线性非齐次方程初值问题(3.5)–(3.6),能依次地得到解$ \overline{T}_{i}(t, x)\ (i = 1, 2, \cdots) $.将得到的$ \overline{T}_{i}(t, x)\ (i = 1, 2, \cdots) $代入(3.4)式.我们便得到外部解$ \overline{T}(t, x) $.但外部解$ \overline{T}(t, x) $未必满足边界条件(2.5),为此我们尚需构造边界层校正项$ \widetilde{T} $.

首先引入一个伸长变量变换

(4.1) $ \begin{align} \xi = \frac{x}{\varepsilon}. \end{align} $

考虑到变换(4.1),则(2.4)–(2.6)式为

(4.2) $ \begin{align} \frac{\partial^{2}T}{\partial \xi^{2}} = D\frac{\partial^{\alpha+1}T}{\partial t^{\alpha+1}} +\frac{\partial T}{\partial t}+f(T, \varepsilon), \end{align} $

(4.3) $ \begin{align} \frac{\partial T(t, 0)}{\partial \xi} = -g(t)-D\frac{\partial^{\alpha}}{\partial t^{\alpha}}g(t), \ \ \frac{\partial T(t, l)}{\partial \xi} = 0, \end{align} $

(4.4) $ \begin{align} T|_{t = 0} = -\overline{T}|_{t = 0}, \ \ \frac{\partial T}{\partial t}|_{t = 0} = -\frac{\partial \overline{T}}{\partial t}|_{t = 0}. \end{align} $

再设

(4.5) $ \begin{align} T(t, x) = \overline{T}(t, x)+\widetilde{T}(t, \xi), \end{align} $

其中

(4.6) $ \begin{align} \widetilde{T}(t, \xi) = \sum^{\infty}_{i = 0}\widetilde{T}_i(t, \xi)\varepsilon^{i}. \end{align} $

将(4.5), (4.6)式代入(4.2)–(4.4)式,按$ \varepsilon $展开非线性项,合并$ \varepsilon^{i} $的同次幂项,并令各次幂项的系数为零.由$ \varepsilon^{i} $的同次幂项的系数为零,依次可得

(4.7) $ \begin{align} \frac{\partial^{2}\widetilde{T}_{0}}{\partial \xi^{2}}- D\frac{\partial^{\alpha+1}\widetilde{T}_0}{\partial t^{\alpha+1}} -\frac{\partial\widetilde{T}_{0}}{\partial t} = 0, \end{align} $

(4.8) $ \begin{align} \frac{\partial \widetilde{T}_0(t, 0)}{\partial \xi} = -g(t)-D\frac{\partial^{\alpha}}{\partial t^{\alpha}}g(t), \ \ \frac{\partial \widetilde{T}_0(t, l)}{\partial \xi} = 0, \end{align} $

(4.9) $ \begin{align} \widetilde{T}_{0}|_{t = 0} = \frac{\partial \widetilde{T}_{0}}{\partial t}|_{t = 0} = 0, \end{align} $

(4.10) $ \begin{align} \frac{\partial^{2}\widetilde{T}_{i}}{\partial \xi^{2}} -D\frac{\partial^{\alpha+1}\widetilde{T}_{i}}{\partial t^{\alpha+1}} -\frac{\partial \widetilde{T}_{i}}{\partial t} = \widetilde{F}_{i}, \ \ i = 1, 2, \cdots, \end{align} $

(4.11) $ \begin{align} \frac{\partial \widetilde{T}_{i}(t, 0)}{\partial \xi} = \frac{\partial \widetilde{T}_{i}(t, l)}{\partial \xi} = 0, \ \ i = 1, 2, \cdots, \end{align} $

(4.12) $ \begin{align} \widetilde{T}_{i}|_{t = 0} = \frac{\partial \widetilde{T}_{i}}{\partial t}|_{t = 0} = 0, \ \ i = 1, 2, \cdots, \end{align} $

其中

$ \widetilde{F}_{i} = \frac{1}{i!}\frac{\partial^i}{\partial\varepsilon^i} \bigg[f\bigg(\sum^{\infty}_{i = 0}\widetilde{T}_{i}(t, \xi)\varepsilon^{i}, \varepsilon)\bigg)\bigg]_{\varepsilon = 0}. $

由线性分数阶方程初始边值问题(4.7)–(4.9)和(4.10)–(4.12),能依次地得到解$ \widetilde{T}_{i}(t, \xi) $.利用分数阶线性扰动热波方程初始边值问题的定性理论,不难知道$ \widetilde{T}_{i}(t, \xi)\ (i = 0, 1, \cdots) $具有如下渐近性态

(4.13) $ \begin{align} \widetilde{T}_{i} = O(\exp(-k_{i}\xi)) = O(\exp(-k_{i}\frac{x}{\varepsilon})), \ \ (t, x)\in[0, \infty)\times[0, l], \ \ i = 0, 1, \cdots, \end{align} $

其中$ k_{i} $为正常数.

将得到的$ \widetilde{T}_\varepsilon(t, \xi)\ (i = 0, 1, \cdots) $代入(4.6)式.我们便得到分数阶线性扰动热波方程初始边值问题(2.4)–(2.6)具有边界层校正性态的形式渐近展开式$ T(t, x) $:

(4.14) $ \begin{align} T(t, x)\sim \overline{T}_{0}(t, x) +\sum^{\infty}_{i = 0}\overline{T}_{i}(t, x)\varepsilon^{i} +\sum^{\infty}_{i = 0}\widetilde{T}_{i}(t, \xi)\varepsilon^{i}, \ \ 0<\varepsilon\ll 1. \end{align} $

现证明渐近式(4.14)为分数阶非线性扰动热波方程初始边值问题(2.4)–(2.6)在$ (t, x)\in[0, \infty)\times[0, l] $上为一致有效的渐近展开式.

引入如下泛函分析不动点原理^[11-12]:

定理5.1  设$ N $为由具有范数$ \|p\| $的元素$ p $组成的一个线性赋范空间, $ B $为一个具有范数$ \|q\| $的元素$ q $的Banach空间.设$ G $为一个由$ N $到$ B $的非线性映射, $ G[0] = 0 $.设$ G[p] $可分解为$ G[p] = L[p]+\Psi[p] $,其中$ L $为在$ p = 0 $时$ F $的线性化的算子.算子$ L $满足如下条件:

(I) $ L^{-1} $为$ L $的连续逆映射,且

$ \|L^{-1}[q]\|\leq l^{-1}\|q\|; $

(II) $ K_{N}(r) $表示球$ \{p|p\in N, \|p\|\leq r\} $,存在一个正数$ \overline{r} $,满足Lipschitz条件: $ \|\Psi[p_{2}]-\Psi[p_{1}\| $$ \leq m(r)\|p_{2}-p_{1}\|, \forall p_{1}, p_{2} \in K_{N}(r) $, $ 0\leq r\leq\overline{r} $,并且当$ r\rightarrow 0 $时$ m(r) $单调下降为零: $ { }\lim_{r\rightarrow 0}(r) = 0. $

则对满足$ f\in B, \|f\|\leq \frac{1}{2}lr_{0} $存在$ p\in N $使得$ G[p] = f $且$ \|p\|\leq 2l^{-1}\|f\|\leq r_{0} $,其中$ r_{0} = \sup\limits_{r\geq 0}\{r|0\leq r\leq\overline{r}, $$ m(r)\leq\frac{1}{2}l\} $.

现用上述不动点原理来估计分数阶非线性扰动热波方程初始边值问题(2.4)–(2.6)的形式渐近解(4.14)的余项$ R $.设

$ T(t, x) = \hat{T}(t, x)+R, $

其中

$ \hat{T}(t, x) = \sum^{M}_{i = 1}\overline{T}_{i}(t, x)\varepsilon^{i}+ \sum^{M}_{i = 1}\widetilde{T}_{i}(t, \xi)\varepsilon^{i}, $

这里$ M $为任意大的正整数.利用(5.1)式和性态(4.13)式,有

$ G(R) = \varepsilon^{2}\frac{\partial^{2}R}{\partial x^{2}}- D\frac{\partial^{\alpha+1}R}{\partial t^{\alpha+1}} -\frac{\partial R}{\partial t}-f(\hat{T}+R, \varepsilon)+f(\hat{T}, \varepsilon) = O(\varepsilon^{M+1}), \ (t, x)\in[0, \infty)\times[0, 1], $

$ \frac{\partial R(t, 0)}{\partial x} = \frac{\partial R(t, l)}{\partial x} = O(\varepsilon^{M+1}), $

$ R|_{t = 0} = \frac{\partial R}{\partial t}|_{t = 0} = O(\varepsilon^{M+1}). $

设线性化的微分算子$ \overline{L} $为

$ \overline{L}[p] = \varepsilon^{2}\frac{\partial^{2}p}{\partial x^{2}} -D\frac{\partial^{\alpha+1}p}{\partial t^{\alpha+1}}-\frac{\partial p}{\partial t} -f_{T}(\hat{T}, \varepsilon)p. $

故

$ \Psi[p]\equiv G[p]-\overline{L}[p] = f(\hat{T}, \varepsilon)- f(\hat{T}+p, \varepsilon)+f_{T}(\hat{T}+\theta p, \varepsilon), \ \ 0<\theta<1. $

固定$ \varepsilon $,选择线性赋范空间$ N $

$ \begin{eqnarray*} N& = &\bigg\{p|p\in C^{2}((0, \infty)\times[0, 1]), -\kappa\frac{\partial p}{\partial x}|_{x = 0} = [p+\tau^{\alpha}_{1}\frac{\partial^{\alpha}p}{\partial t^{\alpha}}]_{x = 0}, \\ &&\frac{\partial T(t, l)}{\partial x} = 0, \ p|_{t = 0} = \frac{\partial p}{\partial t}|_{t = 0} = 0\bigg\}, \end{eqnarray*} $

且范数$ { }\|q\| = \sup_{t\in(0, \infty), x\in[0, 1]}|q| $.

由不动点假设,成立

$ \|\overline{L}^{-1}[q]\|\leq l^{-1}\|q\|, \ \ \forall q\in B, $

其中$ l $独立于$ \varepsilon $, $ \overline{L}^{-1} $为$ \overline{L} $的连续逆算子. Lipschitz条件为

$ \begin{eqnarray*} \|\Psi[p_{2}]-\Psi[p_{1}]\|& = &\sup\limits_{t\in(0, \infty), x\in[0, l]} \bigg|\frac{\partial^{2}f}{\partial\hat{T}^{2}}(\hat{T}+\theta_{2}p_{2}, \varepsilon)p^{2}_{2} \\ && -\frac{\partial f}{\partial T}(\hat{T}+\theta_{1}p_{2}, \varepsilon)p^{2}_{1} -\frac{\partial f^{2}}{\partial T^{2}}(\hat{T}+\theta_{1}p_{1}, \varepsilon)p^{2}_{1}\bigg| \\ &<&C_{1}\sup\limits_{t\in(0, \infty), x\in[0, l]}\{(|p_{1}|+|p_{2}|)|p_{2}-p_{1}\} +C_{2}\sup\limits_{t\in(0, \infty), x\in[0, l]}\{|p_{1}|^{2}\cdot|p_{2}-p_{1}|\} \\ &<&Cr\|p_{2}-p_{1}\|, \end{eqnarray*} $

其中$ C_{1}, C_{2} $和$ C $为独立于$ \varepsilon $的常数,并对任意的$ p_{1}, p_{2} $在球$ K_{N}(r), \|r\|\leq l $中成立.再由定理5.1,分数阶非线性扰动热波方程初始边值问题(2.4)–(2.6)解的渐近展开式(4.14)的余项$ R $满足

$ \sup\limits_{t\in(0, \infty), x\in[0, l]}|R(t, x)| = O(\varepsilon^{M+1}), \ \ 0<\varepsilon\ll 1. $

于是有如下定理:

定理5.2  在本文开始叙述的假设下,分数阶非线性扰动热波方程初始边值问题(2.4)–(2.6)存在一个解$ T(t, x) $,在$ (t\in(0, \infty), x\in[0, l]) $上关于$ \varepsilon $成立一致有效的渐近展开式(4.14).

由于泛函分析映射方法得到的分数阶广义非线性扰动热波方程初始边值问题的近似解$ T(t, x) $是近似的解析关系式,因此还可以通过解析运算,譬如进行微分、积分等运算继续对传热模型的相关物理量进一步研究,并可得到相应的物理性态.

例如,在本文所研究的参数估计问题中,用通过求解问题获得的真实温度场和随机误差合成仿真实验数据,介质内部温度的测量值,来研究分数阶阶数和热松弛时间的两参数估计.可以通过得到的近似解析解$ T(t, x) $进行参数的仿真实验.得到相应的目标函数的最优化的变化关系.因此本文为分数阶热波模型的参数估计提供了一种有效的方法.

再如,改变分数阶广义非线性扰动方程相应的参量以达到扰动热波问题的理想结果.

本文用奇摄动和泛函分析方法,选择合理的初始近似函数,能以较快速度的得到较高精度的近似解析解.

由本文用奇摄动和泛函分析方法得到的分数阶广义非线性扰动热波模型初始边值问题的近似解,它不同于单纯用数值模拟方法得到的模拟解,因为它还可进行微分、积分等运算,继续对非线性扰动热波研究.