上海理工大学学报  2022, Vol. 44 Issue (4): 368-372 PDF

An hp-version Legendre spectral collocation method for nonlinear fractional differential equations
LI Shan, AN Xiao, SUN Guilei
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: The hp-version Legendre spectral collocation method for solving nonlinear fractional differential equations was studied. At first, the multifractional differential equation was transformed into an equivalent Volterra integral equation. Then a numerical method to approximate the original equation was constructed. Finally, the correctness of the algorithm theory and the effectiveness of the proposed numerical method were demonstrated by numerical experiments.
Key words: nonlinear fractional differential equations     Legendre spectral collocation method     hp-version error bounds
1 问题的提出

 $\left\{ {\begin{array}{*{20}{l}} {{{\rm{D}}^{\alpha + 1}}u = f(t,u(t)),\quad t \in [0,T]} \\ {u(0) = 0,\quad {u'}(0) = 0} \end{array}} \right.$ (1)

 $\begin{split}& {{\rm{D}}^\alpha }u = \frac{1}{{\Gamma (n - \alpha )}}\int_0^t {{{(t - \tau )}^{n - \alpha - 1}}} {u^{(n)}}(\tau ){\text{d}}\tau, \\& \quad {{t}} > 0,\quad {{n}} - 1 < \alpha < {{n}}{} \end{split}$

Caputo导数 ${{\rm{D}}^\alpha }$ 不同于传统的微分算子，它是一种全局算子。目前分数阶导数主要分为3种类型：Riesz 意义下的分数阶导数、Riemann-Liouville 分数阶导数和Caputo 型分数阶导数。1832年Liouville采用级数的形式给出了分数阶导数，1853 年Riemann 采用定积分的形式给出了另一种分数阶微分，Grünwald 和Krug 将Riemann 和Liouville 的结论进行了统一，得到了Riemann-Liouville 分数阶导数的定义。1967 年Caputo 提出了Caputo 型分数阶导数。Caputo 型分数阶导数因其广泛的应用背景和相对简便的运算被广泛地应用于各个领域。目前在分数阶微分方程理论的研究中，关于其数值解进行了大量的研究。在Caputo 型分数阶定义下的初值条件具有明确的物理意义，因而在工程物理学领域该方程具有重要的应用价值。但由于解的复杂性，不能直接应用于实际工程中，所以，求解Caputo 型分数阶微分方程的数值解受到广泛关注。

2 预备知识

${I_h}$ 是区间 $I = \left[ {0,T} \right]$ 上的一个网格, ${I_h}: = \{ {t_n}: 0 = {t_0} < {t_1} < \cdots < {t_N} = T \}$ ，并且设 ${h_n} = {t_n} - {t_{n - 1}}$ ${I_n} = ({t_{n - 1}},{t_n}]$ 。通常情况下，对于一个给定区间 $\varLambda$ 和某一权函数 $\chi (x)$ ，定义

 ${L}_{\chi }^{2}(\varLambda )=\left\{v|v是可测的并且{\Vert v\Vert }_{{L}_{\chi }^{2}(\varLambda )} < \infty \right\}$

${L_k}(x),x \in ( - 1,1)$ 是标准的阶数为 $k$ 的Legendre多项式。Legendre 多项式集合构成一个完备的 ${L^2}( - 1,1)$ 正交系统，即

 $\int_{ - 1}^1 {{L_k}} (x){L_j}(x){\rm{d}}x = \frac{2}{{2k + 1}}{\delta _{k,j}}$ (2)

 ${L_{n,k}}(t) = {L_k}\left( {\frac{{2t - {t_{n - 1}} - {t_n}}}{{{h_n}}}} \right),\quad t \in {I_n},\quad k \geqslant 0$ (3)

${L_{n,k}}(t),k \geqslant 0$ 的集合构成一个完备的 ${L^2}\left( {{I_n}} \right)$ 正交系统，即

 $\int_{{I_n}} {{L_{n,k}}} (t){L_{n,j}}(t){\rm{d}}t = \frac{{{h_n}}}{{2k + 1}}{\delta _{k,j}}$ (4)

 ${t_{n,j}} = \frac{1}{2}\left( {{h_n}{x_{n,j}} + {t_{n - 1}} + {t_n}} \right),\quad 1 \leqslant n \leqslant N,\quad 0 \leqslant j \leqslant {M_n}$

 $\int_{{I_n}} \phi (t){\rm{d}}t = \frac{{{h_n}}}{2}\sum\limits_{j = 0}^{{M_n}} \phi \left( {{t_{n,j}}} \right){\omega _{n,j}},\quad \phi \in {\mathcal{P}_{2{M_n} + 1}}\left( {{I_n}} \right)$ (5)

 $\begin{split}& \sum\limits_{j = 0}^{{M_n}} {{L_{n,p}}} \left( {{t_{n,j}}} \right){L_{n,q}}\left( {{t_{n,j}}} \right){\omega _{n,j}} = \frac{2}{{2p + 1}}{\delta _{p,q}},\\& \qquad 0 \leqslant p + q \leqslant 2{M_n} + 1 \end{split}$ (6)

 ${\mathcal{I}_{s,{M_n}}}v\left( {{t_{n,j}}} \right) = v\left( {{t_{n,j}}} \right),\quad 0 \leqslant j \leqslant {M_n}$ (7)

 $\mathcal{I}_{\xi ,{M_n}}^tv\left( {{\xi _{n,j}}} \right) = v\left( {{\xi _{n,j}}} \right),\quad 0 \leqslant j \leqslant {M_n}$ (8)

 ${\xi _{n,j}} = \sigma \left( {{t_{n,j}},t} \right): = {t_{n - 1}} + \frac{{\left( {t - {t_{n - 1}}} \right)\left( {{t_{n,j}} - {t_{n - 1}}} \right)}}{{{h_n}}}$

 $\mathcal{I}_{\xi ,{M_n}}^tv(\xi ) = {\mathcal{I}_{\lambda ,{M_n}}}v(\sigma (\lambda ,t))$

 $\begin{split} & \int_{{I_k}} {{{(t - s)}^\nu }} \phi (s){\rm{d}}s = \sum\limits_{j = 0}^{{M_k}} \phi \left( {{t_{k,j}}} \right)\tilde \omega _{k,j}^v(t),\\&\qquad t \in {I_n},\quad k < n \end{split}$ (9)

 $\tilde \omega _{k,j}^v(t) = \int_{{I_k}} {{{(t - s)}^\nu }} {l_{k,j}}(s){\rm{d}}s$ (10)

 $\int_{{t_{n - 1}}}^t {{{(t - \xi )}^\nu }} \psi (\xi )d\xi = \sum\limits_{j = 0}^{{M_n}} \psi \left( {{\xi _{n,j}}} \right)\hat \omega _{n,j}^\nu (t),\quad t \in {I_n}$ (11)

 $\hat \omega _{n,j}^\nu (t) = \int_{{t_{n - 1}}}^t {{{(t - \xi )}^\nu }} l_{{{n}},j}^t(\xi ){\rm{d}}\xi$ (12)

3 hp型谱配置法

 $u(t) = \frac{1}{{\Gamma (1 + \alpha )}}\int_0^t {{{(t - s)}^\alpha }} f(s,u(s)){\rm{d}}s$ (13)

${u^n}(t)$ 是第 $n$ 个区间上式(13)的解，即

 ${u^n}(t): = u(t),\quad t \in {I_n},\quad 1 \leqslant n \leqslant N$

 $\begin{split}& {u^n}(t) = \frac{1}{{\Gamma (1 + \alpha )}}\Bigg( \sum\limits_{k = 1}^{n - 1} {\int_{{I_k}} {{{(t - s)}^\alpha }} } f\left( {s,{u^k}(s)} \right){\rm{d}}s +\Bigg.\\&\quad \Bigg. \int_{{t_{n - 1}}}^t {{{(t - \xi )}^\alpha }} f\left( {\xi ,{u^n}(\xi )} \right){\rm{d}}\xi \Bigg) \end{split}$ (14)

 $\begin{split}& {U^n}(t) =\\& \quad{{\cal I}_{t,{M_n}}}\left[ {\frac{1}{{\Gamma (\alpha + 1)}}\left( {\sum\limits_{k = 1}^{n - 1} {\int_{{I_k}} {{{(t - s)}^\alpha }} } {{\cal I}_{s,{M_k}}}f\left( {s,{U^k}(s)} \right){\rm{d}}s} \right.} \right. +\\& \quad \left. {\left. { \int_{{t_{n - 1}}}^t {{{(t - \xi )}^\alpha }} {\cal I}_{\xi ,{M_n}}^tf\left( {\xi ,{U^n}(\xi )} \right){\rm{d}}\xi } \right)} \right]\\[-12pt] \end{split}$ (15)

 $U(t) = {U^k}(t),\quad t \in {I_k},\quad 1 \leqslant k \leqslant N$

 ${U^n}(t) = \sum\limits_{p = 0}^{{M_n}} {u_p^n} {L_{n,p}}(t)$ (16)

 $\begin{split}& {\mathcal{I}_{t,{M_n}}}\int_{{I_k}} {{{(t - s)}^\alpha }} {\mathcal{I}_{s,{M_k}}}f\left( {s,{U^k}(s)} \right){\rm{d}}s = \\& \quad {\mathcal{I}_{t,{M_n}}}\sum\limits_{{p'} = 0}^{{M_k}} f \left( {{t_{k,{p'}}},{U^k}\left( {{t_{k,{p'}}}} \right)} \right)\tilde \omega _{k,{{\text{p}}'}}^\alpha (t) = \\& \quad \sum\limits_{{p'} = 0}^{{M_k}} f \left( {{t_{k,{p'}}},{U^k}\left( {{t_{k,{p'}}}} \right)} \right){\mathcal{I}_{t,{M_n}}}\tilde \omega _{k,{{\text{p}}'}}^\alpha (t) = \\& \quad \sum\limits_{{p'} = 0}^{{M_k}} f \left( {{t_{k,{p'}}},{U^k}\left( {{t_{k,{p'}}}} \right)} \right)\sum\limits_{p = 0}^{{M_n}} {\omega _{k,{p'},p}^{n,\alpha }} {L_{n,p}}(t) = \\& \quad \sum\limits_{p = 0}^{{M_n}} {a_{k,p}^n} {L_{n,p}}(t) \\[-8pt] \end{split}$ (17)

 $\begin{split}& a_{k,p}^n = \sum\limits_{{p'} = 0}^{{M_k}} f \left( {{t_{k,{p'}}},{U^k}\left( {{t_{k,{p'}}}} \right)} \right)\omega _{k,{p'},p}^{n,\alpha } , \omega _{k,{p^\prime },p}^{n,\alpha } = \\& \qquad \frac{{2p + 1}}{2}\sum\limits_{j = 0}^{{M_k}} {\tilde \omega _{k,{{\text{p}}'}}^\alpha } \left( {{t_{n,j}}} \right){L_{n,p}}\left( {{t_{n,j}}} \right){\omega _{n,j}} \end{split}$ (18)

 $\begin{split}& {\mathcal{I}_{t,{M_n}}}\left( {\int_{{t_{n - 1}}}^t {{{(t - \xi )}^\alpha }} \mathcal{I}_{\xi ,{M_n}}^tf\left( {\xi ,{U^n}(\xi )} \right){\rm{d}}\xi } \right) = \\& \qquad {\mathcal{I}_{t,{M_n}}}\sum\limits_{j = 0}^{{M_n}} f \left( {{\xi _{n,j}},{U^n}\left( {{\xi _{n,j}}} \right)} \right)\hat \omega _{n,j}^\alpha (t) =\\& \qquad \sum\limits_{p = 0}^{{M_n}} {b_p^n} {L_{n,p}}(t) \\[-8pt] \end{split}$ (19)

 $\begin{split} b_p^n =&\frac{{2p + 1}}{2}\cdot\\& \sum\limits_{i,j = 0}^{{M_n}} f \left( {{\xi _{n,j}},{U^n}\left( {{\xi _{n,j}}} \right)} \right)\hat \omega _{n,j}^\alpha \left( {{t_{n,i}}} \right){L_{n,p}}\left( {{t_{n,i}}} \right){\omega _{n,i}} \end{split}$ (20)

 $\begin{split}& \sum\limits_{p = 0}^{{M_n}} {u_p^n} {L_{n,p}}(t) = \frac{1}{{\Gamma (\alpha + 1)}}\sum\limits_{p = 0}^{{M_n}} {\sum\limits_{k = 1}^{n - 1} {a_{k,p}^n} } {L_{n,p}}(t) +\\& \qquad \quad \frac{1}{{\Gamma (\alpha + 1)}}\sum\limits_{p = 0}^{{M_n}} {b_p^n} {L_{n,p}}(t) \end{split}$ (21)

 $u_p^n = \frac{1}{{\Gamma (\alpha + 1)}}\left( {\sum\limits_{k = 1}^{n - 1} {a_{k,p}^n} + b_p^n} \right)$ (22)

4 数值结果

 $\begin{gathered} {E_{{L^2}(0,T)}}: = {\left( {\sum\limits_{k = 1}^N {\frac{{{h_k}}}{2}} \sum\limits_{j = 0}^{{M_k}} {{{\left( {u\left( {{t_{k,j}}} \right) - U\left( {{t_{k,j}}} \right)} \right)}^2}} {\omega _{k,j}}} \right)^{\frac{1}{2}}} \\ {E_{{L^\infty }(0,T)}}: = \mathop {\max }\limits_{_{1 \leqslant k \leqslant N}} \left( {\mathop {\max }\limits_{_{0 \leqslant j \leqslant {M_k}}} \left| {u\left( {{t_{k,j}}} \right) - U\left( {{t_{k,j}}} \right)} \right|} \right) \\ \end{gathered}$

 $\left\{ {\begin{array}{*{20}{l}} {{{\rm{D}}^{1.5}}y(t) + y(t) = {t^\alpha } + \dfrac{{\Gamma (\alpha + 1)}}{{\Gamma (\alpha - 1/2)}}{t^{\alpha - 3/2}}} \\ {u(0) = 0,\quad {u^\prime }(0) = 0} \end{array}} \right.$ (23)

 图 1 例1在 ${L^2}$ 范数和 ${L^\infty }$ 范数下的收敛阶 Fig. 1 The convergence order under L2-norm and ${{\boldsymbol{L}}^\infty }$ -norm for example 1

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