上海理工大学学报  2019, Vol. 41 Issue (5): 422-428 PDF

A Fully Diagonalized Legendre Rational Spectral Method for Solving Fourth Order Equations
LI Shan, LI Qiaoling
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: A fully diagonalized Legendre rational spectral method for solving fourth order elliptic equations on the half line was proposed. Some Sobolev orthogonal Legendre rational basis functions were constructed which leaded to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions could be represented as infinite and truncated Fourier series. Numerical results demonstrate the effectiveness and the spectral accuracy
Key words: fourth order elliptic equation     Legendre rational spectral method     Sobolev orthogonal function     half line
1 问题提出

2 预备知识 2.1 Legendre多项式

 ${\partial _y}((1 - {y^2}){\partial _y}{L_k}(y)) + k(k + 1){L_k}(y) = 0,\;\;k \geqslant 0$ (1)
 $\int_I {{L_k}(y){L_l}(y){\rm{d}}y = \frac{2}{{2k + 1}}{\delta _{k,l}}}$ (2)
 $\int_I {{\partial _y}} {L_k}(y){\partial _y}{L_l}(y)(1 - {y^2}){\rm{d}}y = \frac{{2k(k + 1)}}{{2k + 1}}{\delta _{k,l}}$ (3)

 $\begin{split}&{L_0}(y) = 1,\;{L_1}(y) = y\\ &(k + 1){L_{k + 1}}(y) = (2k + 1)y{L_k}(y) - k{L_{k - 1}}(y)\end{split}$ (4)
 $(2k + 1){L_k}(y) = {\partial _y}{L_{k + 1}}(y) - {\partial _y}{L_{k - 1}}(y)$ (5)
 $\begin{split}&\quad(2k + 1)(1 - {y^2}){\partial _y}{L_k}(y) =\\ & \qquad k(k + 1)({L_{k - 1}}(y) - {L_{k + 1}}(y))\end{split}$ (6)

2.2 Legendre有理函数

 ${R_k}(x) = \frac{{\sqrt 2 }}{{x + 1}}{L_k}\left(\frac{{x - 1}}{{x + 1}}\right),\;\;x \in \varLambda = (0,\infty ),\;\;k \geqslant 0$

Legendre有理函数满足如下递推式：

 $\begin{split}&{R_0}(x) = \frac{{\sqrt 2 }}{{x + 1}},\;\;{R_1}(x) =\dfrac{{\sqrt 2 (x - 1)}}{{{{(x + 1)}^2}}},\;\;(k + 1){R_{k + 1}}(x) =\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\\ & \quad (2k + 1)\frac{{x - 1}}{{x + 1}}{R_k}(x) - k{R_{k - 1}}(x),\;\;k \geqslant 1 \end{split}$ (7)
 $\begin{split}&2(2k + 1){R_k}(x) = (x + 1)({R_{k + 1}}(x) - {R_{k - 1}}(x)) +\\ & \quad {(x + 1)^2}({\partial _x}{R_{k + 1}}(x) - {\partial _x}{R_{k - 1}}(x)),\;\;k \geqslant 1 \end{split}$ (8)

 $\int_\varLambda {{R_k}} (x){R_l}(x){\rm{d}}x = \frac{2}{{2k + 1}}{\delta _{k,l}}$ (9)

 $v(x) = \sum\limits_{k = 0}^\infty {{{\hat v}_k}} {R_k}(x),\;\;{\hat v_k} = \left(k + \frac{1}{2}\right)\int_\varLambda {v(x){R_k}} (x){\rm{d}}x$

 ${{\cal{R}} _N}(\varLambda ) = {\rm{span}}\{ {R_0}(x),{R_1}(x), \cdots ,{R_N}(x)\}$

 $\begin{split}&{\varphi _k}(x) = {R_k}(x) + {R_{k + 1}}(x),{\rm{ }}{\psi _k}(x) = (k + 2)({R_k}(x) + \\ &\quad {R_{k + 1}}(x)) + (k + 1)({R_{k + 1}}(x) + {R_{k + 2}}(x)) \end{split}$

 $\begin{split}&{\partial _x}{R_k}(x) = \dfrac{{7{k^2} - 2}}{{4k + 2}}{R_{k - 1}}(x) - \dfrac{{4{k^2} + 4k + 1}}{{4k + 2}}{R_k}(x) +\\ &\;\;\;\; \dfrac{{{k^2} + 2k + 1}}{{4k + 2}}{R_{k + 1}}(x) \!+\! \sum\limits_{j = 0}^{k \!-\! 2} {{{( - 1)}^{j \!+\! k \!+\! 1}}} (2j \!-\! 3){R_j}(x) \end{split} \!\!\!\!$ (10)
 $\begin{split}&{\partial _x}{\varphi _k}(x) = - \frac{{{k^2}}}{{2(2k + 1)}}{R_{k - 1}}(x) +\frac{{3{k^2} + 6k + 2}}{{2(2k + 3)}}{R_k}(x) -\!\!\!\!\!\!\!\!\!\!\!\!\!\\ &\quad \frac{{3{k^2} + 6k + 2}}{{2(2k + 1)}}{R_{k + 1}}(x) + \frac{{{{(k + 2)}^2}}}{{2(2k + 3)}}{R_{k + 2}}(x)\end{split}$ (11)

 ${\partial _x}{R_0}(x) = - \frac{1}{2}{R_0}(x) + \frac{1}{2}{R_1}(x)$
 ${\partial _x}{R_1}(x) = \frac{5}{6}{R_0}(x) - \frac{3}{2}{R_1}(x) + \frac{2}{3}{R_2}(x)$
 ${\partial _x}{R_2}(x) = - {R_0}(x) + \frac{{13}}{5}{R_1}(x) - \frac{5}{2}{R_2}(x) + \frac{9}{{10}}{R_3}(x)$

$k = 0,1,2$ 时，式 （10）成立。假设 $k \leqslant n$ 时，式（10）成立，只需验证 $k = n + 1$ 时式（10）成立即可。由式（7）可知

 $\begin{split}&(k + 1){\partial _x}{R_{k + 1}}(x) = - k{\partial _x}{R_{k - 1}}(x) + \frac{{2(2k + 1)}}{{{{(x + 1)}^2}}}{R_k}(x) +\\ &\quad (2k + 1){\partial _x}{R_k}(x) - \frac{{2(2k + 1)}}{{x + 1}}{\partial _x}{R_k}(x) \end{split}$

 $\begin{split} &\dfrac{1}{{x + 1}}{R_k}(x) = - \dfrac{k}{{2(2k + 1)}}{R_{k - 1}}(x) + \dfrac{1}{2}{R_k}(x) -\\ &\quad \dfrac{{k + 1}}{{2(2k + 1)}}{R_{k + 1}}(x)\end{split}$

 $\begin{split}&\!\!\!\!\dfrac{1}{{{{(x + 1)}^2}}}{R_k}(x) = \dfrac{{k(k - 1)}}{{2(2k + 1)(2k - 1)}}{R_{k - 2}}(x) - \\ &\dfrac{k}{{2(2k + 1)}}{R_{k - 1}}(x) + \dfrac{{3{k^2} + 3k + 2}}{{2(2k + 3)(2k - 1)}}{R_k}(x)- \\ &\dfrac{{k + 1}}{{2(2k + 1)}}{R_{k + 1}}(x) + \dfrac{{(k + 1)(k + 2)}}{{4(2k + 3)(2k + 1)}}{R_{k + 2}}(x) \end{split}$

 $\begin{split}&{\partial _x}{\psi _k}(x) = - \dfrac{{{k^2}(k + 2)}}{{4k + 2}}{R_{k - 1}}(x) - \dfrac{{3(2{k^3} + 9{k^2} + 11k + 3)}}{{2(2k + 2)(2k + 5)}}\cdot\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\\ &\quad{R_{k + 1}}(x)+\dfrac{{{k^2} + 3k + 1}}{2}{R_k}(x) - \dfrac{{{k^2} + 3k + 1}}{2}{R_{k + 2}}(x) + \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\\ &\quad\dfrac{{{{(k + 3)}^2}(k + 1)}}{{2(2k + 5)}}{R_{k + 3}}(x)\end{split}$ (12)
 $\begin{split}&\partial _x^2{\psi _k}(x) = \dfrac{{({k^3} - 3k + 2){k^2}}}{{4(4{k^2} - 1)}}{R_{k - 2}}(x) - \dfrac{{(3{k^2} + 6k - 1){k^2}}}{{4(2k + 1)}}\cdot\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\\ &\quad {R_{k - 1}}(x)+ \dfrac{{15{k^5} + 75{k^4} + 97{k^3} + {k^2} - 20k - 8}}{{4(2k - 1)(2k + 5)}}{R_k}(x)-\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\\ &\quad \dfrac{{10{k^5} + 75{k^4} + 204{k^3} + 243{k^2} + 124k + 24}}{{2(2k + 1)(2k + 5)}}{R_{k + 1}}(x)+\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\\ &\quad \dfrac{{15{k^5} + 150{k^4} + 547{k^3} + 872{k^2} + 568k + 128}}{{4(2k + 1)(2k + 7)}}{R_{k + 2}}(x)-\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\\ &\quad \dfrac{{{{(k + 3)}^2}(3{k^2} + 12k + 8)}}{{4\left( {2k + 5} \right)}}{R_{k + 3}}(x) + \\ &\quad \dfrac{{{{(k + 3)}^2}(k + 1){{(k + 4)}^2}}}{{4\left( {2k + 5} \right)(2k + 7)}}{R_{k + 4}}(x) \end{split}$ (13)

 $\begin{split} &\!\!\!\!{\partial _x}{\psi _k}(x) = (k + 2){\partial _x}{\varphi _k}(x) + (k + 1){\partial _x}{\varphi _{k + 1}}(x)=\\ & (k + 2)\left( - \frac{{{k^2}}}{{2(2k + 1)}}{R_{k - 1}}(x) + \frac{{3{k^2} + 6k + 2}}{{2(2k + 3)}}{R_k}(x)\right. -\\ &\left. \frac{{3{k^2} + 6k + 2}}{{2(2k + 1)}}{R_{k + 1}}(x)+ \frac{{{{(k + 2)}^2}}}{{2(2k + 3)}}{R_{k + 2}}(x)\right) + \\ \end{split}$
 $\begin{split} &(k + 1)\left( - \frac{{{{(k + 1)}^2}}}{{2(2k + 3)}}{R_k}(x)+\! \frac{{3{{(k \!+\! 1)}^2}\!+ 6(k + 1) + 2}}{{2(2k + 5)}}{R_{k + 1}}(x)-\right. \\ &\left.\frac{{3{{(k + 1)}^2} + 6(k + 1) + 2}}{{2(2k + 3)}}{R_{k + 2}}(x) + \frac{{{{(k + 3)}^2}}}{{2(2k + 5)}}{R_{k + 3}}(x)\right)= \\ & - \frac{{{k^2}(k + 2)}}{{4k + 2}}{R_{k - 1}}(x) + \frac{{{k^2} + 3k + 1}}{2}{R_k}(x) - \\ &\frac{{3(2{k^3} + 9{k^2} + 11k + 3)}}{{2(2k + 1)(2k + 5)}}{R_{k + 1}}(x)- \frac{{{k^2} + 3k + 1}}{2}{R_{k + 2}}(x) +\\ &\frac{{{{(k + 3)}^2}(k + 1)}}{{2(2k + 5)}}{R_{k + 3}}(x) \end{split}$

 $\begin{split} &\partial _x^2{\psi _k}(x) = (k + 2)\partial _x^2{\varphi _k}(x) + (k + 1)\partial _x^2{\varphi _{k + 1}}(x)=\\ & (k + 2)\left( - \dfrac{{{k^2}}}{{2(2k + 1)}}{\partial _x}{R_{k - 1}}(x) + \dfrac{{3{k^2} + 6k + 2}}{{2(2k + 3)}}{\partial _x}{R_k}(x) -\right.\\ & \left.\dfrac{{3{k^2} + 6k + 2}}{{2(2k + 1)}}{\partial _x}{R_{k + 1}}(x)+ \dfrac{{{{(k + 2)}^2}}}{{2(2k + 3)}}{\partial _x}{R_{k + 2}}(x)\right)+\\ & (k + 1)\left( - \dfrac{{{{(k + 1)}^2}}}{{2(2k + 3)}}{\partial _x}{R_k}(x) + \dfrac{{{{(k + 3)}^2}}}{{2(2k + 5)}}{\partial _x}{R_{k + 3}}(x)\right.+\\ & \dfrac{{3{{(k \!+\! 1)}^2} \!+\! 6(k \!+\! 1) \!+\! 2}}{{2(2k\! +\! 5)}}{\partial _x}{R_{k \!+\! 1}}(x) \!-\! \dfrac{{3{{(k \!+\! 1)}^2} \!+\! 6(k \!+\! 1) \!+\! 2}}{{2(2k \!+\! 3)}}\cdot\\ & {\partial _x}{R_{k + 2}}(x){\Bigg{)}}= \dfrac{{({k^3} - 3k + 2){k^2}}}{{4(4{k^2} - 1)}}{R_{k - 2}}(x) -\dfrac{{(3{k^2} + 6k + 1){k^2}}}{{4(2k + 1)}}\cdot\\ &{R_{k - 1}}(x)+\dfrac{{15{k^5} + 75{k^4} + 97{k^3} + {k^2} - 20k - 8}}{{4(2k - 1)(2k + 5)}}{R_k}(x)-\\ & \dfrac{{10{k^5} + 75{k^4} + 204{k^3} + 243{k^2} + 124k + 24}}{{2(2k + 5)(2k + 1)}}{R_{k + 1}}(x)+\\ & \dfrac{{15{k^5} + 150{k^4} + 547{k^3} + 872{k^2}+568k + 128}}{{4(2k + 1)(2k + 7)}}{R_{k + 2}}(x)-\\ & \dfrac{{{{(k \!+\! 3)}^2}(3{k^2} \!+\! 12k \!+\! 8)}}{{4(2k \!+\! 5)}}{R_{k \!+\! 3}}(x) \!+\! \dfrac{{{{(k \!+\! 3)}^2}(k \!+\! 1){{(k \!+\! 4)}^2}}}{{4(2k \!+\! 5)(2k \!+\! 7)}}{R_{k \!+\! 4}}(x)\end{split}$

3 四阶Dirichlet边值问题

 $\left\{\!\!\!\begin{array}{l} {u^{(4)}}(x) - \alpha u''(x) + \beta u(x) = f(x),\alpha ,\beta \geqslant 0,x \in \varLambda \\ u(0) = u'(0) = 0,\mathop {\lim }\limits_{x \to + \infty } u(x) = \mathop {\lim }\limits_{x \to + \infty } u'(x) = 0 \end{array} \right.\!\!\!\!\!$ (14)

 $\begin{split}&{D_{\alpha ,\beta }}(u,v): = (\partial _x^2u,\partial _x^2v) + \alpha ({\partial _x}u,{\partial _x}v) +\\ &\quad\beta (u,v) = (f,v),\;\;\;v \in H_0^2(\varLambda )\end{split}$ (15)

Legendre有理谱方法是寻找到 ${u_N} \in {{\cal{R}} _N}(\varLambda )$ ，使得

 ${D_{\alpha ,\beta }}({u_N},\varphi ) = (f,\varphi ),\;\;\;\varphi \in {{\cal{R}} _N}(\varLambda )$ (16)

 ${D_{\alpha ,\beta }}({P_k},{P_l}) = {\sigma _k}{\delta _{k,l}},\;\;\;k,l \geqslant 0$ (17)

 $\begin{split}&{P_k}(x) = \frac{{{\psi _k}(x)}}{{{{(k + 1)}^2}}} - {a_k}{P_{k - 1}}(x) - {b_k}{P_{k - 2}}(x) -{c_k}{P_{k - 3}}(x) -\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\\ & \quad{d_k}{P_{k - 4}}(x) - {e_k}{P_{k - 5}}(x) - {f_k}{P_{k - 6}}(x),k \geqslant {\rm{6}}\end{split}$ (18)

 $\begin{split} &{\tilde a_{k - i}} = {a_{k - i}}{a_{k - i - 1}} - {b_{k - i}},\;\;{\tilde b_{k - i}} = {a_{k - i}}{b_{k - i - 1}} - {c_{k - i}} ,\;\;{\tilde c_{k - i}} = {a_{k - i}}{c_{k - i - 1}} - {d_{k - i}} ,\;\;{\tilde d_{k - i}} = {a_{k - i}}{d_{k - i - 1}} - {e_{k - i}},\\ &{\hat a_{k - i}} = {a_{k - i - 2}}{\tilde a_{k - i}} - {\tilde b_{k - i}},\;\;{\hat b_{k - i}} = {\tilde a_{k - i}}{\tilde a_{k - i - 2}} - {a_{k - i - 3}}{\tilde b_{k - i}} + {\tilde c_{k - i}},\;\;{\hat c_{k - i}} = {\tilde a_{k - i - 3}}{\tilde b_{k - i}} - {a_{k - i - 4}}{\tilde c_{k - i}} + {\tilde d_{k - i}},\;\;{\bar a_{k - 1}} = {\tilde a_{k - 1}}{\hat a_{k - 3}} - {\hat c_{k - 1}},\\ & {Q_k} = 2k + 1,\;\;{M_k} = {Q_{k + 2}}{Q_k}{Q_{k - 1}}{Q_{k - 2}}{Q_{k - 3}},{\tilde M_k} = {M_k}{Q_{k + 3}},\;\;{\hat M_k} = {M_k}{Q_{k - 4}},\;\;{N_k} = {k^2},{R_k} = k\\ &{S_1} = 3(154{k^{11}} + 2\;541{k^{10}} + 16\;737{k^9} + 54\;432{k^8} + 79\;212{k^7} - 13\;671{k^6})\\ &{S_2} = - 3(216\;427{k^5} + 289\;902{k^4} + 128\;220{k^3} - 31\;752{k^2} - 44\;064k - 10\;368),\;\;S = ({S_1} + {S_2}){Q_{k + 1}}\\ & \tilde A = \frac{{3(33{k^8} + 264{k^7} + 598{k^6} - 108{k^5} - 1\;739{k^4} - 1\;308{k^3} + 532{k^2} + 576k - 192){Q_k}{Q_{k - 3}}}}{{2{{\tilde M}_k}{N_{k + 1}}{N_k}}}\\ & \tilde B = \frac{{3(165{k^{10}} + 825{k^9} - 670{k^8} - 7\;630{k^7} - 4\;931{k^6} + 15\;377{k^5} + 19\;564{k^4} + 2\;948{k^3} - 17\;392{k^2} - 14\;784k - 3\;840){Q_{k - 1}}}}{{8{{\tilde M}_k}{N_{k + 1}}{N_{k - 1}}}} \\ &\tilde C = \frac{{{k^3}(55{k^6} - 570{k^4} + 1\;263k + 980){Q_{k - 2}}}}{{4{M_k}{N_{k + 1}}{N_{k - 2}}}}\\ &\tilde D = \frac{{{k^2}(33{k^8} - 165{k^7} - 158{k^6} + 1\;622{k^5} - 1\;135{k^4} - 2\;965{k^3} + 5\;420{k^2} - 3\;676k + 1\;024){Q_{k - 3}}}}{{4{{\hat M}_k}{N_{k + 1}}{N_{k - 3}}}}\\ &\tilde E = \frac{{(3{k^2} - 6k - 25){R_{k - 1}}{Q_{k + 2}}{N_k}{N_{k - 1}}{N_{k - 2}}}}{{4{M_k}{N_{k + 1}}{N_{k - 4}}}},\;\;\tilde F = \frac{{({k^3} - 3k + 2){R_{k - 5}}{Q_{k + 2}}{N_k}{N_{k - 2}}{N_{k - 3}}}}{{8{{\hat M}_k}{N_{k + 1}}{N_{k - 5}}}},\;\; \tilde G = \frac{{{N_k}{N_{k - 1}}{R_{k + 2}}{R_{k - 3}}}}{{2{Q_k}{Q_{k - 1}}{Q_{k - 2}}{N_{k + 1}}{N_{k - 3}}}}\\ &\tilde H = \frac{{{k^3}({k^2} - 5)}}{{{Q_k}{Q_{k - 1}}{N_{k + 1}}{N_{k + 2}}}},\;\;\tilde I = \frac{{(2{k^6} + 6{k^5} - 11{k^4} - 32{k^3} + 17{k^2} + 34k + 12){Q_{k - 1}}{Q_{k - 3}}}}{{{M_k}{N_{k + 1}}{N_{k - 1}}}},\;\;\tilde J = \frac{{2{R_{k + 2}}{R_{k - 1}}}}{{{Q_k}{N_{k + 1}}{N_{k - 1}}}}\\ &\tilde K = \frac{{{k^5} + 5{k^4} + 9{k^3} + 7{k^2} - 2k + 4}}{{{Q_{k + 2}}{Q_{k - 1}}{N_{k + 1}}{N_k}}},\;\;\tilde L = \frac{{4{R_{k + 1}}}}{{{N_{k + 1}}{N_k}}},\;\;\tilde S = \frac{S}{{2{Q_{k + 2}}{{\tilde M}_{k + 1}}N_{k + 1}^2}}\\ &\tilde T = \frac{{(10{k^7} + 105{k^6} + 417{k^5} + 765{k^4} + 589{k^3} + 18{k^2} - 176k - 48){Q_{k - 2}}{Q_{k - 3}}}}{{{{\tilde M}_k}N_{k + 1}^2}},\;\;\tilde W = \frac{{12(2{k^3} + 9{k^2} + 13k + 6)}}{{{Q_{k + 2}}{Q_k}N_{k + 1}^2}} \end{split}$

a. ${\sigma _k} = - {({a_k})^2}{\sigma _{k - 1}} - {({b_k})^2}{\sigma _{k - 2}} - {({c_k})^2}{\sigma _{k - 3}} - {({d_k})^2}{\sigma _{k - 4}} - {({e_k})^2}{\sigma _{k - 5}} - {({f_k})^2}{\sigma _{k - 6}} + \tilde S + \alpha \tilde T + \beta \tilde W,\;\;k \geqslant 0$

b. ${a_k} = - \dfrac{1}{{{\sigma _{k - 1}}}}(\tilde A + \tilde B{a_{k - 1}} + \tilde C{\tilde a_{k - 1}} + \tilde D{\hat a_{k - 1}} \!+\! \tilde E{\hat b_{k - 1}} \!+\! \tilde F{\bar a_{k - 1}} \!-\! \alpha \tilde G{\hat a_{k - 1}} \!-\! \alpha \tilde H{\tilde a_{k - 1}} \!-\! \alpha \tilde I{a_{k - 1}} + \beta \tilde J{a_{k - 1}} + \alpha \tilde K - \beta \tilde L),\;\;k \geqslant 1$

c. ${b_k} = \dfrac{1}{{{\sigma _{k - 2}}}}(\tilde B + \tilde C{a_{k - 2}} + \tilde D{\tilde a_{k - 2}} + \tilde E{\hat a_{k - 2}} + \tilde F{\hat b_{k - 2}} - \alpha \tilde G{\tilde a_{k - 2}} - \alpha \tilde H{a_{k - 2}} - \alpha \tilde I + \beta \tilde J),\;\;k \geqslant 2$

d. ${c_k} = - \dfrac{1}{{{\sigma _{k - 3}}}}(\tilde C + \tilde D{a_{k - 3}} + \tilde E{\tilde a_{k - 3}} + \tilde F{\hat a_{k - 3}} - \alpha \tilde G{a_{k - 3}} - \alpha \tilde H),\;\;k \geqslant 3$

e. ${d_k} = \dfrac{1}{{{\sigma _{k - 4}}}}(\tilde D + \tilde E{a_{k - 4}} + \tilde F{\tilde a_{k - 4}} - \alpha \tilde G),\;\;k \geqslant 4$

f. ${e_k} = - \dfrac{1}{{{\sigma _{k - 5}}}}(\tilde E + \tilde F{a_{k - 5}}),\;\;k \geqslant 5$

g. ${f_k} = \dfrac{{\tilde F}}{{{\sigma _{k - 6}}}},\;\;k \geqslant 6$

 $\begin{split} & {P_k}(x) = \dfrac{{{\psi _k}(x)}}{{{{(k + 1)}^2}}} - \sum\limits_{m = 0}^{k - 1} {\dfrac{{{D_{\alpha ,\beta }}\left(\dfrac{{{\psi _k}(x)}}{{{{(k + 1)}^2}}},{P_m}(x)\right)}}{{{\sigma _m}}}} {P_m}(x),\\ & \quad k \geqslant 1 \\[-11pt] \end{split}$ (19)

 $\begin{split} &{D_{\alpha ,\beta }}\left(\frac{{{\psi _1}(x)}}{4},{P_0}(x)\right) = \left(\partial _x^2\frac{{{\psi _1}(x)}}{4},\partial _x^2{P_0}(x)\right) + \\ &\quad \alpha \left({\partial _x}\frac{{{\psi _1}(x)}}{4},{\partial _x}{P_0}(x)\right) + \beta \left(\frac{{{\psi _1}(x)}}{4},{P_0}(x)\right)=\\ &\quad - 16 - \frac{{4\alpha }}{7} + 2\beta \end{split}$

 $\begin{split} &{D_{\alpha , \beta }}\left(\frac{{{\psi _2}(x)}}{9},{P_0}(x)\right) = \left(\partial _x^2\frac{{{\psi _2}(x)}}{9},\partial _x^2{P_0}(x)\right) +\\ & \quad \alpha \left({\partial _x}\frac{{{\psi _2}(x)}}{9},{\partial _x}{P_0}(x)\right) + \beta \left(\frac{{{\psi _2}(x)}}{9},{P_0}(x)\right)= \\ & \quad \frac{{6\;112 - 44\alpha + 88\beta }}{{495}}\end{split}$
 $\begin{split}&{D_{\alpha ,\beta }}\left(\frac{{{\psi _2}(x)}}{9},{P_1}(x)\right) = - \frac{{1\;744}}{{99}} - \frac{{6\;112{a_1}}}{{495}} +\\ &\quad \alpha \left(\frac{{4{a_1}}}{{45}} - \frac{{17}}{{81}}\right) + \beta \left(\frac{1}{3} - \frac{{8{a_1}}}{{45}}\right)\end{split}$

 $\begin{split} &{a_3} = - \frac{{43\;072}}{{2\;145{\sigma _2}}} - \frac{{12\;696{a_2}}}{{1\;001{\sigma _2}}} - \frac{{378{{\tilde a}_2}}}{{55{\sigma _2}}} - \alpha \left(\frac{{59}}{{495{\sigma _2}}} - \frac{{17{a_2}}}{{176{\sigma _2}}} - \frac{{27{{\tilde a}_2}}}{{140{\sigma _2}}}\right) + \beta \left(\frac{1}{{9{\sigma _2}}} - \frac{{5{a_2}}}{{112{\sigma _2}}}\right)\\ &{b_3} = \frac{{12\;696}}{{1\;001{\sigma _1}}} + \frac{{378{a_1}}}{{55{\sigma _1}}} - \alpha \left(\frac{{17}}{{176{\sigma _1}}} + \frac{{27{a_1}}}{{140{\sigma _1}}}\right) + \frac{{5\beta }}{{112{\sigma _1}}},{c_3} = - \frac{{378}}{{55{\sigma _0}}} + \frac{{27\alpha }}{{140{\sigma _0}}}\\ & {a_4} = - \frac{{37\;092}}{{1\;625{\sigma _3}}} - \frac{{204\;416{a_3}}}{{14\;625{\sigma _3}}} - \frac{{2\;128{{\tilde a}_3}}}{{325{\sigma _3}}} + ({{\tilde b}_3} - {a_1}{{\tilde a}_3})\frac{{29\;248}}{{11\;375{\sigma _3}}}+ \alpha \left( - \frac{{149}}{{1\;820{\sigma _3}}} + \frac{{9\;892{a_3}}}{{131\;625{\sigma _3}}} + \frac{{176{{\tilde a}_3}}}{{1\;575{\sigma _3}}} -\right. \\ &\quad\quad\left.({{\tilde b}_3} - {a_1}{{\tilde a}_3})\frac{{48}}{{875{\sigma _3}}}\right) + \beta \left(\frac{1}{{20{\sigma _3}}} - \frac{{4{a_3}}}{{225{\sigma _3}}}\right)\\ &{b_4} = \frac{{204\;416}}{{14\;625{\sigma _2}}} + \frac{{2\;128{a_2}}}{{325{\sigma _2}}} + \frac{{29\;248{{\tilde a}_2}}}{{11\;375{\sigma _2}}} - \alpha \left(\frac{{9\;892}}{{131\;625{\sigma _2}}} + \frac{{176{a_2}}}{{1\;575{\sigma _2}}} - \frac{{48{{\tilde a}_2}}}{{875{\sigma _2}}}\right) + \frac{{4\beta }}{{225{\sigma _2}}}\\ &{c_4} = - \frac{{2\;128}}{{325{\sigma _1}}} - \frac{{29\;248{a_1}}}{{11\;375{\sigma _1}}} + \alpha \left(\frac{{176}}{{1\;575{\sigma _1}}} + \frac{{48{a_1}}}{{875{\sigma _1}}}\right),\;\;{d_4} = \frac{{29\;248}}{{11\;375{\sigma _0}}} - \frac{{48\alpha }}{{875{\sigma _0}}}\\ &{a_5} = - \frac{{3\;440\;624}}{{133\;875{\sigma _4}}} - \frac{{611\;441{a_4}}}{{39\;270{\sigma _4}}} - \frac{{6\;200{{\tilde a}_4}}}{{891{\sigma _4}}} + ({\tilde b_4} - {a_2}{\tilde a_4})\frac{{1\;580{{\tilde a}_3}}}{{693{\sigma _4}}} - ({\tilde a_2}{\tilde a_4} - {a_1}{\tilde b_4} + {c_4})\frac{{400}}{{693{\sigma _4}}}+ \beta \left(\frac{2}{{75{\sigma _4}}} - \frac{{7{a_4}}}{{792{\sigma _4}}}\right) -\\ &\quad\quad\alpha \left(\frac{{628}}{{10\;125{\sigma _4}}} - \frac{{43{a_4}}}{{720{\sigma _4}}} - \frac{{625{{\tilde a}_4}}}{{8\;019{\sigma _4}}} + ({\tilde b_4} - {a_2}{\tilde a_4})\frac{{25}}{{891{\sigma _4}}}\right)\\ &{b_5} = \frac{{611\;441}}{{39\;270{\sigma _3}}} + \frac{{6\;200{a_3}}}{{891{\sigma _3}}} + \frac{{1\;580{{\tilde a}_3}}}{{693{\sigma _3}}} - ({\tilde b_3} - {a_1}{\tilde a_3})\frac{{400}}{{693{\sigma _3}}} - \alpha \left(\frac{{43}}{{720{\sigma _3}}} + \frac{{625{a_3}}}{{8\;019{\sigma _3}}} + \frac{{25{{\tilde a}_3}}}{{891{\sigma _3}}}\right) + \frac{{7\beta }}{{792{\sigma _3}}}\\ &{c_5} = - \frac{{6\;200}}{{891{\sigma _2}}} - \frac{{1\;580{a_2}}}{{693{\sigma _2}}} - \frac{{400{{\tilde a}_2}}}{{693{\sigma _2}}} + \alpha \left(\frac{{625}}{{8\;019{\sigma _2}}} + \frac{{25{a_2}}}{{891{\sigma _2}}}\right),\;\;{d_5} = \frac{{1\;580}}{{693{\sigma _1}}} + \frac{{400{a_1}}}{{693{\sigma _1}}} - \frac{{25\alpha }}{{891{\sigma _1}}},\;\;{e_5} = - \frac{{400}}{{693{\sigma _0}}}\\ &{a_6} = - \frac{{6\;411\;544}}{{223\;839{\sigma _5}}} - \frac{{622\;689\;488{a_5}}}{{36\;006\;425{\sigma _5}}} - \frac{{372\;006{{\tilde a}_5}}}{{49\;049{\sigma _5}}} + ({{\tilde b}_5} - {a_3}{{\tilde a}_5})\frac{{2\;508\;320}}{{1\;072\;071{\sigma _5}}} - ({{\tilde a}_3}{{\tilde a}_5} - {a_2}{{\tilde b}_5} + {{\tilde c}_5})\frac{{23\;500}}{{49\;049{\sigma _5}}} +({a_1}{{\tilde a}_3}{{\tilde a}_5} + \\ &\quad\quad {{\tilde a}_5}{{\tilde b}_3} \!+\!{{\tilde a}_2}{{\tilde b}_5} \!+\! {a_1}{{\tilde c}_5} \!-\! {{\tilde d}_5})\frac{{2\;880}}{{49\;049{\sigma _5}}} \!-\! \alpha \left(\!\frac{{587}}{{11\;781{\sigma _5}}} \!-\! \frac{{13\;292{a_5}}}{{270\;725{\sigma _5}}} \!-\! \frac{{837{{\tilde a}_5}}}{{14\;014{\sigma _5}}} \!+\! ({{\tilde b}_5} \!-\! {a_3}{{\tilde a}_5})\frac{{400}}{{21\;021{\sigma _5}}}\!\right) \!+\! \beta \left(\frac{1}{{63{\sigma _5}}} \!-\! \frac{{16}}{{3\;185{\sigma _5}}}\right) \\ &{b_6} = \frac{{622\;689\;488}}{{36\;006\;425{\sigma _4}}} + \frac{{372\;006{a_4}}}{{49\;049{\sigma _4}}} + \frac{{2\;508\;320{{\tilde a}_4}}}{{1\;072\;071{\sigma _4}}} - ({{\tilde b}_4} - {a_2}{{\tilde a}_4})\frac{{23\;500}}{{49\;049{\sigma _4}}} + ({{\tilde a}_2}{{\tilde a}_4} - {a_1}{{\tilde b}_4} + {{\tilde c}_4})\frac{{2\;880}}{{49\;049{\sigma _4}}} +\\ &\quad\quad\alpha \left(\frac{{13\;292}}{{270\;725{\sigma _4}}} - \frac{{837{a_4}}}{{14\;014{\sigma _4}}} - \frac{{400{{\tilde a}_4}}}{{21\;021{\sigma _4}}}\right) + \frac{{16\beta }}{{3\;185{\sigma _4}}} \\ &{c_6} = - \frac{{372\;006}}{{49\;049{\sigma _3}}} - \frac{{2\;508\;320{a_3}}}{{1\;072\;071{\sigma _3}}} - \frac{{23\;500{{\tilde a}_3}}}{{49\;049{\sigma _3}}} + ({\tilde b_3} - {a_1}{\tilde a_3})\frac{{2\;880}}{{49\;049{\sigma _3}}} + \alpha \left(\frac{{837}}{{14\;014{\sigma _3}}} + \frac{{400{a_3}}}{{21\;021{\sigma _3}}}\right)\\ &{d_6} = \frac{{2\;508\;320}}{{1\;072\;071{\sigma _2}}} + \frac{{23\;500{a_2}}}{{49\;049{\sigma _2}}} + \frac{{2\;880{{\tilde a}_2}}}{{49\;049{\sigma _2}}} - \frac{{400\alpha }}{{21\;021{\sigma _2}}},\;\;{e_6} = - \frac{{23\;500}}{{49\;049{\sigma _1}}} - \frac{{2\;880{a_1}}}{{49\;049{\sigma _1}}},\;\;{f_6} = \frac{{2\;880}}{{49\;049{\sigma _0}}} \end{split}$

 $\begin{split}&\!\!\!\!{P_l}(x) = \frac{{{\psi _l}(x)}}{{{{(l + 1)}^2}}} - {a_l}{P_{l - 1}}(x) - {b_l}{P_{l - 2}}(x)- \\ &{c_l}{P_{l - 3}}(x) -{d_l}{P_{l - 4}}(x) - {e_l}{P_{l - 5}}(x) - {f_l}{P_{l - 6}}(x)\end{split}$

 $\begin{split}&{P_k}(x) = \frac{{{\psi _k}(x)}}{{{{(k + 1)}^2}}} - {a_k}{P_{k - 1}}(x) - {b_k}{P_{k - 2}}(x)- {c_k}{P_{k - 3}}(x) -\\ &\quad {d_k}{P_{k - 4}}(x) - {e_k}{P_{k - 5}}(x) - {f_k}{P_{k - 6}}(x)\end{split}$

 $\begin{split} &{D_{\alpha ,\beta }}\left(\frac{{{\psi _k}}}{{{{(1 + k)}^2}}},{P_m}\right)= \left(\partial _x^2\frac{{{\psi _k}}}{{{{(1 + k)}^2}}},\partial _x^2{P_m}\right) + \alpha \left({\partial _x}\frac{{{\psi _k}}}{{{{(1 + k)}^2}}},{\partial _x}{P_m}\right) + \beta \left(\frac{{{\psi _k}}}{{{{(1 + k)}^2}}},{P_m}\right)=\\ &\quad \left(\partial _x^2\frac{{{\psi _k}}}{{{{(1 + k)}^2}}},\partial _x^2\frac{{{\psi _m}}}{{{{(1 + m)}^2}}} - {a_m}\partial _x^2{P_{m - 1}} - {b_m}\partial _x^2{P_{m - 2}} - {c_m}\partial _x^2{P_{m - 3}} - {d_m}\partial _x^2{P_{m - 4}} - {e_m}\partial _x^2{P_{m - 5}} - {f_m}\partial _x^2{P_{m - 6}}\right)+\\ &\quad { \alpha \left({\partial _x}\frac{{{\psi _k}}}{{{{(1 + k)}^2}}},{\partial _x}\frac{{{\psi _m}}}{{{{(1 + m)}^2}}} - {a_m}{\partial _x}{P_{m - 1}} - {b_m}{\partial _x}{P_{m - 2}} - {c_m}{\partial _x}{P_{m - 3}} - {d_m}{\partial _x}{P_{m - 4}} - {e_m}{\partial _x}{P_{m - 5}} - {f_m}{\partial _x}{P_{m - 6}}\right)}+\\ &\quad \beta \left({\frac{{{\psi _k}}}{{{{(1 + k)}^2}}},\frac{{{\psi _m}}}{{{{(1 + m)}^2}}} - {a_m}{P_{m - 1}} - {b_m}{P_{m - 2}} - {c_m}{P_{m - 3}}}- {d_m}{P_{m - 4}} - {e_m}{P_{m - 5}} - {f_m}{P_{m - 6}}\right)=\\ &\quad \left(\partial _x^2\frac{{{\psi _k}}}{{{{(1 + k)}^2}}},\partial _x^2\frac{{{\psi _m}}}{{{{(1 + m)}^2}}} -\right. {a_m}\partial _x^2\frac{{{\psi _{m - 1}}}}{{{m^2}}} + {\tilde a_m}\partial _x^2\frac{{{\psi _{m - 2}}}}{{{{(m - 1)}^2}}}+ ({\tilde b_m} - {a_{m - 2}}{\tilde a_m})\partial _x^2\frac{{{\psi _{m - 3}}}}{{{{(m - 2)}^2}}}+ ({\tilde a_m}{\tilde a_{m - 2}} - {a_{m - 3}}{\tilde b_m} + {\tilde c_m})\partial _x^2\frac{{{\psi _{m - 4}}}}{{{{(m - 3)}^2}}}+\\ &\quad { ( - {a_{m - 4}}{{\tilde a}_m}{{\tilde a}_{m - 2}} + {{\tilde a}_m}{{\tilde b}_{m - 2}} + {{\tilde a}_{m - 3}}{{\tilde b}_m} - {a_{m - 4}}{{\tilde c}_m} + {{\tilde d}_m}){\partial _x}^2\frac{{{\psi _{m - 5}}}}{{{{(m - 4)}^2}}}}+ \beta \left(\frac{{{\psi _k}}}{{{{(k + 1)}^2}}},\frac{{{\psi _m}}}{{{{(m + 1)}^2}}} - {a_m}\frac{{{\psi _{m - 1}}}}{{{m^2}}}\right)+\\ &\quad \alpha \left({{\partial _x}\frac{{{\psi _k}}}{{{{(1 + k)}^2}}},{\partial _x}\frac{{{\psi _m}}}{{{{(1 + m)}^2}}} - {a_m}{\partial _x}\frac{{{\psi _{m - 1}}}}{{{m^2}}} + {{\tilde a}_m}{\partial _x}\frac{{{\psi _{m - 2}}}}{{{{(m - 1)}^2}}}}+ ({\tilde b_m} - {a_{m - 2}}{\tilde a_m}){\partial _x}\frac{{{\psi _{m - 3}}}}{{{{(m - 2)}^2}}}\right)\end{split}$

$m = k - 1,\;\;k - 2,\;\;k - 3,\;\;k - 4,\;\;k - 5,\;\;k - 6$ ，可以得到结论 ${\rm{b}} \sim {\rm{g}}$

 $\begin{split} &{D_{\alpha ,\beta }}\left(\frac{{{\psi _k}}}{{{{(k + 1)}^2}}},\frac{{{\psi _k}}}{{{{(k + 1)}^2}}}\right)= \left(\partial _x^2\frac{{{\psi _k}}}{{{{(k + 1)}^2}}},\partial _x^2\frac{{{\psi _k}}}{{{{(k + 1)}^2}}}\right) + \\ &\quad \alpha \left({\partial _x}\frac{{{\psi _k}}}{{{{(k + 1)}^2}}},{\partial _x}\frac{{{\psi _k}}}{{{{(k + 1)}^2}}}\right) + \beta \left(\frac{{{\psi _k}}}{{{{(k + 1)}^2}}},\frac{{{\psi _k}}}{{{{(k + 1)}^2}}}\right)=\\ &\quad \frac{{3(154{k^{11}} + 2\;541{k^{10}} + 16\;737{k^9} + 54\;432{k^8})}}{{2(2k + 9)(2k + 7)(2k + 5)(2k - 3)(4{k^2} - 1)}}+\\ &\quad \frac{{3( - 128\;220{k^3} + 31\;752{k^2} + 44\;064k + 10\;368)}}{{2(2k + 9)(2k + 7)(2k + 5)(2k - 3)(4{k^2}- 1)}}+\\ &\quad \alpha \frac{{10{k^7} \!+\! 105{k^6} \!+\! 417{k^5} \!+\! 765{k^4} \!+\! 589{k^3} \!+\! 18{k^2} \!-\! 176k \!-\! 48}}{{(2k + 7)(2k + 5)(4{k^2} - 1)}}+ \\ &\quad \beta \frac{{12(2{k^3} + 9{k^2} + 13k + 6)}}{{(2k + 5)(2k + 1)}}=\tilde S + \alpha \tilde T + \beta \tilde W\end{split}$

 $\begin{split} &{D_{\alpha ,\beta }}\left(\frac{{{\psi _k}}}{{{{(k + 1)}^2}}},\frac{{{\psi _k}}}{{{{(k + 1)}^2}}}\right)={D_{\alpha ,\beta }}({P_k} + {a_k}{P_{k - 1}} + {b_k}{P_{k - 2}} + \\ &\quad {c_k}{P_{k - 3}} + {d_k}{P_{k - 4}} + {e_k}{P_{k - 5}} + {f_k}{P_{k - 6}},\;\;{P_k} + {a_k}{P_{k - 1}} +\\ &\quad {b_k}{P_{k - 2}} + {c_k}{P_{k - 3}} + {d_k}{P_{k - 4}} + {e_k}{P_{k - 5}} + {f_k}{P_{k - 6}})= \\ &\quad {\sigma _k} + {({a_k})^2}{\sigma _{k - 1}} + {({b_k})^2}{\sigma _{k - 2}} + {({c_k})^2}{\sigma _{k - 3}}+ {({d_k})^2}{\sigma _{k - 4}} +\\ &\quad {({e_k})^2}{\sigma _{k - 5}} + {({f_k})^2}{\sigma _{k - 6}}\end{split}$

 $\begin{gathered} u\left( x \right) = \sum\limits_{k = 0}^\infty {{{\hat u}_k}{P_k}(x)} ,\;\;\;\;\;{u_N}(x) = \sum\limits_{k = 0}^N {{{\hat u}_k}{P_k}(x)} \\ {{\hat u}_k} = \frac{1}{{{\eta _k}}}{D_{\alpha ,\beta }}\left( {u,{P_k}} \right) = \frac{1}{{{\eta _k}}}(f,{P_k}),\;\;\;k \geqslant 0 \\ \end{gathered}$
4 数值实验

 图 1 指数衰减时谱格式（16）的误差 Fig. 1 Errors of scheme (16) with exponential decay function

 图 2 代数衰减时谱格式（16）的误差 Fig. 2 Errors of scheme (16) with algebraic decay function

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