查看“︁迪姆方程”︁的源代码

'''迪姆方程'''是[[以色列]]数学家哈利·迪姆创建的三阶[[非线性偏微分方程]]:

:<math>u_t = u^3u_{xxx}.\,</math>
==Bäcklund变换解==
[[File:Dym eq Backlund solution animation.gif|thumb|right|300px|Dym eq Backlund transform solution animation]]
通过[[Bäcklund变换]]可得迪姆方程的分析解<ref>^ Fritz Gesztesy and Karl Unterkofler, Isospectral deformations for Sturm–Liouville and Dirac-type operators and associated nonlinear evolution equations, Rep. Math. Phys. 31 (1992), 113–137.</ref>。

: <math> u(t,x) = (- 3 \alpha (x + 4 \alpha^2 t )^{2/3} . </math>


==近似解==
[[File:Harry Dym nlpde 3d animation.gif|thumb|right|400px|Harry Dym nlpde 3d animation]]
迪姆方程有解析解<ref>W HeremanP P, BanerjeeSS and M R Chatterjee, Derivation and implicit solution of the Harry Dym equation and its connections with the Korteweg-de Vries equation;J. Phys. A: Math. Gen. 22 (1989) 241-255. </ref>。  

<math>u(x,t)= \tanh(\frac{\sqrt(V)}{2}*(x-x[0]+V*t+ef(x, t)))^2</math>

其中

<math>ef(x,t)=\frac{2}{\sqrt(V)}*tanh((\frac{\sqrt(V)}{2}*(x-x[0]+V*t+ef(x,t)))</math>

五次迭代后可得近似解：

<math>tanh((1/2)*x*\sqrt(V)-(1/2)*x[0]*\sqrt(V)+(1/2)*V^(3/2)*t+tanh((1/2)*x*\sqrt(V)-(1/2)*x[0]*\sqrt(V)+(1/2)*V^(3/2)*t+tanh((1/2)*x*\sqrt(V)-(1/2)*x[0]*\sqrt(V)+(1/2)*V^(3/2)*t+tanh((1/2)*x*\sqrt(V)-(1/2)*x[0]*\sqrt(V)+(1/2)*V^(3/2)*t+tanh((1/2)*\sqrt(V)*(x-x[0]+V*t))))))</math>
==阿多米安近似法==
[[File:Dym equation Adomian cos plot.gif|thumb|left|200px]]
[[File:Dym equation Admain plot cosh.gif|thumb|left|200px|]]

用[[阿多米安分解法]]可求得迪姆方程的[[柯西问题]]近似解<ref>Inna Shingareve  Carlos Lizarraga Celaya，Solving Nonlinear Partial Differential Equations with Maple and Mathematica  p230-236, Springer</ref>。

;初始条件：u（0）=cos(x)
pa := (.53823*sin(10.*x)+0.16931e-1*sin(4.*x)+.72240*sin(8.*x)+.24408*sin(6.*x)+0.57870e-4*sin(2.*x))*t^9+(0.59326e-2*cos(3.*x)+0.13563e-5*cos(x)+.55850*cos(7.*x)+.46338*cos(9.*x)+.15138*cos(5.*x))*t^8+(-0.13889e-2*sin(2.*x)-.43393*sin(6.*x)-0.88889e-1*sin(4.*x)-.40635*sin(8.*x))*t^7+(-.33908*cos(5.*x)-0.47461e-1*cos(3.*x)-0.10851e-3*cos(x)-.36474*cos(7.*x))*t^6+(.26667*sin(4.*x)+0.20833e-1*sin(2.*x)+.33750*sin(6.*x))*t^5+(.21094*cos(3.*x)+.32552*cos(5.*x)+0.52083e-2*cos(x))*t^4+(-.16667*sin(2.*x)-.33333*sin(4.*x))*t^3+(-.12500*cos(x)-.37500*cos(3.*x))*t^2+.50000*t*sin(2.*x)+cos(x)



;初始条件 u(0)=cosh(x)
pa := (-.5382*sinh(10.*x)-.7224*sinh(8.*x)-.2441*sinh(6.*x)-0.5787e-4*sinh(2.*x)-0.1693e-1*sinh(4.*x))*t^9+(.4634*cosh(9.*x)+0.5933e-2*cosh(3.*x)+.5585*cosh(7.*x)+.1514*cosh(5.*x)+0.1356e-5*cosh(x))*t^8+(-.4063*sinh(8.*x)-0.8889e-1*sinh(4.*x)-.4339*sinh(6.*x)-0.1389e-2*sinh(2.*x))*t^7+(0.1085e-3*cosh(x)+0.4746e-1*cosh(3.*x)+.3647*cosh(7.*x)+.3391*cosh(5.*x))*t^6+(-0.2083e-1*sinh(2.*x)-.2667*sinh(4.*x)-.3375*sinh(6.*x))*t^5+(.3255*cosh(5.*x)+0.5208e-2*cosh(x)+.2109*cosh(3.*x))*t^4+(-.3333*sinh(4.*x)-.1667*sinh(2.*x))*t^3+(.3750*cosh(3.*x)+.1250*cosh(x))*t^2-.5000*t*sinh(2.*x)+cosh(x)

==参考文献==
{{非线性偏微分方程}}
<references/>
# *谷超豪 《[[孤立子]]理论中的[[达布变换]]及其几何应用》 上海科学技术出版社
# *阎振亚著 《复杂非线性波的构造性理论及其应用》 科学出版社 2007年
# 李志斌编著 《非线性数学物理方程的行波解》 科学出版社
#王东明著 《消去法及其应用》 科学出版社 2002
# *何青 王丽芬编著 《[[Maple]] 教程》 科学出版社 2010 ISBN 9787030177445
#Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
# Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
#Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
#Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
#Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
#Dongming Wang, Elimination Practice,Imperial College Press 2004
# David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
# George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759

[[category:非线性偏微分方程]]
[[category:孤立子]]