广义伯格斯-KdV方程 (Generalized Burgers-KdV equation)是一个非线性偏微分方程:[1]
解析解
当 n=7, 有下列特解:
![{\displaystyle u(x,t)=(71280*\alpha *_{C}4^{7}*_{C}1+_{C}5)/(\beta *_{C}4)-665280*\alpha *_{C}4^{6}*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,0,_{C}1)^{3}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/62fe28e9bdd90dc2d5d7774062fcafec240f3e25)
![{\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})/(\beta *_{C}3)-665280*\alpha *_{C}3^{6}*(-1+(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*((-(1/2)*{\sqrt {(}}3)-1/2*I)^{2}-2)*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{4}/\beta +665280*\alpha *_{C}3^{6}*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{6}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a236cdce4ecca3679daa03087c91e01ef648ec8)
![{\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})/(\beta *_{C}3)-665280*\alpha *_{C}3^{6}*(-1+(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*((-(1/2)*{\sqrt {(}}3)-1/2*I)^{2}-2)*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{4}/\beta +665280*\alpha *_{C}3^{6}*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{6}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/532b3ccfc2ad48d4402ddb1afee0aa6f7848e001)
![{\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})/(\beta *_{C}3)-665280*_{C}3^{6}*\alpha *(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2}*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*(1+(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{4}/\beta -665280*\alpha *_{C}3^{6}*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{6}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d951896e97e39c13c8148d495154f5415a245702)
![{\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*(-(1/2)*{\sqrt {(}}3)+1/2*I)^{2})/(\beta *_{C}3)-665280*_{C}3^{6}*\alpha *(-(1/2)*{\sqrt {(}}3)+1/2*I)^{2}*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)-1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*(1+(-(1/2)*{\sqrt {(}}3)+1/2*I)^{2})*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)-1/2*I)^{4}/\beta -665280*\alpha *_{C}3^{6}*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)-1/2*I)^{6}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1790cf0ffdf3e6561ea5946882de60e2677f6ad8)
![{\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*((1/2)*{\sqrt {(}}3)+1/2*I)^{2})/(\beta *_{C}3)-665280*\alpha *_{C}3^{6}*(-1+((1/2)*{\sqrt {(}}3)+1/2*I)^{2})*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*(((1/2)*{\sqrt {(}}3)+1/2*I)^{2}-2)*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{4}/\beta +665280*\alpha *_{C}3^{6}*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{6}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e1f13881c0676b253e59c067f7c3ca54e370a66)
广义伯格斯-KdV方程之部分通解为:
![{\displaystyle u(x,t)=C1^{(}n-1)*(x+C1)*(C1*x+b*C1*C2*t+C3)/(b*(C2+t))+C2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f71d7ece64b0f50c4e5181010374f9a2644df6a2)
![{\displaystyle u(x,t)=C1^{(}n-1)*(x+C1)*(C1*x+b*C1*C2*t+C3)/(b*(C2+t))+C2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f71d7ece64b0f50c4e5181010374f9a2644df6a2)
![{\displaystyle u(x,t)=C1^{(}n-1)*((-1)^{n}*a*(2*n-1)!/(b*(n-1)!*(x+b*C1*t+C2)^{(}n-1))+C1)*(C1*x+b*C1*C2*t+C3)+C2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8856e6a786d713f89fe38eb57bcc29ee05a5ca4)
![{\displaystyle u(x,t)=C1^{(}n-1)*((-1)^{n}*a*(2*n-1)!/(b*(n-1)!*(x+b*C1*t+C2)^{(}n-1))+C1)*(C1*x+b*C1*C2*t+C3)+C2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8856e6a786d713f89fe38eb57bcc29ee05a5ca4)
![{\displaystyle u(x,t)=C1^{(}n-1)*(x+C1)*(C1^{n}*t+C4)/(b*(C2+t))+C2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b675b9934c19bcd39bccae5d30aa06a63da793c4)
![{\displaystyle u(x,t)=C1^{(}n-1)*(x+C1)*(C1^{n}*t+C4)/(b*(C2+t))+C2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b675b9934c19bcd39bccae5d30aa06a63da793c4)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
行波图
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
7阶广义伯格斯-KdV方程行波图
|
10阶广义伯格斯-KdV方程行波图
|
10阶广义伯格斯-KdV方程行波图
|
10阶广义伯格斯-KdV方程行波图
|
10阶广义伯格斯-KdV方程行波图
|
10阶广义伯格斯-KdV方程行波图
|
10阶广义伯格斯-KdV方程行波图
|
10阶广义伯格斯-KdV方程行波图
|
参考文献
- ^ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS,(《非线性偏微分方程手册》) SECOND EDITION p1045 CRC PRESS
- *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