齐次蒙日-安培方程

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齐次蒙日-安培方程(Homogeneous Monge-Ampère equation)是一个常见于黎曼几何的非线性偏微分方程:[1]

sys = {(U[x,t])^2-U[x,x]*U[t,t]=0 };


解析解[编辑]

 u[1] := C1*y^2+C2*x*y+C2^2*x^2/(4*C1)+C3*y+C4*x+C5
 u[2] := (C2*y^2+C3*y+C3^2/(4*C2))/(x+C1)+C4*y+C5*x+C6
 u[3] := (C1*x+C2*y+C3)^(k+1)/(C4*x+C5*y+C6)^k+C7*x+C8*y+C9
 u[4] := (C1*x+C2*y+C3)^k+C4*x+C5*y+C6
 u[5] := -(C1*x+C2*y+C3)^k+C4*x+C5*y+C6
  u[6] := -(C1*x+C2*y+C3)^(k+1)/(C4*x+C5*y+C6)^k+C7*x+C8*y+C9
 u[7] := \sqrt{C1*x^2+2*C1*x*a+C1*a^2+C2*x*y+C2*x*b+C2*a*y+C2*a*b+C3*y^2+2*C3*y*b+C3*b^2}+C5*x+C6*y+C7
 u[8] := C1*(C1*y^2+C2*x*y+(1/4)*C2^2*x^2/C1+C3*y+C4*x+C5)*(C2*x+C3*y+C4)+C5*x+C6*y+C7
 u[9] := C1*((C2*y^2+C3*y+(1/4)*C3^2/C2)/(x+C1)+C4*y+C5*x+C6)*(C2*x+C3*y+C4)+C5*x+C6*y+C7
 u[10] := C1*((C1*x+C2*y+C3)^(k+1)/(C4*x+C5*y+C6)^k+C7*x+C8*y+C9)*(C2*x+C3*y+C4)+C5*x+C6*y+C7

行波图[编辑]

Homogeneous Monge-Ampere equation plot
Homogeneous Monge-Ampere equation plot
Homogeneous Monge-Ampere equation plot
Homogeneous Monge-Ampere equation plot
Homogeneous Monge-Ampere equation plot
Homogeneous Monge-Ampere equation plot
Homogeneous Monge-Ampere equation plot
Homogeneous Monge-Ampere equation plot
Homogeneous Monge-Ampere equation plot
Homogeneous Monge-Ampere equation plot
Homogeneous Monge-Ampere equation plot
Homogeneous Monge-Ampere equation plot

参考文献[编辑]

  1. ^ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p775-776 CRC PRESS
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