KPP type equation

KPP type equation is a nonlinear partial differential equation proposed by Andrey Kolmogorov, Ivan Petrovsky, and Nikolai Piskunov：[1]

u_{t}-\alpha*u-\beta*u^2+\gamma*u^3=0

Analytic solutions

u(x, t) = -(1/2)*\beta/\gamma-(1/2)*\sqrt(-\beta^2+4*\gamma*\alpha)*cot(_C1-(1/4)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)*\gamma)*x/\gamma-(1/4)*\sqrt(-\beta^2+4*\gamma*\alpha)*\beta*t/\gamma)/\gamma
u(x, t) = -(1/2)*\beta/\gamma+(1/2)*\sqrt(-\beta^2+4*\gamma*\alpha)*tan(_C1-(1/4)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)*\gamma)*x/\gamma-(1/4)*\sqrt(-\beta^2+4*\gamma*\alpha)*\beta*t/\gamma)/\gamma
u(x, t) = -(1/2)*\beta/\gamma-(1/2)*\sqrt(\beta^2-4*\gamma*\alpha)*coth(_C1-(1/4)*\sqrt(2)*\sqrt((-\beta^2+4*\gamma*\alpha)*\gamma)*x/\gamma-(1/4)*\sqrt(\beta^2-4*\gamma*\alpha)*\beta*t/\gamma)/\gamma
u(x, t) = -(1/2)*\beta/\gamma-(1/2)*\sqrt(\beta^2-4*\gamma*\alpha)*tanh(_C1-(1/4)*\sqrt(2)*\sqrt((-\beta^2+4*\gamma*\alpha)*\gamma)*x/\gamma-(1/4)*\sqrt(\beta^2-4*\gamma*\alpha)*\beta*t/\gamma)/\gamma
{u(x, t) = -(1/2)*\beta*(16*((1/8)*\sqrt(2)*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)-(1/8*I)*\beta*\sqrt(2)*\sqrt(1/\gamma))^2+\alpha)/(-3*\gamma*\alpha+\beta^2+24*\gamma*((1/8)*\sqrt(2)*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)-(1/8*I)*\beta*\sqrt(2)*\sqrt(1/\gamma))^2)+(1/4)*(-3*\gamma*\alpha+\beta^2+24*\gamma*((1/8)*\sqrt(2)*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)-(1/8*I)*\beta*\sqrt(2)*\sqrt(1/\gamma))^2)*\sqrt(36*\alpha^2*\gamma^2-16*\alpha*\gamma*\beta^2-(12*I)*\gamma^2*\alpha*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)*\beta*\sqrt(1/\gamma)+2*\beta^4+(2*I)*\beta^3*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)*\sqrt(1/\gamma)*\gamma)*tanh(_C1+((1/8)*\sqrt(2)*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)-(1/8*I)*\beta*\sqrt(2)*\sqrt(1/\gamma))*x+(1/8)*\sqrt(36*\alpha^2*\gamma^2-16*\alpha*\gamma*\beta^2-(12*I)*\gamma^2*\alpha*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)*\beta*\sqrt(1/\gamma)+2*\beta^4+(2*I)*\beta^3*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)*\sqrt(1/\gamma)*\gamma)*t/\gamma)/(\gamma*\beta*(9*\gamma*\alpha-2*\beta^2))}
{u(x, t) = -(1/2)*\beta*(16*((1/8)*\sqrt(2)*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)+(1/8*I)*\beta*\sqrt(2)*\sqrt(1/\gamma))^2+\alpha)/(-3*\gamma*\alpha+\beta^2+24*\gamma*((1/8)*\sqrt(2)*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)+(1/8*I)*\beta*\sqrt(2)*\sqrt(1/\gamma))^2)+(1/4)*(-3*\gamma*\alpha+\beta^2+24*\gamma*((1/8)*\sqrt(2)*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)+(1/8*I)*\beta*\sqrt(2)*\sqrt(1/\gamma))^2)*\sqrt(2)*\sqrt(18*\alpha^2*\gamma^2-8*\alpha*\gamma*\beta^2+(6*I)*\gamma^2*\alpha*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)*\beta*\sqrt(1/\gamma)+\beta^4-I*\beta^3*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)*\sqrt(1/\gamma)*\gamma)*coth(_C1+((1/8)*\sqrt(2)*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)+(1/8*I)*\beta*\sqrt(2)*\sqrt(1/\gamma))*x+(1/8)*\sqrt(2)*\sqrt(18*\alpha^2*\gamma^2-8*\alpha*\gamma*\beta^2+(6*I)*\gamma^2*\alpha*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)*\beta*\sqrt(1/\gamma)+\beta^4-I*\beta^3*\sqrt((-\beta^2+4*\gamma*\alpha)/\gamma)*\sqrt(1/\gamma)*\gamma)*t/\gamma)/(\gamma*\beta*(9*\gamma*\alpha-2*\beta^2))}
{u(x, t) = -(1/2)*\beta*(16*(-(1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))^2-\alpha)/(3*\gamma*\alpha-\beta^2+24*\gamma*(-(1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))^2)+(1/4)*(3*\gamma*\alpha-\beta^2+24*\gamma*(-(1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))^2)*\sqrt(16*\alpha*\gamma*\beta^2+6*\gamma^(3/2)*\alpha*\sqrt(2)*\beta*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)-36*\alpha^2*\gamma^2-2*\beta^4-\beta^3*\sqrt(2)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)*\sqrt(\gamma))*cot(_C1+(-(1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))*x-(1/8)*\sqrt(16*\alpha*\gamma*\beta^2+6*\gamma^(3/2)*\alpha*\sqrt(2)*\beta*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)-36*\alpha^2*\gamma^2-2*\beta^4-\beta^3*\sqrt(2)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)*\sqrt(\gamma))*t/\gamma)/(\gamma*\beta*(9*\gamma*\alpha-2*\beta^2))}
{u(x, t) = -(1/2)*\beta*(16*(-(1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))^2-\alpha)/(3*\gamma*\alpha-\beta^2+24*\gamma*(-(1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))^2)-(1/4)*(3*\gamma*\alpha-\beta^2+24*\gamma*(-(1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))^2)*\sqrt(16*\alpha*\gamma*\beta^2+6*\gamma^(3/2)*\alpha*\sqrt(2)*\beta*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)-36*\alpha^2*\gamma^2-2*\beta^4-\beta^3*\sqrt(2)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)*\sqrt(\gamma))*cot(_C1+(-(1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))*x+(1/8)*\sqrt(16*\alpha*\gamma*\beta^2+6*\gamma^(3/2)*\alpha*\sqrt(2)*\beta*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)-36*\alpha^2*\gamma^2-2*\beta^4-\beta^3*\sqrt(2)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)*\sqrt(\gamma))*t/\gamma)/(\gamma*\beta*(9*\gamma*\alpha-2*\beta^2))}
{u(x, t) = -(1/2)*\beta*(16*(-(1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))^2-\alpha)/(3*\gamma*\alpha-\beta^2+24*\gamma*(-(1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))^2)+(1/4)*(3*\gamma*\alpha-\beta^2+24*\gamma*(-(1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))^2)*\sqrt(16*\alpha*\gamma*\beta^2+6*\gamma^(3/2)*\alpha*\sqrt(2)*\beta*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)-36*\alpha^2*\gamma^2-2*\beta^4-\beta^3*\sqrt(2)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)*\sqrt(\gamma))*tan(_C1+(-(1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))*x+(1/8)*\sqrt(16*\alpha*\gamma*\beta^2+6*\gamma^(3/2)*\alpha*\sqrt(2)*\beta*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)-36*\alpha^2*\gamma^2-2*\beta^4-\beta^3*\sqrt(2)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)*\sqrt(\gamma))*t/\gamma)/(\gamma*\beta*(9*\gamma*\alpha-2*\beta^2))}
{u(x, t) = -(1/2)*\beta*(16*((1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))^2-\alpha)/(3*\gamma*\alpha-\beta^2+24*\gamma*((1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))^2)+(1/4)*(3*\gamma*\alpha-\beta^2+24*\gamma*((1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))^2)*\sqrt(16*\alpha*\gamma*\beta^2-6*\gamma^(3/2)*\alpha*\sqrt(2)*\beta*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)-36*\alpha^2*\gamma^2-2*\beta^4+\beta^3*\sqrt(2)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)*\sqrt(\gamma))*cot(_C1+((1/8)*\sqrt(2)*\beta/\sqrt(\gamma)-(1/8)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma))*x-(1/8)*\sqrt(16*\alpha*\gamma*\beta^2-6*\gamma^(3/2)*\alpha*\sqrt(2)*\beta*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)-36*\alpha^2*\gamma^2-2*\beta^4+\beta^3*\sqrt(2)*\sqrt(-(-2*\beta^2+8*\gamma*\alpha)/\gamma)*\sqrt(\gamma))*t/\gamma)/(\gamma*\beta*(9*\gamma*\alpha-2*\beta^2))}

Traveling wave plots

KPP type equation traveling wave plot
KPP type equation traveling wave plot
KPP type equation traveling wave plot
KPP type equation traveling wave plot
KPP type equation traveling wave plot
KPP type equation traveling wave plot
KPP type equation traveling wave plot
KPP type equation traveling wave plot
KPP type equation traveling wave plot
KPP type equation traveling wave plot
KPP type equation traveling wave plot
KPP type equation traveling wave plot
KPP type equation traveling wave plot
KPP type equation traveling wave plot
KPP type equation traveling wave plot
KPP type equation traveling wave plot

References

  1. ↑ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p1633 CRC PRESS
  1. Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential Equations Academy Press
  2. Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
  3. Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
  4. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
  5. Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
  6. Dongming Wang, Elimination Practice,Imperial College Press 2004
  7. David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
  8. George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759