aa tk rat =Ka i+ d choose normalize dimensionless 4 variables variable oA 1+R A-A+2c2 Q +p homogeneous solution =R+1 0/as=0/at=0 =(R+1)22 2 2 2 2 2 x i k a i D t i x a a D i a r k t a i i i a a a a ∂ ∂ = − + ∂ ∂ ∂ ∂ = + − + ∂ ∂ γ γ choose dimensionless variable normalize 4 variables ( ) 2 2 2 2 2 2 1 s I Q A I P τ I s A A I A R τ A ∂ ∂ = − + ∂ ∂ ∂ ∂ = + − + ∂ ∂ 2 ( 1) 1 = + = + I R A R homogeneous solution ∂ / ∂s = ∂ / ∂t = 0 2