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Nonlinear Mathematical Physics 1994, V.1, N 1, 60–64. Printed in the Ukraina.

Antireduction and exact solutions of nonlinear heat equations
WILHELM FUSHCHYCH and RENAT ZHDANOV,
Mathematical Institute of the Ukrainian Academy of Sciences, Tereshchenkivska Street 3, 252004 Kiev, Ukraina Received October 10, 1993 Abstract We construct a number of ansatzes that reduce one-dimensional nonlinearheat equations to systems of ordinary differential equations. Integrating these, we obtain new exact solution of nonlinear heat equations with various nonlinearities.

By the term antireduction for a partial differential equation (PDE) we mean the construction of an ansatz which transforms the PDE to a system of differential equations for several unknown differentiable functions. As a rule, suchprocedure reduces the PDE under consideration to a system of PDE with fewer numbers of independent variables and greater number of dependent variables [1–4]. Antireduction of the nonlinear acoustics equation ux0 x1 − (ux1 u)x1 − ux2 x2 − ux3 x3 = 0 is carried out in the paper [2] with the use of the ansatz 1 1 u = x1 ϕ1 (x0 , x2 , x3 ) − x2 ϕ2 (x0 , x2 , x3 ) + ϕ3 (x0 , x2 , x3 ). 3 6 1 In [3]antireduction of the equation for short waves in gas dynamics 2ux0 x1 − 2(2x1 + ux1 )ux1 x1 + ux2 x2 + 2λux1 = 0 is carried out via the following ansatz: u = x1 ϕ1 + x2 ϕ2 + x1 ϕ3 + ϕ4 , 1
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(2)

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ϕi = ϕi (x0 , x2 ).

(4)

c 1994 by Mathematical Ukraina Publisher.

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ANTIREDUCTION AND EXACT SOLUTIONS

61Ansatzes (2), (4) reduce equations (1), (3) to system of PDE for three and four functions, respectively. In the present paper we adduce some new results on antireduction for the nonlinear heat equations of the form ut = (a(u)ux )x + F (u). The antireduction of equation (5) is performed by means of the ansatz h(t, x, u, ϕ1 (ω), ϕ2 (ω), ..., ϕN (ω)) = 0 (6) (5)

where ω = ω(t, x, u) is a newindependent variable. Ansatz (6) reduces equation (5) to a system of ordinary differential equations (ODE) for the unknown functions ϕi (ω), i = 1, N . Below we list, without derivation, explicit forms of a(u) and F (u), such that equation (5) admits an antireduction of the form (6). For each case the reduced ODE are given. 1. ¨ a(u) = θ(u)θ(u), ˙ θ(u) = ϕ1 (t) + ϕ2 (t)x, ϕ1 = (λ2 + ϕ2 )ϕ1 + λ1 , ˙ 2 2. ˙a(u) = uθ(u), θ(u) = ϕ1 (t) + ϕ2 (t)x, ϕ1 = λ 2 ϕ1 + ϕ 2 + λ 1 , ˙ 2 3. ˙ a(u) = θ(u), θ(u) = ϕ1 (t) + ϕ2 (t)x, ϕ1 = λ 2 ϕ1 + λ 1 , ˙ 4. a(u) = λuk , ϕ2 = λ 2 ϕ2 ; ˙ F (u) = λ1 u + λ2 u1−k , ϕ2 = λ 2 ϕ2 ; ˙ ˙ F (u) = (λ1 + λ2 θ(u))(θ(u))−1 , ˙ ¨ F (u) = (λ1 + λ2 θ(u))(θ(u))−1 , ϕ2 = (λ2 + ϕ2 )ϕ2 ; ˙ 2 ˙ F (u) = (λ1 + λ2 θ(u))(θ(u))−1 ,

uk = ϕ1 (t) + ϕ2 (t)x + ϕ3 (t)x2 , ϕ1 = 2λϕ1 ϕ3 + λk −1 ϕ2+ kλ2 , ˙ 2 ϕ2 = 2λ(1 + 2k −1 )ϕ2 ϕ3 + kλ1 ϕ2 , ˙ ϕ3 = 2λ(1 + 2k −1 )ϕ2 + kλ1 ϕ3 ; ˙ 3 5. a(u) = λeu , F (u) = λ1 + λ2 e−u ,

eu = ϕ1 (t) + ϕ2 (t)x + ϕ3 (t)x2 , ϕ1 = 2λϕ1 ϕ3 + λ1 ϕ1 + λ2 , ˙ ϕ2 = 2λϕ2 ϕ3 + λ1 ϕ2 , ˙ ϕ3 = 2λϕ2 + λ1 ϕ3 ; ˙ 3

62 6. a(u) = λu−3/2 ,

W.FUSHCHYCH and R.ZHDANOV

F (u) = λ1 u + λ2 u5/2 ,

u−3/2 = ϕ1 (t) + ϕ2 (t)x + ϕ3 (t)x2 + ϕ4 (t)x3 , 2 3 3 ϕ1 = 2λϕ1 ϕ3 −λϕ2 − λ1 ϕ1 − λ2 , ˙ 2 3 2 2 2 3 ϕ2 = − λϕ2 ϕ3 + 6λϕ1 ϕ4 − λ1 ϕ2 , ˙ 3 2 2 2 3 ϕ3 = − λϕ3 + 2λϕ2 ϕ4 − λ1 ϕ3 , ˙ 3 2 3 ϕ 4 = − λ 1 ϕ4 ; ˙ 2 7. a(u) = 1, ln u = ϕ1 (t) + ϕ2 (t)x, ϕ1 = βϕ1 + ϕ2 + α, ˙ 2 8. a(u) = 1, ϕ2 = αϕ2 ; ˙ F (u) = (α + β ln u)u,

F (u) = (α + β ln u − γ 2 (ln u)2 )u, ϕ2 = (β + γ 2 − 2γ 2 ϕ1 )ϕ2 ; ˙

ln u = ϕ1 (t) + ϕ2 (t)eγx , ϕ1 = α + βϕ1 − γ 2 ϕ2 , ˙ 1 9. a(u) = 1,
ln uF (u) = −u(1 + ln u2 )(α + β (ln u)−1/2 ), 2ατ + 4βτ 1/2 + ϕ2 (t) ϕ2 = 4β 2 − 2αϕ2 ; ˙ F (u) = −2(u3 + αu2 + βu),
−1/2

dτ = x + ϕ1 (t),

ϕ1 = 0, ˙ 10. a(u) = 1,

(a) α = β = 0 u = (ϕ1 (t) + 2ϕ2 (t)x)(1 + ϕ1 (t)x + ϕ2 (t)x2 )−1 , ϕ1 = −6ϕ1 ϕ2 , ˙ (b) ϕ2 = −6ϕ2 ; ˙ 2
−1

α2 = 4β = 0 α α u = − ϕ1 (t) + 1 − x ϕ2 (t) eαx/2 + ϕ1 (t) + ϕ2 (t)x 2 2 2 α α2 ϕ1 = − ϕ1 − αϕ2 , ˙ ϕ2 = − ϕ2 ; ˙ 4...