Antireduction and exact solutions of nonlinear heat equations

Transcription

1 Nonlinear Mathematical Physics 1994, V.1, N 1, 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, 5004 Kiev, Ukraina Received October 10, 1993 Abstract We construct a number of ansatzes that reduce one-dimensional nonlinear heat 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, such procedure 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 u x0 x 1 u x1 u x1 u x x u x3 x 3 = 0 1 is carried out in the paper [] with the use of the ansatz u = 1 3 x 1ϕ 1 x 0, x, x x 1ϕ x 0, x, x 3 + ϕ 3 x 0, x, x 3. In [3] antireduction of the equation for short waves in gas dynamics u x0 x 1 x 1 + u x1 u x1 x 1 + u x x + λu x1 = 0 3 is carried out via the following ansatz: u = x 1 ϕ 1 + x 1ϕ + x 3/ 1 ϕ 3 + ϕ 4, ϕ i = ϕ i x 0, x. 4 Copyright c 1994 by Mathematical Ukraina Publisher.. All rights of reproduction in any form reserved.

2 ANTIREDUCTION AND EXACT SOLUTIONS 61 Ansatzes, 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 u t = auu x x + F u. 5 The antireduction of equation 5 is performed by means of the ansatz ht, x, u, ϕ 1 ω, ϕ ω,..., ϕ N ω = 0 6 where ω = ωt, x, u is a new independent variable. Ansatz 6 reduces equation 5 to a system of ordinary differential equations ODE for the unknown functions ϕ i ω, i = 1, N. Below we list, without derivation, explicit forms of au and F u, such that equation 5 admits an antireduction of the form 6. For each case the reduced ODE are given. 1. au = θuθu, F u = λ 1 + λ θu θu 1, θu = ϕ 1 t + ϕ tx, ϕ 1 = λ + ϕ ϕ 1 + λ 1, ϕ = λ + ϕ ϕ. au = u θu, F u = λ 1 + λ θu θu 1, θu = ϕ 1 t + ϕ tx, ϕ 1 = λ ϕ 1 + ϕ + λ 1, ϕ = λ ϕ 3. au = θu, F u = λ 1 + λ θu θu 1, θu = ϕ 1 t + ϕ tx, ϕ 1 = λ ϕ 1 + λ 1, ϕ = λ ϕ 4. au = λu k, F u = λ 1 u + λ u 1 k, u k = ϕ 1 t + ϕ tx + ϕ 3 tx, ϕ 1 = λϕ 1 ϕ 3 + λk 1 ϕ + kλ, ϕ = λ1 + k 1 ϕ ϕ 3 + kλ 1 ϕ, ϕ 3 = λ1 + k 1 ϕ 3 + kλ 1 ϕ 3 5. au = λe u, F u = λ 1 + λ e u, e u = ϕ 1 t + ϕ tx + ϕ 3 tx, ϕ 1 = λϕ 1 ϕ 3 + λ 1 ϕ 1 + λ, ϕ = λϕ ϕ 3 + λ 1 ϕ, ϕ 3 = λϕ 3 + λ 1 ϕ 3

3 6 W.FUSHCHYCH and R.ZHDANOV 6. au = λu 3/, F u = λ 1 u + λ u 5/, u 3/ = ϕ 1 t + ϕ tx + ϕ 3 tx + ϕ 4 tx 3, ϕ 1 = λϕ 1 ϕ 3 3 λϕ 3 λ 1ϕ 1 3 λ, ϕ = 3 λϕ ϕ 3 + 6λϕ 1 ϕ 4 3 λ 1ϕ, ϕ 3 = 3 λϕ 3 + λϕ ϕ 4 3 λ 1ϕ 3, ϕ 4 = 3 λ 1ϕ 4 7. au = 1, F u = + β ln uu, ln u = ϕ 1 t + ϕ tx, ϕ 1 = βϕ 1 + ϕ +, ϕ = ϕ 8. au = 1, F u = + β ln u γ ln u u, ln u = ϕ 1 t + ϕ te γx, ϕ 1 = + βϕ 1 γ ϕ 1, ϕ = β + γ γ ϕ 1 ϕ 9. au = 1, F u = u1 + ln u + β ln u 1/, ln u τ + 4βτ 1/ + ϕ t 1/dτ = x + ϕ1 t, ϕ 1 = 0, ϕ = 4β ϕ 10. au = 1, F u = u 3 + u + βu, a = β = 0 u = ϕ 1 t + ϕ tx1 + ϕ 1 tx + ϕ tx 1, ϕ 1 = 6ϕ 1 ϕ, ϕ = 6ϕ b = 4β 0 u = ϕ 1t + 1 x ϕ t e x/ + ϕ 1 t + ϕ tx 1, ϕ 1 = 4 ϕ 1 ϕ, ϕ = 4 ϕ c > 4β u = A + Bϕ 1 te Bx + A Bϕ te Bx e Ax + ϕ 1 te Bx + ϕ te Bx 1, A =, B = 1 4β 1/, ϕ 1 = 3β 4β 1/ ϕ 1,

4 ANTIREDUCTION AND EXACT SOLUTIONS 63 ϕ = d < 4β 3β + 4β 1/ ϕ u = ϕ 1 ta cos Bx B sin Bx + ϕ ta sin Bx + 1, B cos Bx e Ax + ϕ 1 t cos Bx + ϕ t sin Bx ϕ 1 = 3β ϕ 1 4β 1/ ϕ, ϕ = 3β ϕ + 4β 1/ ϕ 1. In the above formulae θ = θu C R 1, R 1 is an arbitrary function λ, λ 1, λ,, β, γ are arbitrary real constants overdot means differentiation with respect to the corresponding argument. Most of above adduced system of ODE can be integrated. As a result, one obtains a number of new exact solutions of the nonlinear heat equation 5. Detailed study of reduced systems of ODE and construction of exact solutions of equation 5 will be a topic of our future paper. Here we present some exact solutions of the nonlinear heat equation u t = u xx + F u obtained with the help of ansatzes 7 10 which are listed above. 1 F u = + β ln u γ ln u u,. a = β + 4γ > 0 u = C cos 1/ t e γx+γt + 1 γ b = β 4γ > 0 u = C ch 1/ t e γx+γt + 1 γ c = β + 4γ = 0 u = Ct e γx+γt + 1 γ βt + t F u = u1 + ln u + βln u 1/, β 1/ tg 1/ t β + 1/ th 1/ t a 0 ln u τ + 4βτ 1/ + Ce t + β 1 1/ dτ = x b = 0 ln u 4βτ 1/ + 4β t 1/ dτ = x

5 64 W.FUSHCHYCH and R.ZHDANOV 3 F u = uu + u + β, a = 4β u = 1 x t x t + C exp x + t 1 b > 4β u = A + BC 1 expa + Bx t + A BC expa Bx t exp3βt + C 1 expa + Bx t + C expa Bx t 1, A =, B = 1 4β 1/ c < 4β u = AC 1 BC cos Bx t + AC + BC 1 sin Bx t exp3βt Ax t + C 1 cos Bx t + C sin Bx t 1, A =, B = 1 4β 1/. In the above formulae C, C 1, C are arbitrary constants. It is worth noting that the above solutions can not be obtained with the use of the classical Lie symmetry reduction technique [6]. That is why they are essentially new. Another impotant feature is that solutions 3a and 3c are soliton-like solutions. Consequently, nonlinear heat equation with cubic nonlinearity admits soliton-like solutions. References [1] Fushchich Fushchych W.I., Conditional symmetry of equations of nonlinear mathematical physics. Ukrainskii Matematicheskii Zhurnal, 1991, V.43, N 11, [] Fushchich W.I., Mironyuk P.I., Conditional symmetry and exact solutions of nonlinear acoustics equation. Dopovidi of the Ukrainian Academy of Sciences, 1991, N 6, 3 9. [3] Fushchich W.I., Repeta V., Exact solutions of equations of gas dynamics and nonlinear acoustics. Dopovidi of the Ukrainian Academy of Sciences, 1991, N 8, [4] Fushchich W.I., Conditional symmetry of equations of mathematical physics. In: Proceedings of the International Workshop Modern Group Analysis, Eds. N.Ibragimov, M.Torrisi and A.Valenti. Kluwer Academic Publishers, 1993, [5] Fushchich W.I., Zhdanov R.Z., Antireduction of the nonlinear wave equations. Dopovidi of the Ukrainian Academy of Sciences, 1993, N 11, [6] Fushchich W., Shtelen W., Serow N., Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics. Kluwer Academic Publishers, 1993, P.450.