Handbook of mathematics for engineers and scienteists part 196 pps

T9.5. HIGHER-ORDER EQUATIONS 1333 6 ◦ . The Boussinesq equation is solved by the inverse scattering method. Any rapidly decaying function F = F(x, y; t)asx → +∞ and satisfying simultaneously the two linear equations 1 √ 3 ∂F ∂t + ∂ 2 F ∂x 2 – ∂ 2 F ∂y 2 = 0, ∂ 3 F ∂x 3 + ∂ 3 F ∂y 3 = 0 generates a solution of the Boussinesq equation in the form w = 12 d dx K(x, x; t), where K(x, y; t) is a solution of the linear Gel’fand–Levitan–Marchenko integral equation K(x, y; t)+F (x, y; t)+  ∞ x K(x, s; t)F (s, y; t) ds = 0. Time t appears here as a parameter. 2. ∂w ∂y ∂ ∂x (Δw) – ∂w ∂x ∂ ∂y (Δw) = νΔΔw,Δw = ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 . There is a two-dimensional stationary equation of motion of a viscous incompressible fluid—it is obtained from the Navier–Stokes equation by the introduction of the stream function w. 1 ◦ . Suppose w(x, y) is a solution of the equation in question. Then the functions w 1 =–w(y, x), w 2 = w(C 1 x + C 2 , C 1 y + C 3 )+C 4 , w 3 = w(x cos α + y sin α,–x sin α + y cos α), where C 1 , , C 4 and α are arbitrary constants, are also solutions of the equation. 2 ◦ . Any solution of the Poisson equation Δw = C is also a solution of the original equation (these are “inviscid” solutions). 3 ◦ . Solutions in the form of a one-variable function or the sum of functions with different arguments: w(y)=C 1 y 3 + C 2 y 2 + C 3 y + C 4 , w(x, y)=C 1 x 2 + C 2 x + C 3 y 2 + C 4 y + C 5 , w(x, y)=C 1 exp(–λy)+C 2 y 2 + C 3 y + C 4 + νλx, w(x, y)=C 1 exp(λx)–νλx + C 2 exp(λy)+νλy + C 3 , w(x, y)=C 1 exp(λx)+νλx + C 2 exp(–λy)+νλy + C 3 , where C 1 , , C 5 and λ are arbitrary constants. 4 ◦ . Generalized separable solutions: w(x, y)=A(kx + λy) 3 + B(kx + λy) 2 + C(kx + λy)+D, w(x, y)=Ae –λ(y+kx) + B(y + kx) 2 + C(y + kx)+νλ(k 2 + 1)x + D, w(x, y)=6νx(y + λ) –1 + A(y + λ) 3 + B(y + λ) –1 + C(y + λ) –2 + D, 1334 NONLINEAR MATH EM ATI CA L PHYSICS EQUATIONS w(x, y)=(Ax + B)e –λy + νλx + C, w(x, y)=  A sinh(βx)+B cosh(βx)  e –λy + ν λ (β 2 + λ 2 )x + C, w(x, y)=  A sin(βx)+B cos(βx)  e –λy + ν λ (λ 2 – β 2 )x + C, w(x, y)=Ae λy+βx + Be γx + νγy + ν λ γ(β – γ)x + C, γ =  λ 2 + β 2 , where A, B, C, D, k, β,andλ are arbitrary constants. 5 ◦ . Generalized separable solution linear in x: w(x, y)=F (y)x + G(y), (1) where the functions F = F (y)andG = G(y) are determined by the autonomous system of fourth-order ordinary differential equations F  y F  yy – FF  yyy = νF  yyyy ,(2) G  y F  yy – FG  yyy = νG  yyyy .(3) Equation (2) has the following particular solutions: F = ay + b, F = 6ν(y + a) –1 , F = ae –λy + λν, where a, b,andλ are arbitrary constants. Let F = F (y) be a solution of equation (2) (F const). Then the corresponding general solution of equation (3) can be written in the form G =  Udy+ C 4 , U = C 1 U 1 + C 2 U 2 + C 3  U 2  U 1 Φ dy – U 1  U 2 Φ dy  , where C 1 , C 2 , C 3 ,andC 4 are arbitrary constants, and U 1 =  F  yy if F  yy 0, F if F  yy ≡ 0, U 2 = U 1  Φ dy U 2 1 , Φ =exp  – 1 ν  Fdy  . 6 ◦ . There is an exact solution of the form (generalizes the solution of Item 5 ◦ ): w(x, y)=F (z)x + G(z), z = y + kx, k is any number. 7 ◦ . Self-similar solution: w =  F (z) dz + C 1 , z =arctan  x y  , where the function F is determined by the first-order autonomous ordinary differential equation 3ν(F  z ) 2 – 2F 3 + 12νF 2 + C 2 F + C 3 = 0 (C 1 , C 2 ,andC 3 are arbitrary constants). 8 ◦ . There is an exact solution of the form (generalizes the solution of Item 7 ◦ ): w = C 1 ln |x| +  V (z) dz + C 2 , z =arctan  x y  . REFERENCES FOR CHAPTER T9 1335 References for Chapter T9 Ablowitz, M. J. and Segur, H., Solitons and the Inverse Scattering Transform, Society for Industrial and Applied Mathematics, Philadelphia, 1981. Andreev, V. K., Kaptsov, O. V., Pukhnachov, V. V., and Rodionov, A. A., Applications of Group-Theoretical Methods in Hydrodynamics, Kluwer Academic, Dordrecht, 1999. Bullough,R.K.andCaudrey,P.J.(Editors),Solitons, Springer-Verlag, Berlin, 1980. Burde, G. I., New similarity reductions of the steady-state boundary-layer equations, J. Physica A: Math. Gen., Vol. 29, No. 8, pp. 1665–1683, 1996. Calogero, F. and Degasperis, A., Spectral Transform and Solitons: Tolls to Solve and Investigate Nonlinear Evolution Equations, North-Holland, Amsterdam, 1982. Cariello, F. and Tabor, M., Painlev ´ e expansions for nonintegrable evolution equations, Physica D, Vol. 39, No. 1, pp. 77–94, 1989. Clarkson, P. A. and Kruskal, M. D., New similarity reductions of the Boussinesq equation, J. Math. Phys., Vol. 30, No. 10, pp. 2201–2213, 1989. Dodd,R.K.,Eilbeck,J.C.,Gibbon,J.D.,andMorris,H.C.,Solitons and Nonlinear Wave Equations, Academic Press, London, 1982. Galaktionov, V. A., Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities, Proc. Roy. Soc. Edinburgh, Vol. 125A, No. 2, pp. 225–448, 1995. Gardner,C.S.,Greene,J.M.,Kruskal,M.D.,andMiura,R.M.,Method for solving the Korteweg–de Vries equation, Phys. Rev. Lett., Vol. 19, No. 19, pp. 1095–1097, 1967. Goursat, E., A Course of Mathematical Analysis, Vol. 3, Part 1 [Russian translation], Gostekhizdat, Moscow, 1933. Grundland, A. M. and Infeld, E., A family of non-linear Klein-Gordon equations and their solutions, J. Math. Phys., Vol. 33, pp. 2498–2503, 1992. Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solutions, Phys. Rev. Lett., Vol. 27, p. 1192, 1971. Ibragimov, N. H. (Editor), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1, Symmetries, Exact Solutions and Conservation Laws, CRC Press, Boca Raton, 1994. Kawahara,T.andTanaka,M.,Interactions of traveling fronts: an exact solution of a nonlinear diffusion equation, Phys. Lett., Vol. 97, p. 311, 1983. Kersner, R., On some properties of weak solutions of quasilinear degenerate parabolic equations, Acta Math. Academy of Sciences, Hung., Vol. 32, No. 3–4, pp. 301–330, 1978. Novikov, S. P., Manakov, S. V., Pitaevskii, L. B., and Zakharov, V. E., Theory of Solitons. The Inverse Scattering Method, Plenum Press, New York, 1984. Pavlovskii, Yu. N., Investigation of some invariant solutions to the boundary layer equations [in Russian], Zhurn. Vychisl. Mat. i Mat. Fiziki, Vol. 1, No. 2, pp. 280–294, 1961. Polyanin, A. D. and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition, Chapman & Hall/CRC Press, Boca Raton, 2004. Polyanin, A. D. and Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2004. Samarskii, A. A., Galaktionov, V. A., Kurdyumov, S. P., and Mikhailov, A. P., Blow-Up in Problems for Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995. Svirshchevskii, S. R., Lie–B ¨ acklund symmetries of linear ODEs and generalized separation of variables in nonlinear equations, Phys. Lett. A, Vol. 199, pp. 344–348, 1995. Chapter T10 Systems of Partial Differential Equations T10.1. Nonlinear Systems of Two First-Order Equations 1. ∂u ∂x = auw, ∂w ∂t = buw. General solution: u =– ψ  t (t) aϕ(x)+bψ(t) , w =– ϕ  x (x) aϕ(x)+bψ(t) , where ϕ(x)andψ(t) are arbitrary functions. 2. ∂u ∂x = auw, ∂w ∂t = bu k . General solution: w = ϕ(x)+E(x)  ψ(t)– 1 2 ak  E(x) dx  –1 , u =  1 b ∂w ∂t  1/k , E(x)=exp  ak  ϕ(x) dx  , where ϕ(x)andψ(t) are arbitrary functions. 3. ∂u ∂x = auw n , ∂w ∂t = bu k w. General solution: u =  –ψ  t (t) bnψ(t)–akϕ(x)  1/k , w =  ϕ  x (x) bnψ(t)–akϕ(x)  1/n , where ϕ(x)andψ(t) are arbitrary functions. 4. ∂u ∂x = uf(w), ∂w ∂t = u k g(w). 1 ◦ . First integral: ∂w ∂x = kg(w)  f(w) g(w) dw + θ(x)g(w), (1) where θ(x) is an arbitrary function. The first integral (1) may be treated as a first-order ordinary differential equation in x.Onfinding its general solution, one should replace the constant of integration C with an arbitrary function of time ψ(t), since w is dependent on x and t. 1337 1338 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 2 ◦ . To the special case θ(x) = const in (1) there correspond special solutions of the form w = w(z), u =[ψ  (t)] 1/k v(z), z = x + ψ(t)(2) involving one arbitrary function ψ(t), with the prime denoting a derivative. The functions w(z)andv(z) are described by the autonomous system of ordinary differential equations v  z = f (w)v, w  z = g(w)v k , the general solution of which can be written in implicit form as  dw g(w)[kF(w)+C 1 ] = z + C 2 , v =[kF (w)+C 1 ] 1/k , F (w)=  f(w) g(w) dw. 5. ∂u ∂x = f(a 1 u + b 1 w), ∂w ∂t = g(a 2 u + b 2 w). Let Δ = a 1 b 2 – a 2 b 1 ≠ 0. Additive separable solution: u = 1 Δ [b 2 ϕ(x)–b 1 ψ(t)], w = 1 Δ [a 1 ψ(t)–a 2 ϕ(x)], where the functions ϕ(x)andψ(t) are determined by the autonomous ordinary differential equations b 2 Δ ϕ  x = f (ϕ), a 1 Δ ψ  t = g(ψ). Integrating yields b 2 Δ  dϕ f(ϕ) = x + C 1 , a 1 Δ  dψ g(ψ) = t + C 2 . 6. ∂u ∂x = f(au + bw), ∂w ∂t = g(au + bw). Solution: u = b(k 1 x – λ 1 t)+y(ξ), w =–a(k 1 x – λ 1 t)+z(ξ), ξ = k 2 x – λ 2 t, where k 1 , k 2 , λ 1 ,andλ 2 are arbitrary constants, and the functions y(ξ)andz(ξ)are determined by the autonomous system of ordinary differential equations k 2 y  ξ + bk 1 = f (ay + bz), –λ 2 z  ξ + aλ 1 = g(ay + bz). 7. ∂u ∂x = f(au – bw), ∂w ∂t = ug(au – bw) + wh(au – bw) + r(au – bw). Here, f(z), g(z), h(z), and r(z) are arbitrary functions. Generalized separable solution: u = ϕ(t)+bθ(t)x, w = ψ(t)+aθ(t)x. Here, the functions ϕ = ϕ(t), ψ = ψ(t), and θ = θ(t) are determined by a system involving one algebraic (transcendental) and two ordinary differential equations: bθ = f(aϕ – bψ), aθ  t = bθg(aϕ – bψ)+aθh(aϕ – bψ), ψ  t = ϕg(aϕ – bψ)+ψh(aϕ – bψ)+r(aϕ – bψ). T10.1. NONLINEAR SYSTEMS OF TWO FIRST-ORDER EQUATIONS 1339 8. ∂u ∂x = f(au – bw) + cw, ∂w ∂t = ug(au – bw) + wh(au – bw) + r(au – bw). Here, f(z), g(z), h(z), and r(z) are arbitrary functions. Generalized separable solution: u = ϕ(t)+bθ(t)e λx , w = ψ(t)+aθ(t)e λx , λ = ac b . Here, the functions ϕ = ϕ(t), ψ = ψ(t), and θ = θ(t) are determined by a system involving one algebraic (transcendental) and two ordinary differential equations: f(aϕ – bψ)+cψ = 0, ψ  t = ϕg(aϕ – bψ)+ψh(aϕ – bψ)+r(aϕ – bψ), aθ  t = bθg(aϕ – bψ)+aθh(aϕ – bψ). 9. ∂u ∂x = e λu f(λu – σw), ∂w ∂t = e σw g(λu – σw). Solutions: u = y(ξ)– 1 λ ln(C 1 t + C 2 ), w = z(ξ)– 1 σ ln(C 1 t + C 2 ), ξ = x + C 3 C 1 t + C 2 , where the functions y(ξ)andz(ξ) are determined by the system of ordinary differential equations y  ξ = e λy f(λy – σz), –C 1 ξz  ξ – C 1 σ = e σz g(λy – σz). 10. ∂u ∂x = u k f(u n w m ), ∂w ∂t = w s g(u n w m ). Self-similar solution with s ≠ 1 and n ≠ 0: u = t m n(s–1) y(ξ), w = t – 1 s–1 z(ξ), ξ = xt m(k–1) n(s–1) , where the functions y(ξ)andz(ξ) are determined by the system of ordinary differential equations y  ξ = y k f(y n z m ), m(k – 1)ξz  ξ – nz = n(s – 1)z s g(y n z m ). 11. ∂u ∂x = u k f(u n w m ), ∂w ∂t = wg(u n w m ). 1 ◦ . Solution: u = e mt y(ξ), w = e –nt z(ξ), ξ = e m(k–1)t x, where the functions y(ξ)andz(ξ) are determined by the system of ordinary differential equations y  ξ = y k f(y n z m ), m(k – 1)ξz  ξ – nz = zg(y n z m ). . 1335 References for Chapter T9 Ablowitz, M. J. and Segur, H., Solitons and the Inverse Scattering Transform, Society for Industrial and Applied Mathematics, Philadelphia, 1981. Andreev, V. K.,. 280–294, 1961 . Polyanin, A. D. and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition, Chapman & Hall/CRC Press, Boca Raton, 2004. Polyanin, A. D. and. Korteweg-de Vries equation for multiple collisions of solutions, Phys. Rev. Lett., Vol. 27, p. 1192, 1971. Ibragimov, N. H. (Editor), CRC Handbook of Lie Group Analysis of Differential Equations,