Handbook of mathematics for engineers and scienteists part 197 pps

1340 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 2 ◦ .Ifk ≠ 1,itismorebeneficial to seek solutions in the form u = x – 1 k–1 ϕ(ζ), w = x n m(k–1) ψ(ζ), ζ = t + a ln |x|, where a is an arbitrary constant, and the functions ϕ(ζ)andψ(ζ) are determined by the autonomous system of ordinary differential equations aϕ  ζ + 1 1 – k ϕ = ϕ k f(ϕ n ψ m ), ψ  ζ = ψg(ϕ n ψ m ). 12. ∂u ∂x = uf (u n w m ), ∂w ∂t = wg(u n w m ). Solution: u = e m(kx–λt) y(ξ), w = e –n(kx–λt) z(ξ), ξ = αx – βt, where k, α, β,andλ are arbitrary constants, and the functions y(ξ)andz(ξ) are determined by the autonomous system of ordinary differential equations αy  ξ + kmy = yf(y n z m ), –βz  ξ + nλz = zg(y n z m ). 13. ∂u ∂x = uf (u n w m ), ∂w ∂t = wg(u k w s ). Let Δ = sn – km ≠ 0. Multiplicative separable solutions: u =  ϕ(x)  s/Δ  ψ(t)  –m/Δ , w =  ϕ(x)  –k/Δ  ψ(t)  n/Δ , where the functions ϕ(x)andψ(t) are determined by the autonomous ordinary differential equations s Δ ϕ  x = ϕf(ϕ), n Δ ψ  t = ψg(ψ). Integrating yields s Δ  dϕ ϕf(ϕ) = x + C 1 , n Δ  dψ ψg(ψ) = t + C 2 . 14. ∂u ∂x = au ln u + uf (u n w m ), ∂w ∂t = wg(u n w m ). Solution: u =exp  Cme ax  y(ξ), w =exp  –Cne ax  z(ξ), ξ = kx – λt, where C, k,andλ are arbitrary constants, and the functions y(ξ)andz(ξ) are determined by the autonomous system of ordinary differential equations ky  ξ = ay ln y + yf(y n z m ), –λz  ξ = zg(y n z m ). T10.2. LINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1341 15. ∂u ∂x = uf (au k + bw), ∂w ∂t = u k . Solution: w = ϕ(x)+C exp  –λt + k  f(bϕ(x)) dx  , u =  ∂w ∂t  1/k , λ = b a , where ϕ(x) is an arbitrary function and C is an arbitrary constant. 16. ∂u ∂x = uf (au n + bw), ∂w ∂t = u k g(au n + bw). Solution: u =(C 1 t + C 2 ) 1 n–k θ(x), w = ϕ(x)– a b (C 1 t + C 2 ) n n–k [θ(x)] n , where C 1 and C 2 are arbitrary constants, and the functions θ = θ(x)andϕ = ϕ(x)are determined by the system of differential-algebraic equations θ  x = θf(bϕ), θ n–k = b(k – n) aC 1 n g(bϕ). 17. ∂u ∂x = cw n u a + bw n , ∂w ∂t = (aw + bw n+1 )u k . General solution with b ≠ 0: w =  ψ(t)e F (x) – be F (x)  e –F (x) ϕ(x) dx  –1/n , u =  w t aw + bw n+1  1/k , F(x)= ck b x – a  ϕ(x) dx, where ϕ(x)andψ(t) are arbitrary functions. T10.2. Linear Systems of Two Second-Order Equations 1. ∂u ∂t = a ∂ 2 u ∂x 2 + b 1 u + c 1 w, ∂w ∂t = a ∂ 2 w ∂x 2 + b 2 u + c 2 w. Constant-coefficient second-order linear system of parabolic type. Solution: u = b 1 – λ 2 b 2 (λ 1 – λ 2 ) e λ 1 t θ 1 – b 1 – λ 1 b 2 (λ 1 – λ 2 ) e λ 2 t θ 2 , w = 1 λ 1 – λ 2  e λ 1 t θ 1 – e λ 2 t θ 2  , where λ 1 and λ 2 are roots of the quadratic equation λ 2 –(b 1 + c 2 )λ + b 1 c 2 – b 2 c 1 = 0, and the functions θ n = θ n (x, t) satisfy the independent linear heat equations ∂θ 1 ∂t = a ∂ 2 θ 1 ∂x 2 , ∂θ 2 ∂t = a ∂ 2 θ 2 ∂x 2 . 1342 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 2. ∂u ∂t = a ∂ 2 u ∂x 2 + f 1 (t)u + g 1 (t)w, ∂w ∂t = a ∂ 2 w ∂x 2 + f 2 (t)u + g 2 (t)w. Variable-coefficient second-order linear system of parabolic type. Solution: u = ϕ 1 (t)U(x, t)+ϕ 2 (t)W (x, t), w = ψ 1 (t)U(x, t)+ψ 2 (t)W (x, t), where the pairs of functions ϕ 1 = ϕ 1 (t), ψ 1 = ψ 1 (t)andϕ 2 = ϕ 2 (t), ψ 2 = ψ 2 (t) are linearly independent (fundamental) solutions to the system of linear ordinary differential equations ϕ  t = f 1 (t)ϕ + g 1 (t)ψ, ψ  t = f 2 (t)ϕ + g 2 (t)ψ, and the functions U = U(x, t)andW = W (x, t) satisfy the independent linear heat equations ∂U ∂t = a ∂ 2 U ∂x 2 , ∂W ∂t = a ∂ 2 W ∂x 2 . 3. ∂ 2 u ∂t 2 = k ∂ 2 u ∂x 2 + a 1 u + b 1 w, ∂ 2 w ∂t 2 = k ∂ 2 w ∂x 2 + a 2 u + b 2 w. Constant-coefficient second-order linear system of hyperbolic type. Solution: u = a 1 – λ 2 a 2 (λ 1 – λ 2 ) θ 1 – a 1 – λ 1 a 2 (λ 1 – λ 2 ) θ 2 , w = 1 λ 1 – λ 2  θ 1 – θ 2  , where λ 1 and λ 2 are roots of the quadratic equation λ 2 –(a 1 + b 2 )λ + a 1 b 2 – a 2 b 1 = 0, and the functions θ n = θ n (x, t) satisfy the linear Klein–Gordon equations ∂ 2 θ 1 ∂t 2 = k ∂ 2 θ 1 ∂x 2 + λ 1 θ 1 , ∂ 2 θ 2 ∂t 2 = k ∂ 2 θ 2 ∂x 2 + λ 2 θ 2 . 4. ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = a 1 u + b 1 w, ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = a 2 u + b 2 w. Constant-coefficient second-order linear system of elliptic type. Solution: u = a 1 – λ 2 a 2 (λ 1 – λ 2 ) θ 1 – a 1 – λ 1 a 2 (λ 1 – λ 2 ) θ 2 , w = 1 λ 1 – λ 2  θ 1 – θ 2  , where λ 1 and λ 2 are roots of the quadratic equation λ 2 –(a 1 + b 2 )λ + a 1 b 2 – a 2 b 1 = 0, and the functions θ n = θ n (x, y) satisfy the linear Helmholtz equations ∂ 2 θ 1 ∂x 2 + ∂ 2 θ 1 ∂y 2 = λ 1 θ 1 , ∂ 2 θ 2 ∂x 2 + ∂ 2 θ 2 ∂y 2 = λ 2 θ 2 . T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1343 T10.3. Nonlinear Systems of Two Second-Order Equations T10.3.1. Systems of the Form ∂u ∂t = a ∂ 2 u ∂x 2 + F (u, w), ∂w ∂t = b ∂ 2 w ∂x 2 + G(u, w) Preliminary remarks. Systems of this form often arise in the theory of heat and mass transfer in chemically reactive media, theory of chemical reactors, combustion theory, mathematical biology, and biophysics. Such systems are invariant under translations in the independent variables (and under the change of x to –x) and admit traveling-wave solutions u = u(kx – λt), w = w(kx – λt). These solutions as well as those with one of the unknown functions being identically zero are not considered further in this section. The functionsf (ϕ), g(ϕ), h(ϕ) appearingbelow arearbitrary functions of their argument, ϕ = ϕ(u, w); the equations are arranged in order of complexity of this argument. T10.3.1-1. Arbitrary functions depend on a linear combination of the unknowns. 1. ∂u ∂t = a ∂ 2 u ∂x 2 +u exp  k w u  f(u), ∂w ∂t = a ∂ 2 w ∂x 2 +exp  k w u  [wf(u) +g(u)]. Solution: u = y(ξ), w =– 2 k ln |bx| y(ξ)+z(ξ), ξ = x + C 3 √ C 1 t + C 2 , where C 1 , C 2 , C 3 ,andb are arbitrary constants, and the functions y = y(ξ)andz = z(ξ)are determined by the system of ordinary differential equations ay  ξξ + 1 2 C 1 ξy  ξ + 1 b 2 ξ 2 y exp  k z y  f(y)=0, az  ξξ + 1 2 C 1 ξz  ξ – 4a kξ y  ξ + 2a kξ 2 y + 1 b 2 ξ 2 exp  k z y  [zf(y)+g(y)] = 0. 2. ∂u ∂t = a 1 ∂ 2 u ∂x 2 + f(bu + cw), ∂w ∂t = a 2 ∂ 2 w ∂x 2 + g(bu + cw). Solution: u = c(αx 2 + βx + γt)+y(ξ), w =–b(αx 2 + βx + γt)+z(ξ), ξ = kx – λt, where k, α, β, γ,andλ arearbitrary constants, and thefunctions y(ξ)andz(ξ) are determined by the autonomous system of ordinary differential equations a 1 k 2 y  ξξ + λy  ξ + 2a 1 cα – cγ + f(by + cz)=0, a 2 k 2 z  ξξ + λz  ξ – 2a 2 bα + bγ + g(by + cz)=0. 1344 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 3. ∂u ∂t = a ∂ 2 u ∂x 2 + f(bu + cw), ∂w ∂t = a ∂ 2 w ∂x 2 + g(bu + cw). Solution: u = cθ(x, t)+y(ξ), w =–bθ(x, t)+z(ξ), ξ = kx – λt, where the functions y(ξ)andz(ξ) are determined by the autonomous system of ordinary differential equations ak 2 y  ξξ + λy  ξ + f(by + cz)=0, ak 2 z  ξξ + λz  ξ + g(by + cz)=0, and the function θ = θ(x, t) satisfies the linear heat equation ∂θ ∂t = a ∂ 2 θ ∂x 2 . 4. ∂u ∂t = a ∂ 2 u ∂x 2 + uf(bu – cw) + g(bu – cw), ∂w ∂t = a ∂ 2 w ∂x 2 + wf(bu – cw) + h(bu – cw). 1 ◦ . Solution: u = ϕ(t)+c exp   f(bϕ – cψ) dt  θ(x, t), w = ψ(t)+b exp   f(bϕ – cψ) dt  θ(x, t), where ϕ = ϕ(t)andψ = ψ(t) are determined by the autonomous system of ordinary differential equations ϕ  t = ϕf(bϕ – cψ)+g(bϕ – cψ), ψ  t = ψf(bϕ – cψ)+h(bϕ – cψ), and the function θ = θ(x, t) satisfies the linear heat equation ∂θ ∂t = a ∂ 2 θ ∂x 2 . 2 ◦ . Let us multiply the first equation by b and the second one by –c and add the results together to obtain ∂ζ ∂t = a ∂ 2 ζ ∂x 2 + ζf(ζ)+bg(ζ)–ch(ζ), ζ = bu – cw.(1) This equation will be considered in conjunction with the first equation of the original system ∂u ∂t = a ∂ 2 u ∂x 2 + uf(ζ)+g(ζ). (2) Equation (1) can be treated separately. An extensive list of exact solutions to equations of this form for various kinetic functions F(ζ)=ζf(ζ)+bg(ζ)–ch(ζ) can be found in the book by Polyanin and Zaitsev (2004). Given a solution ζ = ζ(x, t) to equation (1), the function u = u(x, t) can be determined by solving the linear equation (2) and the function w = w(x, t) is found as w =(bu – ζ)/c. T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1345 Note two important solutions to equation (1): (i) In the general case, equation (1) admits traveling-wave solutions ζ = ζ(z), where z = kx – λt. Then the corresponding exact solutions to equation (2) are expressed as u = u 0 (z)+  e β n t u n (z). (ii) If the condition ζf(ζ)+bg(ζ)–ch(ζ)=k 1 ζ + k 0 holds, equation (1) is linear, ∂ζ ∂t = a ∂ 2 ζ ∂x 2 + k 1 ζ + k 0 , and, hence, can be reduced to the linear heat equation. 5. ∂u ∂t = a ∂ 2 u ∂x 2 + e λu f(λu – σw), ∂w ∂t = b ∂ 2 w ∂x 2 + e σw g(λu – σw). 1 ◦ . Solution: 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 C 1 , C 2 ,andC 3 are arbitrary constants, and the functions y = y(ξ)andz = z(ξ)are determined by the system of ordinary differential equations ay  ξξ + 1 2 C 1 ξy  ξ + C 1 λ + e λy f(λy – σz)=0, bz  ξξ + 1 2 C 1 ξz  ξ + C 1 σ + e σz g(λy – σz)=0. 2 ◦ . Solution with b = a: u = θ(x, t), w = λ σ θ(x, t)– k σ , where k is a root of the algebraic (transcendental) equation λf(k)=σe –k g(k), and the function θ = θ(x, t) is determined by the differential equation ∂θ ∂t = a ∂ 2 θ ∂x 2 + f(k)e λθ . For exact solutions to this equation, see Polyanin and Zaitsev (2004). T10.3.1-2. Arbitrary functions depend on the ratio of the unknowns. 6. ∂u ∂t = a ∂ 2 u ∂x 2 + uf  u w  , ∂w ∂t = b ∂ 2 w ∂x 2 + wg  u w  . 1 ◦ . Multiplicative separable solution: u =[C 1 sin(kx)+C 2 cos(kx)]ϕ(t), w =[C 1 sin(kx)+C 2 cos(kx)]ψ(t), 1346 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS where C 1 , C 2 ,andk are arbitrary constants, and the functions ϕ = ϕ(t)andψ = ψ(t)are determined by the autonomous system of ordinary differential equations ϕ  t =–ak 2 ϕ + ϕf(ϕ/ψ), ψ  t =–bk 2 ψ + ψg(ϕ/ψ). 2 ◦ . Multiplicative separable solution: u =[C 1 exp(kx)+C 2 exp(–kx)]U(t), w =[C 1 exp(kx)+C 2 exp(–kx)]W (t), where C 1 , C 2 ,andk are arbitrary constants, and the functions U = U (t)andW = W (t)are determined by the autonomous system of ordinary differential equations U  t = ak 2 U + Uf(U/W), W  t = bk 2 W + Wg(U/W). 3 ◦ . Degenerate solution: u =(C 1 x + C 2 )U(t), w =(C 1 x + C 2 )W (t), where C 1 and C 2 are arbitrary constants, and the functions U = U(t)andW = W(t)are determined by the autonomous system of ordinary differential equations U  t = Uf(U/W), W  t = Wg(U/W). This autonomous system can be integrated since it is reduced, after eliminating t,toa homogeneous first-order equation. The systems presented in Items 1 ◦ and 2 ◦ can be integrated likewise. 4 ◦ . Multiplicative separable solution: u = e –λt y(x), w = e –λt z(x), where λ is an arbitrary constant and the functions y = y(x)andz = z(x) are determined by the autonomous system of ordinary differential equations ay  xx + λy + yf(y/z)=0, bz  xx + λz + zg(y/z)=0. 5 ◦ . Solution (generalizes the solution of Item 4 ◦ ): u = e kx–λt y(ξ), w = e kx–λt z(ξ), ξ = βx – γt, where k, λ, β,andγ are arbitrary constants, and the functions y = y(ξ)andz = z(ξ)are determined by the autonomous system of ordinary differential equations aβ 2 y  ξξ +(2akβ + γ)y  ξ +(ak 2 + λ)y + yf(y/z)=0, bβ 2 z  ξξ +(2bkβ + γ)z  ξ +(bk 2 + λ)z + zg(y/z)=0. To the special case k = λ = 0 there corresponds a traveling-wave solution. If k = γ = 0 and β = 1, we have the solution of Item 4 ◦ . . Nonlinear Systems of Two Second-Order Equations T10.3.1. Systems of the Form ∂u ∂t = a ∂ 2 u ∂x 2 + F (u, w), ∂w ∂t = b ∂ 2 w ∂x 2 + G(u, w) Preliminary remarks. Systems of this form often arise in. =[C 1 sin(kx)+C 2 cos(kx)]ψ(t), 1346 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS where C 1 , C 2 ,andk are arbitrary constants, and the functions ϕ = ϕ(t )and = ψ(t)are determined by the autonomous system of ordinary differential. first equation of the original system ∂u ∂t = a ∂ 2 u ∂x 2 + uf(ζ)+g(ζ). (2) Equation (1) can be treated separately. An extensive list of exact solutions to equations of this form for various kinetic