Handbook of mathematics for engineers and scienteists part 199 pptx

1354 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 2 ◦ . A periodic solution in time: u = r(x)cos  θ(x)+C 1 t + C 2  , w = r(x)sin  θ(x)+C 1 t + C 2  , where C 1 and C 2 are arbitrary constants, and the functions r = r(x)andθ = θ(x)are determined by the autonomous system of ordinary differential equations ar  xx – ar(θ  x ) 2 + rf(r 2 )=0, arθ  xx + 2ar  x θ  x – C 1 r + rg(r 2 )=0. 3 ◦ . Solution (generalizes the solution of Item 2 ◦ ): u = r(z)cos  θ(z)+C 1 t + C 2  , w = r(z)sin  θ(z)+C 1 t + C 2  , z = x + λt, where C 1 , C 2 ,andλ are arbitrary constants, and the functions r = r(z)andθ = θ(z)are determined by the system of ordinary differential equations ar  zz – ar(θ  z ) 2 – λr  z + rf(r 2 )=0, arθ  zz + 2ar  z θ  z – λrθ  z – C 1 r + rg(r 2 )=0. 20. ∂u ∂t = a ∂ 2 u ∂x 2 + uf  u 2 + w 2  – wg  u 2 + w 2  – w arctan  w u  h  u 2 + w 2  , ∂w ∂t = a ∂ 2 w ∂x 2 + wf  u 2 + w 2  + ug  u 2 + w 2  + u arctan  w u  h  u 2 + w 2  . Functional separable solution (for fixed t,itdefines a structure periodic in x): u = r(t)cos  ϕ(t)x + ψ(t)  , w = r(t)sin  ϕ(t)x + ψ(t)  , where the functions r = r(t), ϕ = ϕ(t), and ψ = ψ(t) are determined by the autonomous system of ordinary differential equations r  t =–arϕ 2 + rf(r 2 ), ϕ  t = h(r 2 )ϕ, ψ  t = h(r 2 )ψ + g(r 2 ). T10.3.1-5. Arbitrary functions depend on the difference of squares of the unknowns. 21. ∂u ∂t = a ∂ 2 u ∂x 2 + uf(u 2 – w 2 ) + wg(u 2 – w 2 ), ∂w ∂t = a ∂ 2 w ∂x 2 + ug(u 2 – w 2 ) + wf(u 2 – w 2 ). 1 ◦ . Solution: u = ψ(t)coshϕ(x, t), w = ψ(t)sinhϕ(x, t), ϕ(x, t)=C 1 x +  g(ψ 2 ) dt + C 2 , where C 1 and C 2 are arbitrary constants, and the function ψ = ψ(t) is described by the separable first-order ordinary differential equation ψ  t = ψf(ψ 2 )+aC 2 1 ψ, T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1355 whose general solution can be represented in implicit form as  dψ ψf(ψ 2 )+aC 2 1 ψ = t + C 3 . 2 ◦ . Solution: u = r(x)cosh  θ(x)+C 1 t + C 2  , w = r(x)sinh  θ(x)+C 1 t + C 2  , where C 1 and C 2 are arbitrary constants, and the functions r = r(x)andθ = θ(x)are determined by the autonomous system of ordinary differential equations ar  xx + ar(θ  x ) 2 + rf(r 2 )=0, arθ  xx + 2ar  x θ  x + rg(r 2 )–C 1 r = 0. 3 ◦ . Solution (generalizes the solution of Item 2 ◦ ): u = r(z)cosh  θ(z)+C 1 t + C 2  , w = r(z)sinh  θ(z)+C 1 t + C 2  , z = x + λt, where C 1 , C 2 ,andλ are arbitrary constants, and the functions r = r(z)andθ = θ(z)are determined by the autonomous system of ordinary differential equations ar  zz + ar(θ  z ) 2 – λr  z + rf(r 2 )=0, arθ  zz + 2ar  z θ  z – λrθ  z – C 1 r + rg(r 2 )=0. 22. ∂u ∂t = a ∂ 2 u ∂x 2 + uf  u 2 – w 2  + wg  u 2 – w 2  + w arctanh  w u  h  u 2 – w 2  , ∂w ∂t = a ∂ 2 w ∂x 2 + wf  u 2 – w 2  + ug  u 2 – w 2  + u arctanh  w u  h  u 2 – w 2  . Functional separable solution: u = r(t)cosh  ϕ(t)x + ψ(t)  , w = r(t)sinh  ϕ(t)x + ψ(t)  , where the functions r = r(t), ϕ = ϕ(t), and ψ = ψ(t) are determined by the autonomous system of ordinary differential equations r  t = arϕ 2 + rf(r 2 ), ϕ  t = h(r 2 )ϕ, ψ  t = h(r 2 )ψ + g(r 2 ). T10.3.1-6. Arbitrary functions depend on the unknowns in a complex way. 23. ∂u ∂t = a ∂ 2 u ∂x 2 + u k+1 f(ϕ), ϕ = u exp  – w u  , ∂w ∂t = a ∂ 2 w ∂x 2 + u k+1 [f(ϕ)lnu + g(ϕ)]. Solution: u =(C 1 t + C 2 ) – 1 k y(ξ), w =(C 1 t + C 2 ) – 1 k  z(ξ)– 1 k ln(C 1 t + C 2 )y(ξ)  , ξ = x + C 3 √ C 1 t + C 2 , 1356 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 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 k y + y k+1 f(ϕ)=0, ϕ = y exp  – z y  , az  ξξ + 1 2 C 1 ξz  ξ + C 1 k z + C 1 k y + y k+1 [f(ϕ)lny + g(ϕ)] = 0. 24. ∂u ∂t = a ∂ 2 u ∂x 2 +uf(u 2 +w 2 )–wg  w u  , ∂w ∂t = a ∂ 2 w ∂x 2 +ug  w u  +wf(u 2 +w 2 ). Solution: u = r(x, t)cosϕ(t), w = r(x, t)sinϕ(t), where the function ϕ = ϕ(t) is determined by the autonomous ordinary differential equation ϕ  t = g(tan ϕ), (1) and the function r = r(x, t) is determined by the differential equation ∂r ∂t = a ∂ 2 r ∂x 2 + rf(r 2 ). (2) The general solution of equation (1) is expressed in implicit form as  dϕ g(tan ϕ) = t + C. Equation (2) admits an exact, traveling-wave solution r = r(z), where z = kx – λt with arbitrary constants k and λ, and the function r(z) is determined by the autonomous ordinary differential equation ak 2 r  zz + λr  z + rf(r 2 )=0. For other exact solutions to equation (2) for various functions f , see Polyanin and Zaitsev (2004). 25. ∂u ∂t = a ∂ 2 u ∂x 2 +uf(u 2 –w 2 )+wg  w u  , ∂w ∂t = a ∂ 2 w ∂x 2 +ug  w u  +wf(u 2 –w 2 ). Solution: u = r(x, t)coshϕ(t), w = r(x, t)sinhϕ(t), where the function ϕ = ϕ(t) is determined by the autonomous ordinary differential equation ϕ  t = g(tanh ϕ), (1) and the function r = r(x, t) is determined by the differential equation ∂r ∂t = a ∂ 2 r ∂x 2 + rf(r 2 ). (2) The general solution of equation (1) is expressed in implicit form as  dϕ g(tanh ϕ) = t + C. Equation (2) admits an exact, traveling-wave solution r = r(z), where z = kx – λt with arbitrary constants k and λ, and the function r(z) is determined by the autonomous ordinary differential equation ak 2 r  zz + λr  z + rf(r 2 )=0. For other exact solutions to equation (2) for various functions f , see Polyanin and Zaitsev (2004). T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1357 T10.3.2. Systems of the Form ∂u ∂t = a x n ∂ ∂x  x n ∂u ∂x  + F (u, w), ∂w ∂t = b x n ∂ ∂x  x n ∂w ∂x  + G(u, w) T10.3.2-1. Arbitrary functions depend on a linear combination of the unknowns. 1. ∂u ∂t = a x n ∂ ∂x  x n ∂u ∂x  + uf(bu – cw) + g(bu – cw), ∂w ∂t = a x n ∂ ∂x  x n ∂w ∂x  + 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 linear heat equation ∂θ ∂t = a x n ∂ ∂x  x n ∂θ ∂x  .(1) 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 x n ∂ ∂x  x n ∂ζ ∂x  + ζf(ζ)+bg(ζ)–ch(ζ), ζ = bu – cw.(2) This equation will be considered in conjunction with the first equation of the original system ∂u ∂t = a x n ∂ ∂x  x n ∂u ∂x  + uf(ζ)+g(ζ). (3) Equation (2) can be treated separately. Given a solution ζ = ζ(x, t) to equation (2), the function u = u(x, t) can be determined by solving the linear equation (3) and the function w = w(x, t) is found as w =(bu – ζ)/c. Note two important solutions to equation (2): (i) In the general case, equation (2) admits steady-state solutions ζ = ζ(x). The corre- sponding exact solutions to equation (3) are expressed as u = u 0 (x)+  e β n t u n (x). (ii) If the condition ζf(ζ)+bg(ζ)–ch(ζ)=k 1 ζ + k 0 holds, equation (2) is linear, ∂ζ ∂t = a x n ∂ ∂x  x n ∂ζ ∂x  + k 1 ζ + k 0 , and hence can bereduced to thelinear heat equation (1) with thesubstitution ζ =e k 1 t ¯ ζ–k 0 k –1 1 . 1358 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 2. ∂u ∂t = a x n ∂ ∂x  x n ∂u ∂x  + e λu f(λu – σw), ∂w ∂t = b x n ∂ ∂x  x n ∂w ∂x  + e σw g(λu – σw). 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 1 t + C 2 , where C 1 and C 2 are arbitrary constants, and the functions y = y(ξ)andz = z(ξ)are determined by the system of ordinary differential equations aξ –n (ξ n y  ξ )  ξ + 1 2 C 1 ξy  ξ + C 1 λ + e λy f(λy – σz)=0, bξ –n (ξ n z  ξ )  ξ + 1 2 C 1 ξz  ξ + C 1 σ + e σz g(λy – σz)=0. T10.3.2-2. Arbitrary functions depend on the ratio of the unknowns. 3. ∂u ∂t = a x n ∂ ∂x  x n ∂u ∂x  + uf  u w  , ∂w ∂t = b x n ∂ ∂x  x n ∂w ∂x  + wg  u w  . 1 ◦ . Multiplicative separable solution: u = x 1–n 2 [C 1 J ν (kx)+C 2 Y ν (kx)]ϕ(t), ν = 1 2 |n – 1|, w = x 1–n 2 [C 1 J ν (kx)+C 2 Y ν (kx)]ψ(t), where C 1 , C 2 ,andk are arbitrary constants, J ν (z)andY ν (z) are Bessel functions, 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 = x 1–n 2 [C 1 I ν (kx)+C 2 K ν (kx)]ϕ(t), ν = 1 2 |n – 1|, w = x 1–n 2 [C 1 I ν (kx)+C 2 K ν (kx)]ψ(t), where C 1 , C 2 ,andk are arbitrary constants, I ν (z)andK ν (z) are modified Bessel functions, 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). 3 ◦ . Multiplicative separable solution: u = e –λt y(x), w = e –λt z(x), T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1359 where λ is an arbitrary constant and the functions y = y(x)andz = z(x) are determined by the system of ordinary differential equations ax –n (x n y  x )  x + λy + yf(y/z)=0, bx –n (x n z  x )  x + λz + zg(y/z)=0. 4 ◦ . This is a special case of equation with b = a.Letk be a root of the algebraic (transcen- dental) equation f(k)=g(k). Solution: u = ke λt θ, w = e λt θ, λ = f (k), where the function θ = θ(x, t) satisfies the linear heat equation ∂θ ∂t = a x n ∂ ∂x  x n ∂θ ∂x  .(1) 5 ◦ . This is a special case of equation with b = a. Solution: u = ϕ(t)exp   g(ϕ(t)) dt  θ(x, t), w =exp   g(ϕ(t)) dt  θ(x, t), where the function ϕ = ϕ(t) is described by the separable first-order nonlinear ordinary differential equation ϕ  t =[f(ϕ)–g(ϕ)]ϕ,(2) and the function θ = θ(x, t) satisfies the linear heat equation (1). To the particular solution ϕ = k = const of equation (2), there corresponds the solution presented in Item 4 ◦ . The general solution of equation (2) is written out in implicit form as  dϕ [f(ϕ)–g(ϕ)]ϕ = t + C. 4. ∂u ∂t = a x n ∂ ∂x  x n ∂u ∂x  + uf  u w  + u w h  u w  , ∂w ∂t = a x n ∂ ∂x  x n ∂w ∂x  + wg  u w  + h  u w  . Solution: u = ϕ(t)G(t)  θ(x, t)+  h(ϕ) G(t) dt  , w = G(t)  θ(x, t)+  h(ϕ) G(t) dt  , G(t)=exp   g(ϕ) dt  , where the function ϕ = ϕ(t) is described by the separable first-order nonlinear ordinary differential equation ϕ  t =[f(ϕ)–g(ϕ)]ϕ,(1) and the function θ = θ(x, t) satisfies the linear heat equation ∂θ ∂t = a x n ∂ ∂x  x n ∂θ ∂x  . The general solution of equation (1) is written out in implicit form as  dϕ [f(ϕ)–g(ϕ)]ϕ = t + C. 1360 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 5. ∂u ∂t = a x n ∂ ∂x  x n ∂u ∂x  + uf 1  w u  + wg 1  w u  , ∂w ∂t = a x n ∂ ∂x  x n ∂w ∂x  + uf 2  w u  + wg 2  w u  . Solution: u=exp   [f 1 (ϕ)+ϕg 1 (ϕ)] dt  θ(x, t), w(x, t)=ϕ(t)exp   [f 1 (ϕ)+ϕg 1 (ϕ)] dt  θ(x, t), where the function ϕ = ϕ(t) is described by the separable first-order nonlinear ordinary differential equation ϕ  t = f 2 (ϕ)+ϕg 2 (ϕ)–ϕ[f 1 (ϕ)+ϕg 1 (ϕ)], and the function θ = θ(x, t) satisfies the linear heat equation ∂θ ∂t = a x n ∂ ∂x  x n ∂θ ∂x  . 6. ∂u ∂t = a x n ∂ ∂x  x n ∂u ∂x  +u k f  u w  , ∂w ∂t = b x n ∂ ∂x  x n ∂w ∂x  +w k g  u w  . Self-similar solution: u =(C 1 t + C 2 ) 1 1–k y(ξ), w =(C 1 t + C 2 ) 1 1–k z(ξ), ξ = x √ C 1 t + C 2 , where C 1 and C 2 are arbitrary constants, and the functions y = y(ξ)andz = z(ξ)are determined by the system of ordinary differential equations aξ –n (ξ n y  ξ )  ξ + 1 2 C 1 ξy  ξ + C 1 k – 1 y + y k f(y/z)=0, bξ –n (ξ n z  ξ )  ξ + 1 2 C 1 ξz  ξ + C 1 k – 1 z + z k g(y/z)=0. 7. ∂u ∂t = a x n ∂ ∂x  x n ∂u ∂x  + uf  u w  ln u + ug  u w  , ∂w ∂t = a x n ∂ ∂x  x n ∂w ∂x  + wf  u w  ln w + wh  u w  . Solution: u = ϕ(t)ψ(t)θ(x, t), w = ψ(t)θ(x, t), where the functions ϕ = ϕ(t)andψ = ψ(t) are determined by solving the autonomous ordinary differential equations ϕ  t = ϕ[g(ϕ)–h(ϕ)+f(ϕ)lnϕ], ψ  t = ψ[h(ϕ)+f(ϕ)lnψ], (1) and the function θ = θ(x, t) is determined by the differential equation ∂θ ∂t = a x n ∂ ∂x  x n ∂θ ∂x  + f(ϕ)θ ln θ.(2) . + C 2 , 1356 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 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. x 1–n 2 [C 1 J ν (kx)+C 2 Y ν (kx)]ψ(t), where C 1 , C 2 ,andk are arbitrary constants, J ν (z)andY ν (z) are Bessel functions, and the functions ϕ = ϕ(t )and = ψ(t) are determined by the autonomous system of ordinary differential. x 1–n 2 [C 1 I ν (kx)+C 2 K ν (kx)]ψ(t), where C 1 , C 2 ,andk are arbitrary constants, I ν (z)andK ν (z) are modified Bessel functions, and the functions ϕ = ϕ(t )and = ψ(t) are determined by the autonomous system of ordinary differential