Arbitrary functions depend on the difference of squares of the unknowns... Arbitrary functions depend on the unknowns in a complex way... For other exact solutions to equation 2 for vari
Trang 12◦ A periodic solution in time:
u = r(x) cos
θ(x) + C1t + C2
, w = r(x) sin
θ(x) + C1t + C2
,
where C1 and C2 are arbitrary constants, and the functions r = r(x) and θ = θ(x) are
determined by the autonomous system of ordinary differential equations
ar
xx – ar(θ x )2+ rf (r2) =0,
arθ
xx+2ar
x θ x – C1r + rg(r2) =0
3◦ Solution (generalizes the solution of Item2◦):
u = r(z) cos
θ(z) + C1t + C2
, w = r(z) sin
θ(z) + C1t + C2
, z = x + λt, where C1, C2, and λ are arbitrary constants, and the functions r = r(z) and θ = θ(z) are
determined by the system of ordinary differential equations
ar
zz – ar(θ z )2– λr z + rf (r2) =0,
arθ
zz+2ar
z θ z – λrθ z – C1r + rg(r2) =0
20. ∂u
∂t = a ∂
2u
∂x2 + uf u2+ w2
– wg u2+ w2
– w arctan w
u
h u2+ w2
,
∂w
∂t = a ∂
2w
∂x2 + wf u2+ w2
+ ug u2+ w2
+ u arctan w
u
h u2+ w2
.
Functional separable solution (for fixed t, it defines 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 = –arϕ2+ rf (r2),
ϕ
t = h(r2)ϕ,
ψ
t = h(r2)ψ + g(r2).
T10.3.1-5 Arbitrary functions depend on the difference of squares of the unknowns
21. ∂u
∂t = a ∂
2u
∂x2 + uf (u2– w2) + wg(u2– w2 ),
∂w
∂t = a ∂
2w
∂x2 + ug(u2– w2) + wf(u2– w2 ).
1◦ Solution:
u = ψ(t) cosh ϕ(x, t), w = ψ(t) sinh ϕ(x, t), ϕ(x, t) = C1x+
g(ψ2) dt + C2,
where C1 and C2 are arbitrary constants, and the function ψ = ψ(t) is described by the
separable first-order ordinary differential equation
ψ
t = ψf (ψ2) + aC12ψ,
Trang 2whose general solution can be represented in implicit form as
dψ
ψf (ψ2) + aC12ψ = t + C3
2◦ Solution:
u = r(x) cosh
θ(x) + C1t + C2
, w = r(x) sinh
θ(x) + C1t + C2
,
where C1 and C2 are arbitrary constants, and the functions r = r(x) and θ = θ(x) are
determined by the autonomous system of ordinary differential equations
ar
xx + ar(θ x )2+ rf (r2) =0,
arθ
xx+2ar
x θ x + rg(r2) – C1r =0
3◦ Solution (generalizes the solution of Item2◦):
u = r(z) cosh
θ(z) + C1t + C2
, w = r(z) sinh
θ(z) + C1t + C2
, z = x + λt, where C1, C2, and λ are arbitrary constants, and the functions r = r(z) and θ = θ(z) are
determined by the autonomous system of ordinary differential equations
ar
zz + ar(θ z )2– λr z + rf (r2) =0,
arθ
zz+2ar
z θ z – λrθ z – C1r + rg(r2) =0
22. ∂u
∂t = a ∂
2u
∂x2 + uf u2– w2
+ wg u2– w2
+ w arctanh w
u
h u2– w2
,
∂w
∂t = a ∂
2w
∂x2 + wf u2– w2
+ ug u2– w2
+ u arctanh w
u
h u2– w2
.
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 = arϕ2+ rf (r2),
ϕ
t = h(r2)ϕ,
ψ
t = h(r2)ψ + g(r2).
T10.3.1-6 Arbitrary functions depend on the unknowns in a complex way
23. ∂u
∂t = a ∂
2u
∂x2 + u k+1 f (ϕ), ϕ = u exp
–w
u
,
∂w
∂t = a ∂
2w
∂x2 + u k+1 [f(ϕ) ln u + g(ϕ)].
Solution:
u = (C1t + C2)–k1y(ξ), w = (C1t + C2)–k1
z(ξ) – 1
k ln(C1t + C2)y(ξ)
, ξ= √ x + C3
C1t + C2,
Trang 3where C1, C2, and C3are arbitrary constants, and the functions y = y(ξ) and z = z(ξ) are
determined by the system of ordinary differential equations
ay
ξξ+ 1
2C1ξy ξ +
C1
k y + y k+1 f (ϕ) =0, ϕ = y exp
–z
y
,
az
ξξ+ 1
2C1ξz ξ +
C1
k z+ C1
k y + y k+1 [f (ϕ) ln y + g(ϕ)] =0
24. ∂u
∂t = a ∂
2u
∂x2+uf (u2+w2)–wg w
u
∂t = a ∂
2w
∂x2 +ug w
u
+wf (u2+w2 ).
Solution:
u = r(x, t) cos ϕ(t), w = r(x, t) sin ϕ(t), where the function ϕ = ϕ(t) is determined by the autonomous ordinary differential equation
ϕ
and the function r = r(x, t) is determined by the differential equation
∂r
∂t = a ∂
2r
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
ak2r
zz + λr z + rf (r2) =0
For other exact solutions to equation (2) for various functions f , see Polyanin and Zaitsev
(2004)
25. ∂u
∂t = a ∂
2u
∂x2+uf (u2–w2)+wg w
u
∂t = a ∂
2w
∂x2+ug w
u
+wf (u2–w2 ).
Solution:
u = r(x, t) cosh ϕ(t), w = r(x, t) sinh ϕ(t), where the function ϕ = ϕ(t) is determined by the autonomous ordinary differential equation
ϕ
and the function r = r(x, t) is determined by the differential equation
∂r
∂t = a ∂2r
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
ak2r
zz + λr z + rf (r2) =0
For other exact solutions to equation (2) for various functions f , see Polyanin and Zaitsev
(2004)
Trang 4T10.3.2 Systems of the Form
∂u
∂t =
a
xn
∂
∂x
xn∂u
∂x
+F (u, w),
∂w
∂t =
b
xn
∂
∂x
xn∂w
∂x
+G(u, w)
T10.3.2-1 Arbitrary functions depend on a linear combination of the unknowns
∂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) and ψ = 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
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
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 = u0(x) +
e β n t u
n (x).
(ii) If the condition ζf (ζ) + bg(ζ) – ch(ζ) = k1ζ + k0holds, equation (2) is linear,
∂ζ
∂t = a
x n
∂
∂x
x n ∂ζ
∂x
+ k1ζ + k0,
and hence can be reduced to the linear heat equation (1) with the substitution ζ = e k1t ¯ζ–k0k–1
1 .
Trang 52. ∂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(C1t + C2), w = z(ξ) – 1
σ ln(C1t + C2), ξ = √ x
C1t + C2, where C1 and C2 are arbitrary constants, and the functions y = y(ξ) and z = z(ξ) are
determined by the system of ordinary differential equations
aξ–n (ξ n y
ξ) ξ+ 1
2C1ξy ξ +
C1
λ + e λy f (λy – σz) =0,
bξ–n (ξ n z
ξ) ξ+ 1
2C1ξz ξ+
C1
σ + e σz g(λy – σz) =0
T10.3.2-2 Arbitrary functions depend on the ratio of the unknowns
∂t = a
x n
∂
∂x
x n ∂u
∂x
+ uf
u w
∂t = b
x n
∂
∂x
x n ∂w
∂x
+ wg
u w
.
1◦ Multiplicative separable solution:
u = x1–2n [C1J ν (kx) + C2Y ν (kx)]ϕ(t), ν= 12|n–1|,
w = x1–2n [C1J ν (kx) + C2Y ν (kx)]ψ(t),
where C1, C2, and k are arbitrary constants, J ν (z) and Y ν (z) are Bessel functions, and the functions ϕ = ϕ(t) and ψ = ψ(t) are determined by the autonomous system of ordinary
differential equations
ϕ
t = –ak2ϕ + ϕf (ϕ/ψ),
ψ
t = –bk2ψ + ψg(ϕ/ψ).
2◦ Multiplicative separable solution:
u = x1–2n [C1I ν (kx) + C2K ν (kx)]ϕ(t), ν = 12|n–1|,
w = x1–2n [C1I ν (kx) + C2K ν (kx)]ψ(t),
where C1, C2, and k are arbitrary constants, I ν (z) and K ν (z) are modified Bessel functions, and the functions ϕ = ϕ(t) and ψ = ψ(t) are determined by the autonomous system of
ordinary differential equations
ϕ
t = ak2ϕ + ϕf (ϕ/ψ),
ψ
t = bk2ψ + ψg(ϕ/ψ).
3◦ Multiplicative separable solution:
u = e–λt y(x), w = e–λt z(x),
Trang 6where λ is an arbitrary constant and the functions y = y(x) and z = 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 Let k 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
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
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 Item4◦ The general solution of equation (2) is written out in implicit form as
dϕ [f (ϕ) – g(ϕ)]ϕ = t + C.
∂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:
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
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.
Trang 75. ∂u
∂t = a
x n
∂
∂x
x n ∂u
∂x
+ uf1
w u
+ wg1
w u
,
∂w
∂t = a
x n
∂
∂x
x n ∂w
∂x
+ uf2
w u
+ wg2
w u
.
Solution:
u= exp
[f1(ϕ)+ϕg1(ϕ)] dt
θ(x, t), w(x, t) = ϕ(t) exp
[f1(ϕ)+ϕg1(ϕ)] dt
θ(x, t),
where the function ϕ = ϕ(t) is described by the separable first-order nonlinear ordinary
differential equation
ϕ
t = f2(ϕ) + ϕg2(ϕ) – ϕ[f1(ϕ) + ϕg1(ϕ)],
and the function θ = θ(x, t) satisfies the linear heat equation
∂θ
∂t = a
x n
∂
∂x
x n ∂θ
∂x
∂t = a
x n
∂
∂x
x n ∂u
∂x
+ u k f
u w
∂t = b
x n
∂
∂x
x n ∂w
∂x
+ w k g
u w
.
Self-similar solution:
u = (C1t + C2)1 –1k y(ξ), w = (C1t + C2)1 –1k z(ξ), ξ= √ x
C1t + C2, where C1 and C2 are arbitrary constants, and the functions y = y(ξ) and z = z(ξ) are
determined by the system of ordinary differential equations
aξ–n (ξ n y
ξ) ξ+ 1
2C1ξy ξ +
C1
k–1y + y k f (y/z) =0,
bξ–n (ξ n z
ξ) ξ+ 1
2C1ξz ξ +
C1
k–1z + z k g(y/z) =0.
∂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) and ψ = ψ(t) are determined by solving the autonomous
ordinary differential equations
ϕ
t = ϕ[g(ϕ) – h(ϕ) + f (ϕ) ln ϕ],
ψ
and the function θ = θ(x, t) is determined by the differential equation
∂θ
∂t = a
x n
∂
∂x
x n ∂θ
∂x