1340 SYSTEMS OFPARTIALDIFFERENTIALEQUATIONS2◦... Linear Systems of Two Second-Order Equations 1.. Constant-coefficient second-order linear system of parabolic type... Variable-coefficien
Trang 11340 SYSTEMS OFPARTIALDIFFERENTIALEQUATIONS
2◦ If k≠ 1, it is more beneficial to seek solutions in the form
u = x– k–11ϕ (ζ), w = x m(k– n1 )ψ (ζ), ζ = t + a ln|x|,
where a is an arbitrary constant, and the functions ϕ(ζ) and ψ(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, and λ are arbitrary constants, and the functions y(ξ) and z(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).
LetΔ = 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) and ψ(t) are determined by the autonomous ordinary differential
equations
s
Δϕ x = ϕf (ϕ),
n
Δψ t = ψg(ψ).
Integrating yields
s
Δ
dϕ
ϕf (ϕ) = x + C1,
n
Δ
dψ
ψg (ψ) = t + C2.
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, and λ are arbitrary constants, and the functions y(ξ) and z(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).
Trang 215. ∂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 = (C1t + C2)n–k1 θ (x), w = ϕ(x) – a
b (C1t + C2)n–k n [θ(x)] n,
where C1 and C2 are arbitrary constants, and the functions θ = θ(x) and ϕ = ϕ(x) are
determined by the system of differential-algebraic equations
θ
x = θf (bϕ),
θ n–k = b (k – n)
aC1n g (bϕ).
17. ∂u
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) and ψ(t) are arbitrary functions.
T10.2 Linear Systems of Two Second-Order Equations
1. ∂u
∂t = a ∂
2u
∂x2 + b1u + c1w, ∂w
∂t = a ∂
2w
∂x2 + b2u + c2w.
Constant-coefficient second-order linear system of parabolic type.
Solution:
u= b1– λ2
b2(λ1– λ2)e
λ1t θ
1– b b1– λ1
2(λ1– λ2)e
λ2t θ
2,
λ1– λ2 e
λ1t θ1– e λ2t θ2
,
where λ1and λ2are roots of the quadratic equation
λ2– (b
1+ c2)λ + b1c2– b2c1 =0,
and the functions θ n = θ n (x, t) satisfy the independent linear heat equations
∂θ1
∂t = a ∂
2θ 1
∂x2 ,
∂θ2
∂t = a ∂
2θ 2
∂x2 .
Trang 31342 SYSTEMS OFPARTIALDIFFERENTIALEQUATIONS
2. ∂u
∂t = a ∂
2u
∂x2 + f1(t)u + g1(t)w, ∂w
∂t = a ∂
2w
∂x2 + f2(t)u + g2(t)w.
Variable-coefficient second-order linear system of parabolic type.
Solution:
u = ϕ1(t)U (x, t) + ϕ2(t)W (x, t),
w = ψ1(t)U (x, t) + ψ2(t)W (x, t), where the pairs of functions ϕ1= ϕ1(t), ψ1= ψ1(t) and ϕ2 = ϕ2(t), ψ2 = ψ2(t) are linearly
independent (fundamental) solutions to the system of linear ordinary differential equations
ϕ
t = f1(t)ϕ + g1(t)ψ,
ψ
t = f2(t)ϕ + g2(t)ψ,
and the functions U = U (x, t) and W = W (x, t) satisfy the independent linear heat equations
∂U
∂t = a ∂
2U
∂x2,
∂W
∂t = a ∂
2W
∂x2 .
3. ∂
2u
∂t2 = k ∂
2u
∂x2 + a1u + b1w, ∂
2w
∂t2 = k ∂
2w
∂x2 + a2u + b2w.
Constant-coefficient second-order linear system of hyperbolic type.
Solution:
u= a1– λ2
a2(λ1– λ2)θ1–
a1– λ1
a2(λ1– λ2)θ2, w=
1
λ1– λ2 θ1– θ2
,
where λ1and λ2are roots of the quadratic equation
λ2– (a
1+ b2)λ + a1b2– a2b1=0,
and the functions θ n = θ n (x, t) satisfy the linear Klein–Gordon equations
∂2θ1
∂t2 = k
∂2θ1
∂x2 + λ1θ1,
∂2θ2
∂t2 = k
∂2θ2
∂x2 + λ2θ2.
4. ∂
2u
∂x2 + ∂
2u
∂y2 = a1u + b1w, ∂
2w
∂x2 + ∂
2w
∂y2 = a2u + b2w.
Constant-coefficient second-order linear system of elliptic type.
Solution:
u= a1– λ2
a2(λ1– λ2)θ1–
a1– λ1
a2(λ1– λ2)θ2, w=
1
λ1– λ2 θ1– θ2
,
where λ1and λ2are roots of the quadratic equation
λ2– (a
1+ b2)λ + a1b2– a2b1=0,
and the functions θ n = θ n (x, y) satisfy the linear Helmholtz equations
∂2θ 1
∂x2 +
∂2θ 1
∂y2 = λ1θ1,
∂2θ 2
∂x2 +
∂2θ 2
∂y2 = λ2θ2.
Trang 4T10.3 Nonlinear Systems of Two Second-Order
Equations
T10.3.1 Systems of the Form
∂u
∂t = a ∂
2u
∂x2 +F (u, w),
∂w
∂t = b ∂
2w
∂x2 +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 functions f (ϕ), g(ϕ), h(ϕ) appearing below are arbitrary 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 ∂
2u
∂x2 + u exp
k w u
f (u), ∂w
∂t = a ∂
2w
∂x2 + exp
k w u
[wf(u)+g(u)].
Solution:
u = y(ξ), w= –2
k ln|bx|y (ξ) + z(ξ), ξ = √ x + C3
C1t + C2, where C1, C2, C3, and b are 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+
1
b2ξ2yexp
k z y
f (y) =0,
az
ξξ+ 1
2C1ξz ξ – 4a
kξ y
ξ+ 2a
kξ2y+
1
b2ξ2 exp
k z y
[zf (y) + g(y)] =0
2. ∂u
∂t = a1∂
2u
∂x2 + f (bu + cw), ∂w
∂t = a2∂
2w
∂x2 + g(bu + cw).
Solution:
u = c(αx2+ βx + γt) + y(ξ), w = –b(αx2+ βx + γt) + z(ξ), ξ = kx – λt,
where k, α, β, γ, and λ are arbitrary constants, and the functions y(ξ) and z(ξ) are determined
by the autonomous system of ordinary differential equations
a1k2y
ξξ + λy ξ+2a1cα – cγ + f (by + cz) =0,
a2k2z
ξξ + λz ξ –2a2bα + bγ + g(by + cz) =0
Trang 51344 SYSTEMS OFPARTIALDIFFERENTIALEQUATIONS
3. ∂u
∂t = a ∂
2u
∂x2 + f (bu + cw), ∂w
∂t = a ∂
2w
∂x2 + g(bu + cw).
Solution:
u = cθ(x, t) + y(ξ), w = –bθ(x, t) + z(ξ), ξ = kx – λt, where the functions y(ξ) and z(ξ) are determined by the autonomous system of ordinary
differential equations
ak2y
ξξ + λy ξ + f (by + cz) =0,
ak2z
ξξ + λz ξ + g(by + cz) =0,
and the function θ = θ(x, t) satisfies the linear heat equation
∂θ
∂t = a ∂
2θ
∂x2.
4. ∂u
∂t = a ∂
2u
∂x2 + uf (bu – cw) + g(bu – cw),
∂w
∂t = a ∂
2w
∂x2 + 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 the linear heat equation
∂θ
∂t = a ∂
2θ
∂x2.
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 ∂
2ζ
∂x2 + 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 ∂
2u
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.
Trang 6Note 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 = u0(z) +
e β n t u
n (z).
(ii) If the condition ζf (ζ) + bg(ζ) – ch(ζ) = k1ζ + k0holds, equation (1) is linear,
∂ζ
∂t = a ∂
2ζ
∂x2 + k1ζ + k0,
and, hence, can be reduced to the linear heat equation
5. ∂u
∂t = a ∂
2u
∂x2 + e λu f (λu – σw), ∂w
∂t = b ∂
2w
∂x2 + e σw g(λu – σw).
1◦ Solution:
u = y(ξ) – 1
λ ln(C1t + C2), w = z(ξ) – 1
σ ln(C1t + C2), ξ = √ x + C3
C1t + C2, where 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
λ + e λy f (λy – σz) =0,
bz
ξξ+
1
2C1ξz ξ +
C1
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 ∂2θ
∂x2 + 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 ∂
2u
∂x2 + uf
u w
, ∂w
∂t = b ∂
2w
∂x2 + wg
u w
.
1◦ Multiplicative separable solution:
u = [C1sin(kx) + C2cos(kx)]ϕ(t),
w = [C1sin(kx) + C2cos(kx)]ψ(t),
Trang 71346 SYSTEMS OFPARTIALDIFFERENTIALEQUATIONS
where C1, C2, and k are arbitrary constants, 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 = [C1exp(kx) + C2exp(–kx)]U (t),
w = [C1exp(kx) + C2exp(–kx)]W (t), where C1, C2, and k are arbitrary constants, and the functions U = U (t) and W = W (t) are
determined by the autonomous system of ordinary differential equations
U
t = ak2U + U f (U/W ),
W
t = bk2W + W g(U/W ).
3◦ Degenerate solution:
u = (C1x + C2)U (t),
w = (C1x + C2)W (t), where C1and C2 are arbitrary constants, and the functions U = U (t) and W = W (t) are
determined by the autonomous system of ordinary differential equations
U
t = U f (U/W ),
W
t = W g(U/W ).
This autonomous system can be integrated since it is reduced, after eliminating t, to a
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) and z = 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 Item4◦):
u = e kx–λt y (ξ), w = e kx–λt z (ξ), ξ = βx – γt, where k, λ, β, and γ are arbitrary constants, and the functions y = y(ξ) and z = z(ξ) are
determined by the autonomous system of ordinary differential equations
aβ2y
ξξ+ (2akβ + γ)y ξ + (ak2+ λ)y + yf (y/z) =0,
bβ2z
ξξ+ (2bkβ + γ)z ξ + (bk2+ λ)z + zg(y/z) =0
To the special case k = λ =0there corresponds a traveling-wave solution If k = γ =0and
β=1, we have the solution of Item4◦.