In addition, it has some interesting properties and other solutions, which are given below... This system can be integrated directly, since the first equation is a Bernoulli equation and
Trang 16a. ∂u
∂t = a ∂
2u
∂x2 + uf
u w
∂t = a ∂
2w
∂x2 + wg
u w
.
This system is a special case of system T10.3.1.6 with b = a and hence it admits the above
solutions given in Items 1◦–5◦ In addition, it has some interesting properties and other
solutions, which are given below
Suppose u = u(x, t), w = w(x, t) is a solution of the system Then the functions
u1 = Au( x + C1, t + C2), w1= Aw( x + C1, t + C2);
u2 = exp(λx + aλ2t )u(x +2aλt , t), w2= exp(λx + aλ2t )w(x +2aλt , t),
where A, C1, C2, and λ are arbitrary constants, are also solutions of these equations.
6◦ Point-source solution:
u= exp
– x
2
4at
ϕ (t), w= exp
– x
2
4at
ψ (t),
where the functions ϕ = ϕ(t) and ψ = ψ(t) are determined by the autonomous system of
ordinary differential equations
ϕ
t= – 1
2t ϕ + ϕf
ϕ
ψ
,
ψ
t= – 1
2t ψ + ψg
ϕ
ψ
7◦ Functional separable solution:
u= exp kxt+ 23ak2t3– λt
y (ξ),
w= exp kxt+ 23ak2t3– λt
z (ξ), ξ = x + akt
2,
where k and λ are arbitrary constants, and the functions y = y(ξ) and z = z(ξ) are determined
by the autonomous system of ordinary differential equations
ay
ξξ + (λ – kξ)y + yf (y/z) =0,
az
ξξ + (λ – kξ)z + zg(y/z) =0
8◦ Let k be a root of the algebraic (transcendental) equation
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 ∂
2θ
∂x2.
9◦ Periodic solution:
u = Ak exp(–μx) sin(βx –2aβμt + B),
w = A exp(–μx) sin(βx –2aβμt + B), β = μ2+
1
a f (k), where A, B, and μ are arbitrary constants, and k is a root of the algebraic (transcendental)
equation (1)
Trang 210◦ 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
∂θ
∂t = a ∂
2θ
∂x2.
To the particular solution ϕ = k = const of equation (2) there corresponds the solution
given in Item8◦ The general solution of equation (2) is written out in implicit form as
dϕ [f (ϕ) – g(ϕ)]ϕ = t + C.
11◦ The transformation
u = a1U + b1W, w = a2U + b2W,
where a n and b n are arbitrary constants (n =1,2), leads to an equation of similar form for
U and W
7. ∂u
∂t = a ∂
2u
∂x2 + uf
u w
+ g
u w
∂t = a ∂
2w
∂x2 + wf
u w
+ h
u w
.
Let k be a root of the algebraic (transcendental) equation
g (k) = kh(k).
1◦ Solution with f (k)≠ 0:
u (x, t) = k
exp[f (k)t]θ(x, t) – h (k)
f (k)
, w (x, t) = exp[f (k)t]θ(x, t) – h (k)
f (k), where the function θ = θ(x, t) satisfies the linear heat equation
∂θ
∂t = a ∂
2θ
2◦ Solution with f (k) =0:
u (x, t) = k[θ(x, t) + h(k)t], w (x, t) = θ(x, t) + h(k)t, where the function θ = θ(x, t) satisfies the linear heat equation (1).
Trang 38. ∂u
∂t = a ∂
2u
∂x2 + uf
u w
+ u
w h
u w
∂t = a ∂
2w
∂x2 + 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
and the function θ = θ(x, t) satisfies the linear heat equation
∂θ
∂t = a ∂2θ
∂x2.
The general solution of equation (1) is written out in implicit form as
dϕ [f (ϕ) – g(ϕ)]ϕ = t + C.
9. ∂u
∂t = a ∂
2u
∂x2+ uf1
w u
+ wg1
w u
∂t = a ∂
2w
∂x2 + 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 ∂
2θ
∂x2.
10. ∂u
∂t = a ∂
2u
∂x2 + u3f
u w
∂t = a ∂
2w
∂x2 + u3g
u w
.
Solution:
u = (x + A)ϕ(z), w = (x + A)ψ(z), z = t + 1
6a (x + A)2+ B, where A and B are arbitrary constants, and the functions ϕ = ϕ(z) and ψ = ψ(z) are
determined by the autonomous system of ordinary differential equations
ϕ
zz+9aϕ3f (ϕ/ψ) =0,
ψ
zz +9aϕ3g (ϕ/ψ) =0
Trang 411. ∂u
∂t = ∂
2u
∂x2 + au – u3f
u w
∂t = ∂
2w
∂x2 + aw – u3g
u w
.
1◦ Solution with a >0:
u=
C1exp 12√
2a x+ 32at
– C2exp –12√
2a x+ 32at
ϕ (z),
w=
C1exp 12√
2a x+ 32at
– C2exp –12√
2a x+ 32at
ψ (z),
z = C1exp 12√
2a x+ 32at
+ C2exp –12√
2a x+ 32at
+ C3,
where C1, C2, and C3are arbitrary constants, and the functions ϕ = ϕ(z) and ψ = ψ(z) are
determined by the autonomous system of ordinary differential equations
aϕ
zz =2ϕ3f (ϕ/ψ),
aψ
zz =2ϕ3g (ϕ/ψ).
2◦ Solution with a <0:
u= exp 32at
sin 12
2|a|x + C1
U (ξ),
w= exp 32at
sin 12
2|a|x + C1
W (ξ),
ξ= exp 32at
cos 12
2|a|x + C1
+ C2,
where C1 and C2 are arbitrary constants, and the functions U = U (ξ) and W = W (ξ) are
determined by the autonomous system of ordinary differential equations
aU
ξξ= –2U3f (U/W ),
aW
ξξ= –2U3g (U/W ).
12. ∂u
∂t = a ∂
2u
∂x2 + u n f
u w
∂t = b ∂
2w
∂x2 + w n g
u w
.
If f (z) = kz–m and g(z) = –kz n–m , the system describes an nth-order chemical reaction (of order n – m in the component u and of order m in the component w).
1◦ Self-similar solution with n≠ 1:
u = (C1t + C2)1 –1n y (ξ), w = (C1t + C2)1 –1n z (ξ), 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
n–1y + y n f
y
z
=0,
bz
ξξ+
1
2C1ξz ξ +
C1
n–1z + z n g
y
z
=0
2◦ Solution with b = a:
u (x, t) = kθ(x, t), w (x, t) = θ(x, t), where k is a root of the algebraic (transcendental) equation
k n–1f (k) = g(k),
and the function θ = θ(x, t) satisfies the heat equation with a power-law nonlinearity
∂θ
∂t = a ∂
2θ
∂x2 + g(k)θ n.
Trang 513. ∂u
∂t = a ∂
2u
∂x2 + uf
u w
ln u + ug
u w
,
∂w
∂t = a ∂
2w
∂x2 + wf
u w
ln w + wh
u w
.
Solution:
u (x, t) = ϕ(t)ψ(t)θ(x, t), w (x, t) = ψ(t)θ(x, t), where the functions ϕ = ϕ(t) and ψ = ψ(t) are determined by solving the first-order
autonomous ordinary differential equations
ϕ
ψ
and the function θ = θ(x, t) is determined by the differential equation
∂θ
∂t = a ∂
2θ
The separable equation (1) can be solved to obtain a solution in implicit form
Equa-tion (2) is easy to integrate—with the change of variable ψ = e ζ, it is reduced to a linear equation Equation (3) admits exact solutions of the form
θ= exp
σ2(t)x2+ σ1(t)x + σ0(t)
,
where the functions σ n (t) are described by the equations
σ
2= f (ϕ)σ2+4aσ2
2,
σ
1= f (ϕ)σ1+4aσ1σ2,
σ
0= f (ϕ)σ0+ aσ12+2aσ2.
This system can be integrated directly, since the first equation is a Bernoulli equation and the second and third ones are linear in the unknown Note that the first equation has a
particular solution σ2=0
Remark. Equation (1) has a special solution ϕ = k = const, where k is a root of the algebraic (transcendental) equation g(k) – h(k) + f (k) ln k =0
14. ∂u
∂t = a ∂
2u
∂x2 + uf w
u
– wg w
u
+ √ u
u2+ w2h w
u
,
∂w
∂t = a ∂
2w
∂x2 + wf w
u
+ ug w u
+ √ w
u2+ w2h w
u
.
Solution:
u = r(x, t) cos ϕ(t), w = r(x, t) sin ϕ(t), where the function ϕ = ϕ(t) is determined from the separable first-order ordinary differential
equation
ϕ
t = g(tan ϕ), and the function r = r(x, t) satisfies the linear equation
∂r
∂t = a ∂
2r
Trang 6The change of variable
r = F (t)
*
Z (x, t) +
h (tan ϕ) dt
F (t)
+
, F (t) = exp
*
f (tan ϕ) dt
+
brings (1) to the linear heat equation
∂Z
∂t = a ∂
2Z
∂x2.
15. ∂u
∂t = a ∂
2u
∂x2 + uf w
u
+ wg w
u
+ √ u
u2– w2h w
u
,
∂w
∂t = a ∂
2w
∂x2 + wf w
u
+ ug w u
+ √ w
u2– w2h w
u
.
Solution:
u = r(x, t) cosh ϕ(t), w = r(x, t) sinh ϕ(t), where the function ϕ = ϕ(t) is determined from the separable first-order ordinary differential
equation
ϕ
t = g(tanh ϕ), and the function r = r(x, t) satisfies the linear equation
∂r
∂t = a ∂
2r
The change of variable
r = F (t)*
Z (x, t) +
h (tanh ϕ) dt
F (t)
+
, F (t) = exp*
f (tanh ϕ) dt+
brings (1) to the linear heat equation
∂Z
∂t = a ∂
2Z
∂x2.
T10.3.1-3 Arbitrary functions depend on the product of powers of the unknowns
16. ∂u
∂t = a ∂
2u
∂x2 + uf (u n w m), ∂w
∂t = b ∂
2w
∂x2 + wg(u n w m).
Solution:
u = e m(kx–λt) y (ξ), w = e–n(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+ (2akmβ + γ)y ξ + m(ak2m + λ)y + yf (y n z m) =0,
bβ2z
ξξ+ (–2bknβ + γ)z ξ + n(bk2n – λ)z + zg(y n z m) =0
To the special case k = λ =0there corresponds a traveling-wave solution
Trang 717. ∂u
∂t = a ∂
2u
∂x2 + u1+kn f u n w m
∂t = b ∂
2w
∂x2 + w1–km g u n w m
.
Self-similar solution:
u = (C1t + C2)–kn1 y (ξ), w = (C1t + C2)km1 z (ξ), 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
kn y + y1+kn f y n z m
=0,
bz
ξξ+
1
2C1ξz ξ –
C1
km z + z1–km g y n z m
=0
18. ∂u
∂t = a ∂
2u
∂x2 + cu ln u + uf (u n w m), ∂w
∂t = b ∂
2w
∂x2 + cw ln w + wg(u n w m).
Solution:
u = exp(Ame ct )y(ξ), w = exp(–Ane ct )z(ξ), ξ = kx – λt,
where A, k, and λ are arbitrary constants, and the functions y = y(ξ) and z = z(ξ) are
determined by the autonomous system of ordinary differential equations
ak2y
ξξ + λy ξ + cy ln y + yf (y n z m) =0,
bk2z
ξξ + λz ξ + cz ln z + zg(y n z m) =0
To the special case A = 0 there corresponds a traveling-wave solution For λ =0, we
have a solution in the form of the product of two functions dependent on time t and the coordinate x.
T10.3.1-4 Arbitrary functions depend on the sum of squares of the unknowns
19. ∂u
∂t = a ∂
2u
∂x2 + uf (u2+ w2) – wg(u2+ w2 ),
∂w
∂t = a ∂
2w
∂x2 + ug(u2+ w2) + wf(u2+ w2 ).
1◦ A periodic solution in the spatial coordinate:
u = ψ(t) cos ϕ(x, t), w = ψ(t) sin 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,
whose general solution can be represented in implicit form as
dψ
ψf (ψ2) – aC12ψ = t + C3
... solution For λ =0, wehave a solution in the form of the product of two functions dependent on time t and the coordinate x.
T10.3.1-4 Arbitrary functions depend on the sum of. .. kz–m and g(z) = –kz n–m , the system describes an nth-order chemical reaction (of order n – m in the component u and of order m in the component w).... 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