Handbook of mathematics for engineers and scienteists part 198 pdf

T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1347
6a.
∂u
∂t
= a
∂
2
u
∂x
2
+ uf

u
w

,
∂w
∂t
= a
∂
2
w
∂x
2
+ 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
u
1
= Au( x + C
1
, t + C
2
), w
1
= Aw( x + C
1
, t + C
2
);
u
2
=exp(λx + aλ
2
t)u(x + 2aλt, t), w
2
=exp(λx + aλ
2
t)w(x + 2aλt, t),
where A, C

1
, C
2
,andλ 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)andψ = 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 +
2
3
ak
2
t
3

– λt

y(ξ),
w =exp

kxt +
2
3
ak
2
t
3
– λt

z(ξ),
ξ = x + akt
2
,
where k and λ are arbitrary constants, and the functions y = y(ξ)andz = 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
◦

.Letk be a root of the algebraic (transcendental) equation
f(k)=g(k). (1)
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
∂
2
θ
∂x
2
.
9
◦
. Periodic solution:
u = Ak exp(–μx)sin(βx – 2aβμt + B),
w = A exp(–μx)sin(βx – 2aβμt + B),
β =

μ
2
+
1
a

f(k),
where A, B,andμ are arbitrary constants, and k is a root of the algebraic (transcendental)
equation (1).
1348 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS
10
◦
. 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
∂θ
∂t
= a
∂
2

θ
∂x
2
.
To the particular solution ϕ = k = const of equation (2) there corresponds the solution
given in Item 8
◦
. 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 = a
1
U + b
1
W , w = a
2
U + b
2
W ,
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
∂
2
u
∂x
2
+ uf

u
w

+ g

u
w

,
∂w
∂t
= a
∂
2
w
∂x
2
+ 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
∂
2

θ
∂x
2
.(1)
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).
T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1349
8.
∂u
∂t
= a
∂
2
u
∂x
2
+ uf

u
w

+
u
w
h

u

w

,
∂w
∂t
= a
∂
2
w
∂x
2
+ 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
∂
2
θ
∂x
2

.
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
∂
2
u
∂x
2
+uf
1

w
u

+wg
1

w
u

,
∂w
∂t

= a
∂
2
w
∂x
2
+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
∂
2
θ

∂x
2
.
10.
∂u
∂t
= a
∂
2
u
∂x
2
+ u
3
f

u
w

,
∂w
∂t
= a
∂
2
w
∂x
2
+ u
3

g

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)andψ = ψ(z)are
determined by the autonomous system of ordinary differential equations
ϕ

zz
+ 9aϕ
3
f(ϕ/ψ)=0,
ψ

zz
+ 9aϕ
3
g(ϕ/ψ)=0.
1350 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS
11.
∂u

∂t
=
∂
2
u
∂x
2
+ au – u
3
f

u
w

,
∂w
∂t
=
∂
2
w
∂x
2
+ aw – u
3
g

u
w


.
1
◦
. Solution with a > 0:
u =

C
1
exp

1
2
√
2ax+
3
2
at

– C
2
exp

–
1
2
√
2ax+
3
2
at


ϕ(z),
w =

C
1
exp

1
2
√
2ax+
3
2
at

– C
2
exp

–
1
2
√
2ax+
3
2
at

ψ(z),

z = C
1
exp

1
2
√
2ax+
3
2
at

+ C
2
exp

–
1
2
√
2ax+
3
2
at

+ C
3
,
where C
1

, C
2
,andC
3
are arbitrary constants, and the functions ϕ = ϕ(z)andψ = ψ(z)are
determined by the autonomous system of ordinary differential equations
aϕ

zz
= 2ϕ
3
f(ϕ/ψ),
aψ

zz
= 2ϕ
3
g(ϕ/ψ).
2
◦
. Solution with a < 0:
u =exp

3
2
at

sin

1

2

2|a| x + C
1

U(ξ),
w =exp

3
2
at

sin

1
2

2|a| x + C
1

W (ξ),
ξ =exp

3
2
at

cos

1

2

2|a| x + C
1

+ C
2
,
where C
1
and C
2
are arbitrary constants, and the functions U = U(ξ)andW = W (ξ)are
determined by the autonomous system of ordinary differential equations
aU

ξξ
=–2U
3
f(U/W),
aW

ξξ
=–2U
3
g(U/W).
12.
∂u
∂t
= a

∂
2
u
∂x
2
+ u
n
f

u
w

,
∂w
∂t
= b
∂
2
w
∂x
2
+ 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 =(C
1
t + C
2
)
1
1–n
y(ξ), w =(C
1
t + C
2
)
1
1–n
z(ξ), 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
n – 1
y + y
n
f

y
z


= 0,
bz

ξξ
+
1
2
C
1
ξz

ξ
+
C
1
n – 1
z + 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–1
f(k)=g(k),
and the function θ = θ(x, t) satisfies the heat equation with a power-law nonlinearity
∂θ
∂t
= a
∂
2
θ
∂x
2
+ g(k)θ
n
.
T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1351
13.
∂u
∂t
= a
∂
2
u
∂x
2
+ uf

u
w


ln u + ug

u
w

,
∂w
∂t
= a
∂
2
w
∂x
2
+ 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)andψ = ψ(t) are determined by solving the first-order
autonomous ordinary differential equations
ϕ


t
= ϕ[g(ϕ)–h(ϕ)+f(ϕ)lnϕ], (1)
ψ

t
= ψ[h(ϕ)+f (ϕ)lnψ], (2)
and the function θ = θ(x, t) is determined by the differential equation
∂θ
∂t
= a
∂
2
θ
∂x
2
+ f(ϕ)θ ln θ.(3)
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

σ
2
(t)x
2
+ σ
1

(t)x + σ
0
(t)

,
where the functions σ
n
(t) are described by the equations
σ

2
= f (ϕ)σ
2
+ 4aσ
2
2
,
σ

1
= f (ϕ)σ
1
+ 4aσ
1
σ
2
,
σ

0

= f (ϕ)σ
0
+ aσ
2
1
+ 2aσ
2
.
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 σ
2
= 0.
Remark. Equation(1) hasa special solutionϕ =k =const, where k isa rootof thealgebraic (transcendental)
equation g(k)–h(k)+f(k)lnk = 0.
14.
∂u
∂t
= a
∂
2
u
∂x
2
+ uf

w
u

– wg


w
u

+
u
√
u
2
+ w
2
h

w
u

,
∂w
∂t
= a
∂
2
w
∂x
2
+ wf

w
u


+ ug

w
u

+
w
√
u
2
+ w
2
h

w
u

.
Solution:
u = r(x, t)cosϕ(t), w = r(x, t)sinϕ(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

∂
2
r
∂x
2
+ rf(tan ϕ)+h(tan ϕ). (1)
1352 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS
The 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
∂
2
Z
∂x
2
.

15.
∂u
∂t
= a
∂
2
u
∂x
2
+ uf

w
u

+ wg

w
u

+
u
√
u
2
– w
2
h

w
u


,
∂w
∂t
= a
∂
2
w
∂x
2
+ wf

w
u

+ ug

w
u

+
w
√
u
2
– w
2
h

w

u

.
Solution:
u = r(x, t)coshϕ(t), w = r(x, t)sinhϕ(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
∂
2
r
∂x
2
+ rf(tanh ϕ)+h(tanh ϕ). (1)
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
∂
2
Z
∂x
2
.
T10.3.1-3. Arbitrary functions depend on the product of powers of the unknowns.
16.
∂u
∂t
= a
∂
2
u
∂x
2
+ uf(u
n
w
m
),
∂w

∂t
= b
∂
2
w
∂x
2
+ 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,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
+(2akmβ + γ)y

ξ
+ m(ak
2

m + λ)y + yf(y
n
z
m
)=0,
bβ
2
z

ξξ
+(–2bknβ + γ)z

ξ
+ n(bk
2
n – λ)z + zg(y
n
z
m
)=0.
To the special case k = λ = 0 there corresponds a traveling-wave solution.
T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1353
17.
∂u
∂t
= a
∂
2
u
∂x

2
+ u
1+kn
f

u
n
w
m

,
∂w
∂t
= b
∂
2
w
∂x
2
+ w
1–km
g

u
n
w
m

.
Self-similar solution:

u =(C
1
t + C
2
)
–
1
kn
y(ξ), w =(C
1
t + C
2
)
1
km
z(ξ), 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
kn
y + y
1+kn
f

y
n
z
m

= 0,
bz

ξξ

+
1
2
C
1
ξz

ξ
–
C
1
km
z + z
1–km
g

y
n
z
m

= 0.
18.
∂u
∂t
= a
∂
2
u
∂x

2
+cu ln u +uf(u
n
w
m
),
∂w
∂t
= b
∂
2
w
∂x
2
+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,andλ are arbitrary constants, and the functions y = y(ξ)andz = z(ξ)are
determined by the autonomous system of ordinary differential equations
ak
2
y


ξξ
+ λy

ξ
+ cy ln y + yf(y
n
z
m
)=0,
bk
2
z

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
∂
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
◦
. A periodic solution in the spatial coordinate:
u = ψ(t)cosϕ(x, t), w = ψ(t)sinϕ(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,
whose general solution can be represented in implicit form as


dψ
ψf(ψ
2
)–aC
2
1
ψ
= t + C
3
.