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)