Handbook of mathematics for engineers and scienteists part 202 ppsx

T10.4. SYSTEMS OF GENERAL FORM 1375
1
◦
. Solution:
u = ϕ(t)+c exp


f(t, bϕ – cψ) dt

θ(x, t), w = ψ(t)+b exp


f(t, bϕ – cψ) dt

θ(x, t),
where ϕ = ϕ(t)andψ = ψ(t) are determined by the system of ordinary differential equations
ϕ

t
= ϕf (t, bϕ – cψ)+g(t, bϕ – cψ),
ψ

t
= ψf(t, bϕ – cψ)+h(t, bϕ – cψ),
and the function θ = θ(x
1
, , x
n
, t) satisfi es linear equation
∂θ
∂t

= L[θ].
Remark 1. The coefficients of the linear differential operator L can be dependent on x
1
, , x
n
, t.
2
◦
. Let us multiply the first equation by b and the second one by –c and add the results
together to obtain
∂ζ
∂t
= L[ζ]+ζf(t, ζ)+bg(t, ζ)–ch(t, ζ), ζ = bu – cw.(1)
This equation will be considered in conjunction with the first equation of the original system
∂u
∂t
= L[u]+uf (t, ζ)+g(t, ζ). (2)
Equation (1) can be treated separately. Given a solution of this equation, ζ = ζ(x
1
, , x
n
, t),
the function u = u(x
1
, , x
n
, t) can be determined by solving the linear equation (2) and
the function w = w(x
1
, , x

n
, t) is found as w =(bu – ζ)/c.
Remark 2. Let L be a constant-coefficient differential operator with respect to the independent variable
x = x
1
and let the condition
∂
∂t

ζf(t, ζ)+bg(t, ζ)–ch(t, ζ)

= 0
hold true (for example, it is valid if the functions f , g, h are not implicitly dependent on t). Then equation (1)
admits an exact, traveling-wave solution ζ = ζ(z), where z = kx – λt with arbitrary constants k and λ.
2.
∂u
∂t
= L
1
[u] + uf

u
w

,
∂w
∂t
= L
2
[w] + wg


u
w

.
Here, L
1
and L
2
are arbitrary constant-coefficient linear differential operators (of any order)
with respect to x.
1
◦
. Solution:
u = e
kx–λt
y(ξ), w = e
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 system of ordinary differential equations
M
1
[y]+λy + yf(y/z)=0, M
2
[z]+λz + zg(y/z)=0,
M
1
[y]=e
–kx

L
1
[e
kx
y(ξ)], M
2
[z]=e
–kx
L
2
[e
kx
z(ξ)].
To the special case k = λ = 0 there corresponds a traveling-wave solution.
2
◦
. If the operators L
1
and L
2
contain only even derivatives, there are solutions of the form
u =[C
1
sin(kx)+C
2
cos(kx)]ϕ(t), w =[C
1
sin(kx)+C
2
cos(kx)]ψ(t);

u =[C
1
exp(kx)+C
2
exp(–kx)]ϕ(t), w =[C
1
exp(kx)+C
2
exp(–kx)]ψ(t);
u =(C
1
x + C
2
)ϕ(t), w =(C
1
x + C
2
)ψ(t),
where C
1
, C
2
,andk are arbitrary constants. Note that the third solution is degenerate.
1376 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS
3.
∂u
∂t
= L[u] + uf

t,

u
w

,
∂w
∂t
= L[w] + wg

t,
u
w

.
Here, L is an arbitrary linear differential operator with respect to the coordinates x
1
, , x
n
(of any order in derivatives), whose coefficients can be dependent on x
1
, , x
n
, t:
L[u]=

A
k
1
k
n
(x

1
, , x
n
, t)
∂
k
1
+···+k
n
u
∂x
k
1
1
∂x
k
n
n
.(1)
1
◦
. Solution:
u = ϕ(t)exp


g(t, ϕ(t)) dt

θ(x
1
, , x

n
, t),
w =exp


g(t, ϕ(t)) dt

θ(x
1
, , x
n
, t),
(2)
where the function ϕ = ϕ(t) is described by the first-order nonlinear ordinary differential
equation
ϕ

t
=[f(t, ϕ)–g(t, ϕ)]ϕ,(3)
and the function θ = θ(x
1
, , x
n
, t) satisfi es the linear equation
∂θ
∂t
= L[θ].
2
◦
. The transformation

u = a
1
(t)U + b
1
(t)W , w = a
2
(t)U + b
2
(t)W ,
where a
n
(t)andb
n
(t) are arbitrary functions (n = 1, 2), leads to an equation of the similar
form for U and W .
Remark. The coefficients of the operator (1) can also depend on the ratio of the unknowns u/w, A
k
1
k
n
=
A
k
1
k
n
(x
1
, , x
n

, t, u/w) (in this case, L will be a quasilinear operator). Then there also exists a solution
of the form (2), where ϕ = ϕ(t) is described by the ordinary differential equation (3) and θ = θ(x
1
, , x
n
, t)
satisfies the linear equation
∂θ
∂t
= L
◦
[θ], L
◦
= L
u/w=ϕ
.
4.
∂u
∂t
= L[u] + uf

u
w

+ g

u
w

,

∂w
∂t
= L[w] + wf

u
w

+ h

u
w

.
Here, L is an arbitrary linear differential operator with respect to x
1
, , x
n
(of any order
in derivatives), whose coefficients can be dependent on x
1
, , x
n
, t:
L[u]=

A
k
1
k
n

(x
1
, , x
n
, t)
∂
k
1
+···+k
n
u
∂x
k
1
1
∂x
k
n
n
,
where k
1
+ ···+ k
n
≥ 1.
Let k be a root of the algebraic (transcendental) equation
g(k)=kh(k). (1)
1
◦
. Solution if 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)
,
T10.4. SYSTEMS OF GENERAL FORM 1377
where the function θ = θ(x
1
, , x
n
, t) satisfi es the linear equation
∂θ
∂t
= L[θ]. (2)
2
◦
. Solution if 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
1
, , x
n
, t) satisfi es the linear equation (2).
5.
∂u

∂t
= L[u]+uf

t,
u
w

+
u
w
h

t,
u
w

,
∂w
∂t
= L[w]+wg

t,
u
w

+h

t,
u
w


.
Solution:
u = ϕ(t)G(t)

θ(x
1
, , x
n
, t)+

h(t, ϕ)
G(t)
dt

, G(t)=exp


g(t, ϕ) dt

,
w = G(t)

θ(x
1
, , x
n
, t)+

h(t, ϕ)

G(t)
dt

,
where the function ϕ = ϕ(t) is described by the first-order nonlinear ordinary differential
equation
ϕ

t
=[f(t, ϕ)–g(t, ϕ)]ϕ,
and the function θ = θ(x
1
, , x
n
, t) satisfi es the linear equation
∂θ
∂t
= L[θ].
6.
∂u
∂t
= L[u] + uf

t,
u
w

ln u + ug

t,

u
w

,
∂w
∂t
= L[w] + wf

t,
u
w

ln w + wh

t,
u
w

.
Solution:
u(x, t)=ϕ(t)ψ(t)θ(x
1
, , x
n
, t), w(x, t)=ψ(t)θ(x
1
, , x
n
, t),
where the functions ϕ=ϕ(t)andψ =ψ(t) are determined by solving the ordinary differential

equations
ϕ

t
= ϕ[g(t, ϕ)–h(t, ϕ)+f(t, ϕ)lnϕ],
ψ

t
= ψ[h(t, ϕ)+f(t, ϕ)lnψ],
(1)
and the function θ = θ(x
1
, , x
n
, t) is determined by the differential equation
∂θ
∂t
= L[θ]+f(t, ϕ)θ ln θ.(2)
Givenasolutiontothefirst equation in (1), the second equation can be solved with the change
of variable ψ = e
ζ
by reducing it to a linear equation for ζ.IfL is a constant-coefficient
one-dimensional operator (n = 1)andf = const, then equation (2) has a traveling-wave
solution θ = θ(kx – λt).
1378 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS
7. F
1

w,
∂w

∂x
, ,
∂
m
w
∂x
m
,
1
u
k
∂w
∂t
,
1
u
∂u
∂x
, ,
1
u
∂
n
u
∂x
n

=0,
F
2


w,
∂w
∂x
, ,
∂
m
w
∂x
m
,
1
u
k
∂w
∂t
,
1
u
∂u
∂x
, ,
1
u
∂
n
u
∂x
n


=0.
Solution:
w = W (z), u =[ϕ

(t)]
1/k
U(z), z = x + ϕ(t),
where ϕ(t) is an arbitrary function, and the functions W (z)andU (z) are determined by the
autonomous system of ordinary differential equations
F
1

W , W

z
, , W
(m)
z
, W

z
/U
k
, U

z
/U, , U
(n)
z
/U


= 0,
F
2

W , W

z
, , W
(m)
z
, W

z
/U
k
, U

z
/U, , U
(n)
z
/U

= 0.
T10.4.3. Nonlinear Systems of Two Equations Involving the Second
Derivatives in
t
1.
∂

2
u
∂t
2
= L[u] + uf(t, au – bw) + g(t, au – bw),
∂
2
w
∂t
2
= L[w] + wf(t, au – bw) + h(t, au – bw).
Here, L is an arbitrary linear differential operator (of any order) with respect to the spatial
variables x
1
, , x
n
, whose coefficients can be dependent on x
1
, , x
n
, t. It is assumed
that L[const] = 0.
1
◦
. Solution:
u = ϕ(t)+aθ(x
1
, , x
n
, t), w = ψ(t)+bθ(x

1
, , x
n
, t),
where ϕ = ϕ(t)andψ = ψ(t) are determined by the system of ordinary differential equations
ϕ

tt
= ϕf (t, aϕ – bψ)+g(t, aϕ – bψ),
ψ

tt
= ψf(t, aϕ – bψ)+h(t, aϕ – bψ),
and the function θ = θ(x
1
, , x
n
, t) satisfi es linear equation
∂
2
θ
∂t
2
= L[θ]+f(t, aϕ – bψ)θ.
2
◦
. Let us multiply the fi rst equation by a and the second one by –b and add the results
together to obtain
∂
2

ζ
∂t
2
= L[ζ]+ζf(t, ζ)+ag(t, ζ)–bh(t, ζ), ζ = au – bw.(1)
This equation will be considered in conjunction with the first equation of the original system
∂
2
u
∂t
2
= L[u]+uf(t, ζ)+g(t, ζ). (2)
T10.4. SYSTEMS OF GENERAL FORM 1379
Equation (1) can be treated separately. Given a solution ζ = ζ(x, t) to equation (1), the
function u = u(x
1
, , x
n
, t) can be determined by solving the linear equation (2) and the
function w = w(x
1
, , x
n
, t) is found as w =(au – ζ)/b.
Note three important cases where equation (1) admits exact solutions:
(i) Equation (1) admits a spatially homogeneous solution ζ = ζ(t).
(ii) Suppose the coeffi cients of L and the functions f , g, h are not implicitly dependent
on t. Then equation (1) admits a steady-state solution ζ = ζ(x
1
, , x
n

).
(iii) If the condition ζf(t, ζ)+bg(t, ζ)–ch(t, ζ)=k
1
ζ + k
0
holds, equation (1) is linear.
If the operator L is constant-coefficient, the method of separation of variables can be used
to obtain solutions.
2.
∂
2
u
∂t
2
= L
1
[u] + uf

u
w

,
∂
2
w
∂t
2
= L
2
[w] + wg


u
w

.
Here, L
1
and L
2
are arbitrary constant-coefficient linear differential operators (of any order)
with respect to x. It is assumed that L
1
[const] = 0 and L
2
[const] = 0.
1
◦
. Solution in the form of the product of two waves traveling at different speeds:
u = e
kx–λt
y(ξ), w = e
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 system of ordinary differential equations
γ
2
y

ξξ

+ 2λγy

ξ
+ λ
2
y = M
1
[y]+yf(y/z), γ
2
z

ξξ
+ 2λγz

ξ
+ λ
2
z = M
2
[z]+zg(y/z),
M
1
[y]=e
–kx
L
1
[e
kx
y(ξ)], M
2

[z]=e
–kx
L
2
[e
kx
z(ξ)].
To the special case k = λ = 0 there corresponds a traveling-wave solution.
2
◦
. Periodic multiplicative separable solution:
u =[C
1
sin(kt)+C
2
cos(kt)]ϕ(x), w =[C
1
sin(kt)+C
2
cos(kt)]ψ(x),
where C
1
, C
2
,andk are arbitrary constants and the functions ϕ = ϕ(x)andψ = ψ(x)are
determined by the system of ordinary differential equations
L
1
[ϕ]+k
2

ϕ + ϕf(ϕ/ψ)=0,
L
2
[ψ]+k
2
ψ + ψg(ϕ/ψ)=0.
3
◦
. Multiplicative separable solution:
u =[C
1
sinh(kt)+C
2
cosh(kt)]ϕ(x), w =[C
1
sinh(kt)+C
2
cosh(kt)]ψ(x),
where C
1
, C
2
,andk are arbitrary constants and the functions ϕ = ϕ(x)andψ = ψ(x)are
determined by the system of ordinary differential equations
L
1
[ϕ]–k
2
ϕ + ϕf(ϕ/ψ)=0,
L

2
[ψ]–k
2
ψ + ψg(ϕ/ψ)=0.
4
◦
. Degenerate multiplicative separable solution:
u =(C
1
t + C
2
)ϕ(x), w =(C
1
t + C
2
)ψ(x),
1380 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS
where C
1
and C
2
are arbitrary constants and the functions ϕ = ϕ(x)andψ = ψ(x)are
determined by the system of ordinary differential equations
L
1
[ϕ]+ϕf(ϕ/ψ)=0, L
2
[ψ]+ψg(ϕ/ψ)=0.
Remark 1. The coefficients of L
1

, L
2
and the functions f and g in Items 2
◦
–4
◦
can be dependent on x.
Remark 2. If L
1
and L
2
contain only even derivatives, there are solutions of the form
u =[C
1
sin(kx)+C
2
cos(kx)]U(t), w =[C
1
sin(kx)+C
2
cos(kx)]W (t);
u =[C
1
exp(kx)+C
2
exp(–kx)]U(t), w =[C
1
exp(kx)+C
2
exp(–kx)]W (t);

u =(C
1
x + C
2
)U(t), w =(C
1
x + C
2
)W (t),
where C
1
, C
2
,andk are arbitrary constants. Note that the third solution is degenerate.
3.
∂
2
u
∂t
2
= L[u] + uf

t,
u
w

,
∂
2
w

∂t
2
= L[w] + wg

t,
u
w

.
Here, L is an arbitrary linear differential operator with respect to the coordinates x
1
, , x
n
(of any order in derivatives), whose coefficients can be dependent on the coordinates.
Solution:
u = ϕ(t)θ(x
1
, , x
n
),
w = ψ(t)θ(x
1
, , x
n
),
where the functions ϕ = ϕ(t)andψ = ψ(t) are described by the nonlinear system of
second-order ordinary differential equations
ϕ

tt

= aϕ + ϕf(t, ϕ/ψ),
ψ

tt
= aψ + ψg(t, ϕ/ψ),
a is an arbitrary constant, and the function θ = θ(x
1
, , x
n
) satisfies the linear steady-state
equation
L[θ]=aθ.
4.
∂
2
u
∂t
2
= L[u] + uf

u
w

+ g

u
w

,
∂

2
w
∂t
2
= L[w] + wf

u
w

+ h

u
w

.
Here, L is an arbitrary linear differential operator with respect to the coordinates x
1
, , x
n
(of any order in derivatives), whose coefficients can be dependent on x
1
, , x
n
, t.
Solution:
u = kθ(x
1
, , x
n
, t), w = θ(x

1
, , x
n
, t),
where k is a root of the algebraic (transcendental) equation g(k)=kh(k) and the function
θ = θ(x, t) satisfies the linear equation
∂
2
θ
∂t
2
= L[θ]+f(k)w + h(k).
5.
∂
2
u
∂t
2
= L[u] + au ln u + uf

t,
u
w

,
∂
2
w
∂t
2

= L[w] + aw ln w + wg

t,
u
w

.
Here, L is an arbitrary linear differential operator with respect to the coordinates x
1
, , x
n
(of any order in derivatives), whose coefficients can be dependent on the coordinates.
T10.4. SYSTEMS OF GENERAL FORM 1381
Solution:
u = ϕ(t)θ(x
1
, , x
n
),
w = ψ(t)θ(x
1
, , x
n
),
where the functions ϕ = ϕ(t)andψ = ψ(t) are described by the nonlinear system of
second-order ordinary differential equations
ϕ

tt
= aϕ ln ϕ + bϕ + ϕf(t, ϕ/ψ),

ψ

tt
= aψ ln ψ + bψ + ψg(t, ϕ/ψ),
b is an arbitrary constant, and the function θ = θ(x
1
, , x
n
) satisfies the steady-state
equation
L[θ]+aθ ln θ – bθ = 0.
T10.4.4. Nonlinear Systems of Many Equations Involving the First
Derivatives in
t
1.
∂u
m
∂t
= L[u
m
] + u
m
f(t, u
1
– b
1
u
n
, , u
n–1

– b
n–1
u
n
)
+ g
m
(t, u
1
– b
1
u
n
, , u
n–1
– b
n–1
u
n
), m =1, , n.
The system involves n + 1 arbitrary functions f, g
1
, , g
n
that depend on n arguments;
L is an arbitrary linear differential operator with respect to the spatial variables x
1
, , x
n
(of any order in derivatives), whose coefficients can be dependent on x

1
, , x
n
, t.Itis
assumed that L[const] = 0.
Solution:
u
m
= ϕ
m
(t)+exp


f(t, ϕ
1
– b
1
ϕ
n
, , ϕ
n–1
– b
n–1
ϕ
n
) dt

θ(x
1
, , x

n
, t).
Here, the functions ϕ
m
= ϕ
m
(t) are determined by the system of ordinary differential
equations
ϕ

m
= ϕ
m
f(t, ϕ
1
– b
1
ϕ
n
, , ϕ
n–1
– b
n–1
ϕ
n
)+g
m
(t, ϕ
1
– b

1
ϕ
n
, , ϕ
n–1
– b
n–1
ϕ
n
),
where m = 1, , n, the prime denotes the derivative with respect to t, and the function
θ = θ(x
1
, , x
n
, t) satisfies the linear equation
∂θ
∂t
= L[θ].
2.
∂u
m
∂t
= L[u
m
] + u
m
f
m


t,
u
1
u
n
, ,
u
n–1
u
n

+
u
m
u
n
g

t,
u
1
u
n
, ,
u
n–1
u
n

,

∂u
n
∂t
= L[u
n
] + u
n
f
n

t,
u
1
u
n
, ,
u
n–1
u
n

+ g

t,
u
1
u
n
, ,
u

n–1
u
n

.
Here, m = 1, , n – 1 and the system involves n + 1 arbitrary functions f
1
, , f
n
, g
that depend on n arguments; L is an arbitrary linear differential operator with respect to
the spatial variables x
1
, , x
n
(of any order in derivatives), whose coefficients can be
dependent on x
1
, , x
n
, t. It is assumed that L[const] = 0.