G., Green’s Functions and Transfer Functions Handbook, Halstead Press–John Wiley & Sons, New York, 1982.. C., Conduction of Heat in Solids, Clarendon Press, Oxford, 1984.. D., Handbook o
Trang 11298 LINEAREQUATIONS ANDPROBLEMS OFMATHEMATICALPHYSICS
T8.4.3-4 Boundary value problem for a circle
Domain: 0 ≤r ≤a Boundary conditions in the polar coordinate system:
w = f (ϕ) at r = a, ∂ r w = g(ϕ) at r = a.
Solution:
w (r, ϕ) = 1
2πa(r2– a2)2
2π
0
[a – r cos(η – ϕ)]f (η) dη [r2+ a2–2arcos(η – ϕ)]2 –
1 2
2π
0
g (η) dη
r2+ a2–2arcos(η – ϕ)
T8.4.4 Nonhomogeneous Biharmonic EquationΔΔw = Φ(x, y)
T8.4.4-1 Domain: –∞ < x < ∞, –∞ < y < ∞.
Solution:
w (x, y) =
∞
–∞
∞
–∞ Φ(ξ, η) (x – ξ, y – η) dξ dη, (x, y) = 1
8π(x2+ y2) ln
x2+ y2.
T8.4.4-2 Domain: –∞ < x < ∞,0 ≤y<∞ Boundary value problem.
The upper half-plane is considered The derivatives are prescribed at the boundary:
∂ x w = f (x) at y=0, ∂ y w = g(x) at y=0 Solution:
w (x, y) = 1
π
∞ –∞ f (ξ)
arctan
x – ξ
y
+ y (x – ξ)
(x – ξ)2+ y2
dξ+ y 2
π
∞ –∞
g (ξ) dξ (x – ξ)2+ y2
8π
∞ –∞ dξ
∞
0
1
2(R2+– R2–) – R2–lnR+
R–
Φ(ξ, η) dη + C,
where C is an arbitrary constant,
R2
+= (x – ξ)2+ (y + η)2, R2
–= (x – ξ)2+ (y – η)2
T8.4.4-3 Domain: 0 ≤x≤l1, 0 ≤y≤l2 The sides of the plate are hinged
A rectangle is considered Boundary conditions are prescribed:
w = ∂ xx w=0 at x=0, w = ∂ xx w=0 at x = l1,
w = ∂ yy w=0 at y=0, w = ∂ yy w=0 at y = l2
Solution:
w (x, y) =
l1
0
l2
0 Φ(ξ, η)G(x, y, ξ, η) dη dξ,
where
G (x, y, ξ, η) = 4
l1l2
∞
n=1
∞
m=1
1
(p2n + q m2 )2 sin(p n x ) sin(q m y ) sin(p n ξ ) sin(q m η ), p n=
πn
l1 , q m=πm
l2
Trang 2R EFERENCES FOR C HAPTER T8 1299
References for Chapter T8
Butkovskiy, A G., Green’s Functions and Transfer Functions Handbook, Halstead Press–John Wiley & Sons,
New York, 1982.
Carslaw, H S and Jaeger, J C., Conduction of Heat in Solids, Clarendon Press, Oxford, 1984.
Miller, W., Jr., Symmetry and Separation of Variables, Addison-Wesley, London, 1977.
Polyanin, A D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman &
Hall/CRC Press, Boca Raton, 2002.
Sutton, W G L., On the equation of diffusion in a turbulent medium, Proc Roy Soc., Ser A, Vol 138, No 988,
pp 48–75, 1943.
Tikhonov, A N and Samarskii, A A., Equations of Mathematical Physics, Dover Publications, New York,
1990.
Trang 4Chapter T9
Nonlinear Mathematical
Physics Equations
T9.1 Parabolic Equations
T9.1.1 Nonlinear Heat Equations of the Form ∂w
∂t =
∂2w
∂x2 +f (w)
Equations of this form admit traveling-wave solutions w = w(z), z = kx + λt, where
k and λ are arbitrary constants, and the function w(z) is determined by the second-order autonomous ordinary differential equation ak2w
zz – λw z + f (w) =0.
∂t = ∂
2w
∂x2 + aw(1 – w).
Fisher’s equation This equation arises in heat and mass transfer, biology, and ecology Traveling-wave solutions (C is an arbitrary constant):
w (x, t) =
1+ C exp –56at 1
6
√
6a x –2,
w (x, t) = 1+2Cexp –56at 1
6
√
–6a x
1+ C exp –56at 1
6
√
–6a x 2.
∂t = ∂
2w
∂x2 + aw – bw3
1◦ Solutions with a >0and b >0:
w (x, t) = a
b
C1exp 12√
2a x – C2exp –12√
2a x
C1exp 12√
2a x + C2exp –12√
2a x + C3exp –32at ,
w (x, t) = a
b
2C
1exp 2a x + C2exp 12√
2a x– 32at
C1exp 2a x + C2exp 12√
2a x– 32at
+ C3 –1
,
where C1, C2, and C3are arbitrary constants
2◦ Solutions with a <0and b >0:
w (x, t) = |a|
b
sin 12√
2|a|x + C1
cos 12√2|a|
x + C1
+ C2exp –32at
3◦ Solutions with a =0and b >0:
w (x, t) = 2
b
2C1x + C2
C1x2+ C2x+6C1t + C3.
1301
Trang 51302 NONLINEARMATHEMATICALPHYSICSEQUATIONS
4◦ Solution with a =0(generalizes the solution of Item3◦):
w (x, y) = xu(z), z = t + 16x2,
where the function u(z) is determined by the autonomous ordinary differential equation
u
zz–9bu3=0
5◦ For a =0, there is a self-similar solution of the form
w (x, t) = t–1 2f (ξ), ξ = xt–1 2,
where the function f (ξ) is determined by the ordinary differential equation f ξξ + 12ξf
ξ+
1
2f – bf3=0
∂t = ∂
2w
∂x2 – w(1 – w)(a – w).
Fitzhugh–Nagumo equation This equation arises in genetics, biology, and heat and mass
transfer
Solutions:
w (x, t) = A exp(z1) + aB exp(z2)
A exp(z1) + B exp(z2) + C,
z1= √22x+ 12 – a
t, z2= √22ax + a 12a–1 t,
where A, B, and C are arbitrary constants.
∂t = ∂
2w
∂x2 + aw + bw m.
1◦ Traveling-wave solutions (the signs are chosen arbitrarily):
w (x, t) =
β + C exp(λt μx) 2
1 –m,
where C is an arbitrary constant and β = –b
a , λ = a(1– m)(m +3)
2(m +1) , μ =
a(1– m)2
2(m +1) .
2◦ For a = 0, there is a self-similar solution of the form w(x, t) = t1/(1–m) U (z), where
z = xt–1 2
∂t = ∂
2w
∂x2 + a + be λw.
Traveling-wave solutions (the signs are chosen arbitrarily):
w (x, t) = –2
λln
β + C exp μx– 12aλt
a , μ = aλ
2 ,
where C is an arbitrary constant.
Trang 6T9.1 P ARABOLIC E QUATIONS 1303
∂t = ∂
2w
∂x2 + aw ln w.
Functional separable solutions:
w (x, t) = exp
Ae at x+ A2
a e
2at + Be at
,
w (x, t) = exp1
2 – 14a (x + A)2+ Be at
,
w (x, t) = exp
– a (x + A)
2
4(1+ Be–at) +
1 2Be atln(1+ Be–at ) + Ce at
,
where A, B, and C are arbitrary constants.
T9.1.2 Equations of the Form ∂w
∂t =
∂
∂x
*
f (w) ∂w
∂x
+
+g(w)
Equations of this form admit traveling-wave solutions w = w(z), z = kx + λt, where
k and λ are arbitrary constants and the function w(z) is determined by the second-order autonomous ordinary differential equation k2[f (w)w z ] z – λw z + f (w) =0.
∂t = a ∂
∂x
w m ∂w
∂x
.
This equation occurs in nonlinear problems of heat and mass transfer and flows in porous media
1◦ Solutions:
w (x, t) = ( kx + kλt + A)1/m, k = λm/a,
w (x, t) =
m (x – A)2
2a(m +2)(B – t)
1
m
,
w (x, t) =
A|t + B|–m+ m2 – m
2a(m +2)
(x + C)2
t + B
1
m
,
w (x, t) =
m (x + A)2
ϕ (t) + B|x+ A|m+ m1|ϕ(t)|–m(2 (m+2m+1 )3)
1
m , ϕ(t) = C –2a(m +2)t,
where A, B, C, and λ are arbitrary constants The second solution for B >0corresponds
to blow-up regime (the solution increases without bound on a finite time interval).
2◦ There are solutions of the following forms:
w (x, t) = (t + C)–1/m F (x) (multiplicative separable solution);
w (x, t) = t λ G (ξ), ξ = xt– mλ+2 1 (self-similar solution);
w (x, t) = e–2λt H (η), η = xe λmt (generalized self-similar solution);
w (x, t) = (t + C)–1/m U (ζ), ζ = x + λ ln(t + C),
where C and λ are arbitrary constants.
Trang 71304 NONLINEARMATHEMATICALPHYSICSEQUATIONS
∂t = a ∂
∂x
w m ∂w
∂x
+ bw.
By the transformation w(x, t) = e bt v (x, τ ), τ = 1
bm e
bmt + C the original equation can be
reduced to an equation of the form T9.1.2.1:
∂v
∂τ = a ∂
∂x
v m ∂v
∂x
∂t = a ∂
∂x
w m ∂w
∂x
+ bw m+1.
1◦ Multiplicative separable solution (a = b =1, m >0):
w (x, t) =
⎧
⎪
⎪
⎩
2
(m +1)
m (m +2)
cos2(πx/L) (t0– t)
1/m
for |x| ≤ L2,
2,
where L = 2π(m +1)1 2/m This solution describes a blow-up regime that exists on a
limited time interval t[0, t0) The solution is localized in the interval|x|< L/2
2◦ Multiplicative separable solution:
w (x, t) =
Ae μx + Be–μx + D
mλt + C
1/m
,
B = λ
2(m +1)2
4b2A (m +2)2, D= –
λ (m +1)
b (m +2), μ = m
!
a (m +1),
where A, C, and λ are arbitrary constants, ab(m +1) <0
3◦ Functional separable solutions [it is assumed that ab(m +1) <0]:
w (x, t) =
*
F (t) + C2|F(t)|m+ m+21e λx+1/m
C1– bmt, λ = m
!
–b
a (m +1),
where C1and C2are arbitrary constants
4◦ There are functional separable solutions of the following forms:
w (x, t) =
f (t) + g(t)(Ae λx + Be–λx)1/m
!
–b
a (m +1);
w (x, t) =
f (t) + g(t) cos(λx + C)1/m
!
b
a (m +1),
where A, B, and C are arbitrary constants.
... London, 1977.Polyanin, A D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman &
Hall/CRC...
Carslaw, H S and Jaeger, J C., Conduction of Heat in Solids, Clarendon Press, Oxford, 1984.
Miller, W., Jr., Symmetry and Separation of Variables, Addison-Wesley,... data-page="2">
R EFERENCES FOR C HAPTER T8 1299
References for Chapter T8
Butkovskiy, A G., Green’s Functions and Transfer Functions Handbook, Halstead