On solving these equations for the derivatives, we obtain linear separable equations, which are easy to integrate.. Lopez, G., Partial Differential Equations of First Order and Their App
Trang 1T7.3 N ONLINEAR E QUATIONS 1263
10. F1
x, ∂w
∂x
= F2
y, ∂w
∂y
.
A separable equation Complete integral:
w = ϕ(x) + ψ(y) + C1,
where the functions ϕ = ϕ(x) and ψ = ψ(y) are determined from the ordinary differential
equations
F1 x , ϕ x
= C2, F2 y , ψ y
= C2
11. F1
x, ∂w
∂x
+ F2
y, ∂w
∂y
+ aw = 0.
A separable equation Complete integral:
w = ϕ(x) + ψ(y),
where the functions ϕ = ϕ(x) and ψ = ψ(y) are determined from the ordinary differential
equations
F1 x , ϕ x
+ aϕ = C1, F2 y , ψ y
+ aψ = –C1,
where C1is an arbitrary constant If a≠ 0, one can set C1 =0in these equations
12. F1
x, 1
w
∂w
∂x
+ w k F2
y, 1
w
∂w
∂y
= 0.
A separable equation Complete integral:
w (x, y) = ϕ(x)ψ(y).
The functions ϕ = ϕ(x) and ψ = ψ(y) are determined by solving the ordinary differential
equations
ϕ–k F
1 x , ϕ x /ϕ
= C, ψ k F
2 y , ψ y /ψ
= –C, where C is an arbitrary constant.
13. F1
x, ∂w
∂x
+ e λw F2
y, ∂w
∂y
= 0.
A separable equation Complete integral:
w (x, y) = ϕ(x) + ψ(y).
The functions ϕ = ϕ(x) and ψ = ψ(y) are determined by solving the ordinary differential
equations
e–λϕ F1 x , ϕ
x
= C, e λψ F2 y , ψ
y
= –C, where C is an arbitrary constant.
Trang 21264 FIRST-ORDERPARTIALDIFFERENTIALEQUATIONS
14. F1
x, 1
w
∂w
∂x
+ F2
y, 1
w
∂w
∂y
= k ln w.
A separable equation Complete integral:
w (x, y) = ϕ(x)ψ(y).
The functions ϕ = ϕ(x) and ψ = ψ(y) are determined by solving the ordinary differential
equations
F1 x , ϕ x /ϕ
– k ln ϕ = C, F2 y , ψ y /ψ
– k ln ψ = –C, where C is an arbitrary constant.
15. ∂w
∂x + yF1
x, ∂w
∂y
+ F2
x, ∂w
∂y
= 0.
Complete integral:
w = ϕ(x)y –
F2 x , ϕ(x)
dx + C1,
where the function ϕ(x) is determined by solving the ordinary differential equation ϕ x+
F1(x, ϕ) =0
16. F
∂w
∂x + ay, ∂w
∂y + ax
= 0.
Complete integral: w = –axy + C1x + C2y + C3, where F (C1, C2) =0
17.
∂w
∂x
2
+
∂w
∂y
2
= F
x2+ y2, y ∂w
∂x – x ∂w
∂y
.
Complete integral: w = –C1arctan y
x+1 2
ξF (ξ, C1) – C12 dξ
ξ +C2, where ξ = x2+y2
18. F
x, ∂w
∂x, ∂w
∂y
= 0.
Complete integral: w = C1y +ϕ(x, C1)+C2, where the function ϕ = ϕ(x, C1) is determined
from the ordinary differential equation F x , ϕ x , C1
=0
19. F
ax + by, ∂w
∂x, ∂w
∂y
= 0.
For b =0, see equation T7.3.3.18 Complete integral for b≠ 0:
w = C1x + ϕ(z, C1) + C2, z = ax + by, where the function ϕ = ϕ(z) is determined from the nonlinear ordinary differential equation
F z , aϕ z + C1, bϕ z
=0
20. F
∂x, ∂w
∂y
= 0.
Complete integral:
w = w(z), z = C1x + C2y,
where C1and C2 are arbitrary constants and w = w(z) is determined by the autonomous ordinary differential equation F w , C1w
z , C2w
z
=0
Trang 3R EFERENCES FOR C HAPTER T7 1265
21. F
ax + by + cw, ∂w
∂y
= 0.
For c =0, see equation T7.3.3.19 If c≠ 0, then the substitution cu = ax + by + cw leads to
an equation of the form T7.3.3.20: F
cu, ∂u
∂x – a
c, ∂u
∂y – b
c
=0
22. F
x, ∂w
∂x, ∂w
∂y , w – y ∂w
∂y
= 0.
Complete integral: w = C1y + ϕ(x), where the function ϕ(x) is determined from the ordinary differential equation F x , ϕ x , C1, ϕ
=0
23. F
∂y , x ∂w
∂x + y ∂w
∂y
= 0.
Complete integral:
w = ϕ(ξ), ξ = C1x + C2y,
where the function ϕ(ξ) is determined by solving the nonlinear ordinary differential equation
F ϕ , C1ϕ
ξ , C2ϕ ξ , ξϕ ξ
=0
24. F
ax + by, ∂w
∂y , w – x ∂w
∂x – y ∂w
∂y
= 0.
Complete integral:
w = C1x + C2y + ϕ(ξ), ξ = ax + by, where the function ϕ(ξ) is determined by solving the nonlinear ordinary differential equation
F ξ , aϕ ξ + C1, bϕ ξ + C2, ϕ – ξϕ ξ
=0
25. F
x, ∂w
∂x , G
y, ∂w
∂y
= 0.
Complete integral:
w = ϕ(x, C1) + ψ(y, C1) + C2,
where the functions ϕ and ψ are determined by the ordinary differential equations
F (x, ϕ x , C1) =0, G (y, ψ y ) = C1
On solving these equations for the derivatives, we obtain linear separable equations, which are easy to integrate
References for Chapter T7
Kamke, E., Differentialgleichungen: L¨osungsmethoden und L ¨osungen, II, Partielle Differentialgleichungen
Erster Ordnung f¨ur eine gesuchte Funktion, Akad Verlagsgesellschaft Geest & Portig, Leipzig, 1965.
Lopez, G., Partial Differential Equations of First Order and Their Applications to Physics, World Scientific
Publishing Co., Singapore, 2000.
Polyanin, A D., Zaitsev, V F., and Moussiaux, A., Handbook of First Order Partial Differential Equations,
Taylor & Francis, London, 2002.
Rhee, H., Aris, R., and Amundson, N R., First Order Partial Differential Equations, Vols 1 and 2, Prentice
Hall, Englewood Cliffs, New Jersey, 1986 and 1989.
Tran, D V., Tsuji, M., and Nguyen, D T S., The Characteristic Method and Its Generalizations for First-Order
Nonlinear Partial Differential Equations, Chapman & Hall, London, 1999.
Trang 5Chapter T8
Linear Equations and Problems
of Mathematical Physics
T8.1 Parabolic Equations
∂2w
∂x2
T8.1.1-1 Particular solutions
w (x) = Ax + B,
w (x, t) = A(x2+ 2at ) + B,
w (x, t) = A(x3+ 6atx ) + B,
w (x, t) = A(x4+ 12atx2 + 12a2t2) + B,
w (x, t) = x2n+
n
k=1
( 2n)( 2n– 1) (2n– 2k+ 1 )
k x2n–2k,
w (x, t) = x2n+1+
n
k=1
( 2n+ 1 )( 2n ) (2n– 2k+ 2 )
k x2n–2k+1 ,
w (x, t) = A exp(aμ2t μx ) + B,
w (x, t) = A √1
texp
– x
2
4at
+ B,
w (x, t) = A x
t3/2 exp
– x
2
4at
+ B,
w (x, t) = A exp(–aμ2t ) cos(μx + B) + C,
w (x, t) = A exp(–μx) cos(μx –2aμ2t + B) + C,
w (x, t) = A erf
x
2√ at
+ B,
where A, B, C, and μ are arbitrary constants, n is a positive integer, and erf z ≡ 2
√
π
z
0 exp(–ξ
2) dξ is the error function (probability integral).
T8.1.1-2 Formulas allowing the construction of particular solutions
Suppose w = w(x, t) is a solution of the heat equation Then the functions
w1 = Aw( λx + C1, λ2t + C2) + B,
w2 = A exp(λx + aλ2t )w(x +2aλt + C1, t + C2),
w3 = √| A
δ + βt| exp
– βx
2
4a (δ + βt)
w
x
δ + βt,
γ + λt
δ + βt
, λδ – βγ =1,
1267
Trang 61268 LINEAREQUATIONS ANDPROBLEMS OFMATHEMATICALPHYSICS
where A, B, C1, C2, β, δ, and λ are arbitrary constants, are also solutions of this equation The last formula with β =1, γ = –1, δ = λ =0was obtained with the Appell transformation
T8.1.1-3 Cauchy problem and boundary value problems
For solutions of the Cauchy problem and various boundary value problems, see Subsec-tion T8.1.2 withΦ(x, t)≡ 0
∂2w
∂x2 +Φ(x, t)
T8.1.2-1 Domain: –∞ < x < ∞ Cauchy problem.
An initial condition is prescribed:
w = f (x) at t =0 Solution:
w (x, t) =
∞ –∞ f (ξ)G(x, ξ, t) dξ +
t 0
∞ –∞ Φ(ξ, τ)G(x, ξ, t – τ) dξ dτ,
where
G (x, ξ, t) = 1
2√ πat exp
–(x – ξ)
2
4at
T8.1.2-2 Solutions of boundary value problems in terms of the Green’s function
We consider boundary value problems on an interval 0 ≤ x ≤ l with the general initial condition
w = f (x) at t =0
and various homogeneous boundary conditions The solution can be represented in terms
of the Green’s function as
w (x, t) =
l
0 f (ξ)G(x, ξ, t) dξ +
t 0
l
0 Φ(ξ, τ)G(x, ξ, t – τ) dξ dτ.
Here, the upper limit l can be finite or infinite; if l = ∞, there is no boundary condition
corresponding to it
Paragraphs T8.1.2-3 through T8.1.2-8 present the Green’s functions for various types
of homogeneous boundary conditions
Remark Formulas from Section 14.7 should be used to obtain solutions to corresponding nonhomoge-neous boundary value problems.
T8.1.2-3 Domain: 0 ≤x<∞ First boundary value problem.
A boundary condition is prescribed:
w=0 at x=0 Green’s function:
G (x, ξ, t) = 1
2√ πat
exp
–(x – ξ)
2
4at
– exp
–(x + ξ)
2
4at
Trang 7
T8.1 P ARABOLIC E QUATIONS 1269
T8.1.2-4 Domain: 0 ≤x<∞ Second boundary value problem.
A boundary condition is prescribed:
∂ x w=0 at x=0 Green’s function:
G (x, ξ, t) = 1
2√ πat
exp
–(x – ξ)
2
4at
+ exp
–(x + ξ)
2
4at
T8.1.2-5 Domain: 0 ≤x<∞ Third boundary value problem.
A boundary condition is prescribed:
∂ x w – kw =0 at x=0 Green’s function:
G (x, ξ, t) = 1
2√ πat
exp
–(x – ξ)
2
4at
+ exp
–(x + ξ)
2
4at
– 2k
∞
0 exp
–(x + ξ + η)
2
4at – kη
dη
.
T8.1.2-6 Domain: 0 ≤x≤l First boundary value problem
Boundary conditions are prescribed:
Two forms of representation of the Green’s function:
G (x, ξ, t) = 2
l
∞
n=1 sin
nπx l
sin
nπξ l
exp
–an2π2t
l2
= 1
2√ πat
∞
n=–∞
exp
–(x – ξ +2nl)2
4at
– exp
–(x + ξ +2nl)2
4at
The first series converges rapidly at large t and the second series at small t.
T8.1.2-7 Domain: 0 ≤x≤l Second boundary value problem
Boundary conditions are prescribed:
∂ x w=0 at x=0, ∂ x w=0 at x = l.
Two forms of representation of the Green’s function:
G (x, ξ, t) = 1
l + 2
l
∞
n=1 cos
nπx l
cos
nπξ l
exp
–an2π2t
l2
= 1
2√ πat
∞
n=–∞
exp
–(x – ξ +2nl)2
4at
+ exp
–(x + ξ +2nl)2
4at
The first series converges rapidly at large t and the second series at small t.