A model equation of gas dynamics.. This equation is also encountered in hydrodynamics, multiphase flows, wave theory, acoustics, chemical engineering, and other applications... In Subsec
Trang 14. a ∂w
∂x + b ∂w
∂y = f (x) + g(y).
General solution: w = 1
a
f (x) dx + 1
b
g (y) dy + Φ(bx – ay).
5. ∂w
∂x + a ∂w
∂y = f (x)g(y).
General solution: w =
x
x0 f (t)g(y – ax + at) dt + Φ(y – ax), where x0 can be taken
arbitrarily
6. ∂w
∂x + a ∂w
∂y = f (x, y).
General solution: w =
x
x0 f (t, y – ax + at) dt + Φ(y –ax), where x0can be taken arbitrarily
7. ∂w
∂x + [ay + f (x)] ∂w
∂y = g(x).
General solution: w =
g (x) dx + Φ(u), where u = e–ax y–
f (x)e–ax dx
8. ∂w
∂x +
ay + f (x) ∂w
∂y = g(x)h(y).
General solution:
w=
g (x) h
e ax u + e ax
f (x)e–ax dx
dx+Φ(u), where u = e–ax y–
f (x)e–ax dx
In the integration, u is treated as a parameter.
9. ∂w
∂x +
f (x)y + g(x)y k ∂w
∂y = h(x).
General solution: w =
h (x) dx + Φ(u), where
u = e–F y1 –k– (1– k)
e–F g (x) dx, F = (1– k)
f (x) dx.
10. ∂w
∂x +
f (x) + g(x)e λy ∂w
∂y = h(x).
General solution: w =
h (x) dx + Φ(u), where
u = e–λy F (x) + λ
g (x)F (x) dx, F (x) = exp
λ
f (x) dx
Trang 2
1250 FIRST-ORDERPARTIALDIFFERENTIALEQUATIONS
11. ax ∂w
∂x + by ∂w
∂y = f (x, y).
General solution:
w= 1
a
1
x f x , u1/a x b/a
dx+Φ(u), where u = y a x–b.
In the integration, u is treated as a parameter.
12. f (x) ∂w
∂x + g(y) ∂w
∂y = h1(x) + h2(y).
General solution: w =
h1(x)
f (x) dx+
h2(y)
g (y) dy+Φ
dx
f (x) –
dy
g (y)
13. f (x) ∂w
∂x + g(y) ∂w
∂y = h(x, y).
The transformation ξ =
dx
f (x) , η =
dy
g (y) leads to an equation of the form T7.1.2.6 for
w = w(ξ, η).
14. f (y) ∂w
∂x + g(x) ∂w
∂y = h(x, y).
The transformation ξ =
g (x) dx, η =
f (y) dy leads to an equation of the form T7.1.2.6 for w = w(ξ, η).
T7.1.3 Equations of the Form
f (x, y) ∂w
∂x +g(x, y) ∂w ∂y = h(x, y)w + r(x, y)
In the solutions of equations T7.1.3.1–T7.1.3.10, Φ(z) is an arbitrary composite function
whose argument z can depend on both x and y.
1. a ∂w
∂x + b ∂w
∂y = f (x)w.
General solution: w = exp
1
a
f (x) dx
Φ(bx – ay).
2. a ∂w
∂x + b ∂w
∂y = f (x)w + g(x).
General solution: w = exp
1
a
f (x) dx
Φ(bx–ay)+1
a
g (x) exp
–1
a
f (x) dx
dx
3. a ∂w
∂x + b ∂w
∂y =
f (x) + g(y)
w.
General solution: w = exp
1
a
f (x) dx + 1
b
g (y) dy
Φ(bx – ay).
Trang 34. ∂w
∂x + a ∂w
∂y = f (x, y)w.
General solution: w = exp
x
x0 f (t, y – ax + at) dt
Φ(y – ax), where x0 can be taken
arbitrarily
5. ∂w
∂x + a ∂w
∂y = f (x, y)w + g(x, y).
General solution:
w = F (x, u)
Φ(u) +
g (x, u + ax)
F (x, u) dx
, F (x, u) = exp
f (x, u + ax) dx
,
where u = y – ax In the integration, u is treated as a parameter.
6. ax ∂w
∂x + by ∂w
∂y = f (x)w + g(x).
General solution: w= exp
1
a
f (x) dx
x
Φ x–b/a y
a
g (x)
x exp
– 1
a
f (x) dx
x
dx
7. ax ∂w
∂x + by ∂w
∂y = f (x, y)w.
General solution:
w= exp
1
a
1
x f x , u1/a x b/a
dx
Φ(u), where u = y a x–b.
In the integration, u is treated as a parameter.
8. x ∂w
∂x + ay ∂w
∂y = f (x, y)w + g(x, y).
General solution:
w = F (x, u)
Φ(u) +
g (x, ux a)
xF (x, u) dx
, F (x, u) = exp
1
x f (x, ux a ) dx
,
where u = yx–a In the integration, u is treated as a parameter.
9. f (x) ∂w
∂x + g(y) ∂w
∂y =
h1(x) + h2(y)
w.
General solution: w = exp
h1(x)
f (x) dx+
h2(y)
g (y) dy
Φ
dx
f (x) dx–
dy
g (y) dy
Trang 4
1252 FIRST-ORDERPARTIALDIFFERENTIALEQUATIONS
10. f1(x) ∂w
∂x + f2(y) ∂w
∂y = aw + g1(x) + g2(y).
General solution:
w = E1(x) Φ(u) + E1(x)
g1(x) dx
f1(x)E1(x) + E2(y)
g2(y) dy
f2(y)E2(y),
where
E1(x) = exp
a
dx
f1(x)
, E2(y) = exp
a
dy
f2(y)
, u=
dx
f1(x) –
dy
f2(y).
11. f (x) ∂w
∂x + g(y) ∂w
∂y = h(x, y)w + r(x, y).
The transformation ξ =
dx
f (x) , η =
dy
g (y) leads to an equation of the form T7.1.3.5 for
w = w(ξ, η).
12. f (y) ∂w
∂x + g(x) ∂w
∂y = h(x, y)w + r(x, y).
The transformation ξ =
g (x) dx, η =
f (y) dy leads to an equation of the form T7.1.3.5 for w = w(ξ, η).
T7.2 Quasilinear Equations
T7.2.1 Equations of the Form f (x, y) ∂w
∂x +g(x, y) ∂w ∂y = h(x, y, w)
In the solutions of equations T7.2.1.1–T7.2.1.12, Φ(z) is an arbitrary composite function
whose argument z can depend on both x and y.
1. ∂w
∂x + a ∂w
∂y = f (x)w + g(x)w k.
General solution:
w1 –k = F (x) Φ(y – ax) + (1– k)F (x)
g (x)
F (x) dx , where F (x) = exp
(1– k)
f (x) dx
2. ∂w
∂x + a ∂w
∂y = f (x) + g(x)e λw.
General solution:
e–λw = F (x) Φ(y – ax) – λF (x)
g (x)
F (x) dx , where F (x) = exp
–λ
f (x) dx
Trang 5
3. a ∂w
∂x + b ∂w
∂y = f (w).
General solution:
dw
f (w) =
x
a +Φ(bx – ay).
4. a ∂w
∂x + b ∂w
∂y = f (x)g(w).
General solution:
dw
g (w) =
1
a
f (x) dx + Φ(bx – ay).
5. ∂w
∂x + a ∂w
∂y = f (x)g(y)h(w).
General solution:
dw
h (w) =
x
x0 f (t)g(y – ax + at) dt + Φ(y – ax), where x0can be taken
arbitrarily
6. ax ∂w
∂x + by ∂w
∂y = f (w).
General solution:
dw
f (w) =
1
aln|x|+Φ |x|b|y|–a
7. ay ∂w
∂x + bx ∂w
∂y = f (w).
General solution:
dw
f (w) =
1
√
abln√ ab x + ay+Φ ay2– bx2
, ab>0
8. ax n ∂w
∂x + by k ∂w
∂y = f (w).
General solution:
dw
f (w)=
1
a(1– n) x
1 –n+Φ(u), where u = 1
a(1– n) x
1 –n 1
b(1– k) y
1 –k.
9. ay n ∂w
∂x + bx k ∂w
∂y = f (w).
General solution:
a
dw
f (w) =
b a
n+1
k+1x k+1– u
– n n+1
dx , where u = b
a
n+1
k+1x k+1– y n+1.
In the integration, u is treated as a parameter.
10. ae λx ∂w
∂x + be βy ∂w
∂y = f (w).
General solution:
dw
f (w) = –
1
aλ e
–λx+Φ(u), where u = aλe–βy – bβe–λx.
Trang 61254 FIRST-ORDERPARTIALDIFFERENTIALEQUATIONS
11. ae λy ∂w
∂x + be βx ∂w
∂y = f (w).
General solution:
dw
f (w) =
c (βx – λy)
u +Φ(u), where u = aβe λy – bλe βx.
12. f (x) ∂w
∂x + g(y) ∂w
∂y = h(w).
General solution:
dw
h (w) =
dx
f (x) +Φ(u), where u =
dx
f (x) –
dy
g (y).
13. f (y) ∂w
∂x + g(x) ∂w
∂y = h(w).
The transformation ξ =
g (x) dx, η =
f (y) dy leads to an equation of the form T7.2.1.5:
∂w
∂ξ + ∂w
∂η = F (ξ)G(η)h(w), where F (ξ) = 1
g (x) , G(η) =
1
f (y).
T7.2.2 Equations of the Form ∂w
∂x +f (x, y, w) ∂w ∂y = 0
In the solutions of equations T7.2.2.1–T7.2.2.10, Φ(w) is an arbitrary function.
1. ∂w
∂x +
aw + yf (x) ∂w
∂y = 0.
General solution: yF (x) – aw
F (x) dx = Φ(w), where F (x) = exp
–
f (x) dx
2. ∂w
∂x +
aw + f (y) ∂w
∂y = 0.
General solution: x =
y
y0
dt
f (t) + aw +Φ(w).
3. ∂w
∂x + f (w) ∂w
∂y = 0.
A model equation of gas dynamics This equation is also encountered in hydrodynamics,
multiphase flows, wave theory, acoustics, chemical engineering, and other applications
1◦ General solution:
y = xf (w) + Φ(w),
whereΦ is an arbitrary function
2◦ The solution of the Cauchy problem with the initial condition
w = ϕ(y) at x =0
can be represented in the parametric form
y = ξ + F(ξ)x, w = ϕ(ξ),
whereF(ξ) = f ϕ (ξ)
Trang 7
3◦ Consider the Cauchy problem with the discontinuous initial condition
w(0, y) =w
1 for y <0,
w2 for y >0
It is assumed that x≥ 0, f >0, and f >0for w >0, w1 >0, and w2>0
Generalized solution for w1< w2:
w (x, y) =
w
1 for y/x < V1,
f–1(y/x) for V1 ≤y/x≤V2,
w2 for y/x > V2,
where V1= f (w1), V2= f (w2)
Here f– 1is the inverse of the function f , i.e., f– 1 f (w)
≡w This solution is continuous
in the half-plane x >0and describes a “rarefaction wave.”
Generalized solution for w1> w2:
w (x, y) =
w1 for y/x < V ,
w2 for y/x > V , where V =
1
w2– w1
w2
w1 f (w) dw.
This solution undergoes a discontinuity along the line y = V x and describes a “shock wave.”
4◦ In Subsection 13.1.3, qualitative features of solutions to this equation are considered,
including the wave-breaking effect and shock waves This subsection also presents general formulas that permit one to construct generalized (discontinuous) solutions for arbitrary initial conditions
4. ∂w
∂x +
f (w) + ax ∂w
∂y = 0.
General solution: y = xf (w) + 12ax2+Φ(w).
5. ∂w
∂x +
f (w) + ay ∂w
∂y = 0.
General solution: x = 1
alnay + f (w)+Φ(w).
6. ∂w
∂x +
f (w) + g(x) ∂w
∂y = 0.
General solution: y = xf (w) +
g (x) dx + Φ(w).
7. ∂w
∂x +
f (w) + g(y) ∂w
∂y = 0.
General solution: x =
y
y0
dt
g (t) + f (w) +Φ(w).
8. ∂w
∂x +
yf (w) + g(x) ∂w
∂y = 0.
General solution: y exp
–xf (w)
–
x
x0 g (t) exp
–tf (w)
dt=Φ(w), where x0can be taken arbitrarily