Equation of Unsteady Transonic Gas Flows

7.1.3. Equation of Unsteady Transonic Gas Flows

This is an equation of an unsteady transonic gas flow; see Lin, Reissner, and Tsien (1948). This is a special case of equation 7.1.3.2 with f (t) = a and g(t) = −b.

2 = w(ξ, η, t) + ϕ ′′ tt ( t)y +2 bϕ ′ t ( t)x + ψ(t)y + χ(t), ξ = x + λy + bλ 2 t − 2abϕ(t), η = y + 2bλt,

where the C n and λ are arbitrary constants and ϕ = ϕ(t), ψ = ψ(t), and χ = χ(t) are arbitrary functions, are also solutions of the equation.

2 ◦ . Solution:

w(x, y, t) =

b 6 b t +2 12 aαγ)y

2 ( γ tt ′′ +6 aγγ ′ t +4 a γ ) y +

2 b t +2 aγ ) x+β t +2 aβγ]y +( αx + δ)y + γx + βx + µ, where α = α(t), β = β(t), γ = γ(t), µ = µ(t), and δ = δ(t) are arbitrary functions.

3 ◦ . “Two-dimensional” solution:

w(x, y, t) = U (z, t) + ϕ(t)y + ψ(t), z = x + λy,

where ϕ = ϕ(t) and ψ = ψ(t) are arbitrary functions, λ is an arbitrary constant, and the function U = U (z, t) is determined by the first-order partial differential equation

A complete integral of this equation is given by

1 z + bλ C 1 − U=C 1 2 aC 1 t+C 2 ,

where C 1 and C 2 are arbitrary constants. The general solution of equation (1) can be written out in parametric form (Polyanin, Zaitsev, and Moussiaux, 2002):

2 1 U = sz + bλ 2 s−

2 as t + f (s),

z + bλ 2 − as t+f

s ′ (s) = 0,

where f = f (s) is an arbitrary function and s is the parameter.

4 ◦ . “Two-dimensional” solution of a more general form: w(x, y, t) = U (z, t) + ϕ(t)y 2 + ψ(t)y + χ(t)x + θ(t), z = x + λy,

where ϕ = ϕ(t), ψ = ψ(t), χ = χ(t), and θ = θ(t) are arbitrary functions, λ is an arbitrary constant, and the function U = U (z, t) is determined by the first-order partial differential equation [σ(t) is an arbitrary function]:

∂U 2 a ∂U + 2 + aχ(t) − bλ

= 2 bϕ(t) − χ ′ ( t) z + σ(t). ∂t

2 ∂z t ∂z

This equation can be fully integrated—a complete integral is sought in the form U = f (t)z + g(t).

5 ◦ . “Two-dimensional” generalized separable solution cubic in x:

3 w(x, y, t) = f (y, t)x 2 + g(y, t)x + h(y, t)x + r(y, t), where the functions f = f (y, t), g = g(y, t), h = h(y, t), and r = r(y, t) are determined by the

differential equations

The subscripts y and t denote the corresponding partial derivatives. Setting f = 0 and g = ϕ(t)y +ψ(t), where ϕ = ϕ(t) and ψ = ψ(t) are arbitrary functions, one can integrate the system with respect to y to obtain a solution dependent on six arbitrary functions.

w(x, y, t) = v(x, r)t −1 , r = yt /2 ,

where the function v = v(x, r) is determined by the differential equation

7 ◦ . “Two-dimensional” solution:

2 w(x, y, t) = v(p, t) + 2

where γ = γ(t), µ = µ(t), λ = λ(t), and δ = δ(t) are arbitrary functions, and the function v = v(p, t) is determined by the differential equation

2 v 3 b[p(γ ′ +4 b) − aγ δ] γ t p + aγ

References for equation 7.1.3.1: E. V. Mamontov (1969), E. M. Vorob’ev, N. V. Ignatovich, and E. O. Semenova (1989), A. D. Polyanin and V. F. Zaitsev (2002).

∂w ∂ 2 w

2. + f (t)

2 ∂x∂t + g(t) ∂x ∂x ∂y 2 = 0.

1 ◦ . Suppose w(x, y, t) is a solution of this equation. Then the functions

w(ξ, η, t) − 2 y + ψ(t)y + ϕ(t)x + χ(t),

2 g(t)

g(t) dt, where C 1 , ...,C 7 and λ are arbitrary constants and ϕ = ϕ(t), ψ = ψ(t), and χ = χ(t) are arbitrary

ξ = x + λy − 2 λ g(t) + f (t)ϕ(t) dt, η = y − 2λ

functions, are also solutions of the equation.

2 ◦ . Generalized separable solution in the form of a polynomial of degree 4 in y:

4 3 2 w(x, y, t) = a(t)y 2 + b(t)y +[ c(t)x + d(t)]y +[ α(t)x + β(t)]y + γ(t)x + µ(t)x + δ(t), where α = α(t), β = β(t), γ = γ(t), µ = µ(t), and δ = δ(t) are arbitrary functions, and the functions

a = a(t), b = b(t), c = c(t), and d = d(t) are given by

3 ◦ . “Two-dimensional” solution: w(x, y, t) = U (z, t) + ϕ(t)y 2 + ψ(t)y + χ(t)x + θ(t), z = x + λy,

where ϕ = ϕ(t), ψ = ψ(t), χ = χ(t), and θ = θ(t) are arbitrary functions, λ is an arbitrary constant, and the function U = U (z, t) is determined by the first-order partial differential equation [σ(t) is an arbitrary function]:

∂U 2 1 ∂U

2 ∂U

= −[2 g(t)ϕ(t) + χ ′ ∂t

f (t)

+[ f (t)χ(t) + λ g(t)]

2 t ( t)]z + σ(t).

∂z

∂z

This equation can be fully integrated; a complete integral is sought in the form U = f (t)z + g(t).

3 w(x, y, t) = ϕ(y, t)x 2 + ψ(y, t)x + χ(y, t)x + θ(y, t), where the functions ϕ = ϕ(y, t), ψ = ψ(y, t), χ = χ(y, t), and θ = θ(y, t) are determined by the

differential equations

The subscripts y and t denote the corresponding partial derivatives, f = f (t) and g = g(t). These equations can be treated as ordinary differential equations for y with parameter t; the constants of integration will be functions of t. The first equation has the following particular solutions: ϕ = 0

g and ϕ=−

, where h = h(t) is an arbitrary function.

3 f (y + h) 2

5 ◦ . “Two-dimensional” solution:

4 2 w(x, y, t) = u(p, t) + a(t)y 2 +[ b(t)p + c(t)]y + µ(t)y + λ(t), p=y + γ(t)x. Here, c = c(t), γ = γ(t), µ = µ(t), and λ = λ(t) are arbitrary functions; and the function u = u(p, t) is

determined by the differential equation

3 ∂u ∂ u

∂u

+ γ t ′ p+fγ

t ′ +2 g)

+2 g(bp + c) = 0,

where the functions a = a(t) and b = b(t) are given by

Reference : A. D. Polyanin and V. F. Zaitsev (2002).

7.1.4. Equations of the Form ∂w 2 ∂ 2 w

– ∂w ∂ w

= F x, y, ∂w

, ∂w

∂y ∂x∂y

∂x

∂y 2

∂x ∂y

∂y ∂x∂y

∂x ∂y 2

General solution:

w(x, y) = F y + G(x) ,

where

F (z) and G(x) are arbitrary functions.

Reference : D. Zwillinger (1989, p. 397).

= f (x).

∂y ∂x∂y

∂x ∂y 2

1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions

w −1 1 = ✔ C

1 w(x, C 1 y + ϕ(x)) + C 2 ,

where C 1 and C 2 are arbitrary constants and ϕ(x) is an arbitrary function, are also solutions of the equation.

w(x, y) = ✕ y 2 f (x) dx + C 1 + ϕ(x),

w(x, y) = C 1 2 y 2 + ϕ(x)y + ϕ ( x) − 2

f (x) dx + C 2 ,

where ϕ(x) is an arbitrary function and C 1 and C 2 are arbitrary constants.

3 ◦ . The von Mises transformation

∂w

ξ = x, η = w, U (ξ, η) =

where w = w(x, y), (1)

∂y

brings the original equation to the first-order nonlinear equation

which is independent of η. On integrating (2) and taking into account the relations of (1), we obtain the first-order equation

∂w 2

f (x) dx + ψ(w),

∂y

where ψ(w) is an arbitrary function. Integrating (3) yields the general solution in implicit form:

Z dw

= ✕ y + ϕ(x),

2 F (x) + ψ(w)

where ϕ(x) and ψ(w) are arbitrary functions, F (x) =

1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions

w −1

1 = C 1 w(x, C 1 y+C 2 )+ C 3 ,

w 2 = w x, y + ϕ(x) ,

where C 1 , C 2 , and C 3 are arbitrary constants and ϕ(x) is an arbitrary function, are also solutions of the equation.

2 ◦ . Generalized separable solutions:

w(x, y) = y

f (x) dx + C + ϕ(x),

w(x, y) = ϕ(x)e λy 1

f (x) dx + C,

where ϕ(x) is an arbitrary function and C and λ are arbitrary constants.

3 ◦ . The equation can be rewritten as the relation that the Jacobian of the functions Z w and v = w y −

f (x) dx is equal to zero. It follows that w and v are functionally dependent, which means that v is expressible in terms of w:

Z ∂w

f (x) dx = ϕ(w),

∂y ∂y

dw = y + ψ(x),

where ψ(x) and ϕ(w) are arbitrary functions. ∂w ∂ 2 w

+ g(y)

∂y ∂x∂y

First integral:

∂w

= ϕ(w) −

g(y) dy +

f (x) dx,

∂y

where ϕ(w) is an arbitrary function. This equation can be treated as a first-order ordinary differential equation in the independent variable y with parameter x.

= f (x)g

1 ◦ . Suppose w(x, y) is a solution of this equation. Then the functions

w 1 = −1 C 1 w(x, C 1 y+C 2 )+ C 3 , w 2 = w x, y + ϕ(x) ,

where C 1 , C 2 , and C 3 are arbitrary constants and ϕ(x) is an arbitrary function, are also solutions of the equation.

2 ◦ . First integral:

where ϕ(w) is an arbitrary function. This equation can be treated as a first-order ordinary differential equation in the independent variable y with parameter x.