Equations of the Form ∂ 2 = a∂ w 2 + f (x, t, w)

3.2.2. Equations of the Form ∂ 2 = a∂ w 2 + f (x, t, w)

∂t

∂x

This is a special case of equation 3.4.1.7 with f (w) = ae λw .

1 ◦ . Solutions:

ln

2 + w(x, t) = − 1 t− C 1 C 2 x C

λ 2 λ 2 λa t ,

λa

w(x, t) = − t−

ln C 1 e σt + C 2 e σx −

σt

where C 1 , C 2 , and σ are arbitrary constants.

2 ◦ . The substitution λU = λw + βt leads to an equation of the form 3.2.1.1:

∂t ∂x This is a special case of equation 3.4.1.6 with f (w) = ae λw .

1 ◦ . Solutions:

w(x, t) = − x−

w(x, t) = − x−

e σx ,

where C 1 , C 2 , and σ are arbitrary constants.

2 ◦ . The substitution λU = λw + βx leads to an equation of the form 3.2.1.1:

2 + aλe λU ∂t . ∂x

+ ce ax+bt e λw .

∂t 2 ∂x 2

This is a special case of equation 3.4.1.8 with f (w) = ce λw .

Functional separable solutions:

w(x, t) = − ln

C is an arbitrary constant.

Reference : A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

Functional separable solutions:

w(x, t) = − ln

C is an arbitrary constant. ∂ 2 w

1 ◦ . Functional separable solutions for k 2 γ−β 2 λ ≠ 0:

αβλ γ w(x, t) = − ln

k 2 γ−β 2 λ β where

2 kγ

2 kγ

C is an arbitrary constant.

2 ◦ . Generalized traveling-wave solutions for k 2 γ−β 2 λ = 0:

w(x, t) = − ln Ce +

( t x)e −

Reference : A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

2 + kβe λw +( αe kt + λβ ) e λw ∂t . ∂x Generalized traveling-wave solutions:

w(x, t) = − ln Ce kt +

( t ✵ x)e kt − ,

k where

C is an arbitrary constant. ∂ 2 w

1 ◦ . Functional separable solutions for k 2 γ+β 2 λ ≠ 0:

γ w(x, t) = − ln

C is an arbitrary constant.

2 ◦ . Generalized traveling-wave solutions for k 2 γ+β 2 λ = 0:

w(x, t) = − ln Ce kx +

Reference : A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

Functional separable solutions:

w(x, t) = − ln Ce kx

( x ✸ t)e kx λβ − ,

where

C is an arbitrary constant. ∂ 2 w

∂t ∂x Functional separable solutions:

α w(x, t) = − ln

4 k 2 α−β 2 λ β where

4 kα

4 kα

C is an arbitrary constant. ∂ 2 w

∂t ∂x Functional separable solutions:

α w(x, t) = − ln

4 k 2 α+β 2 λ β where

4 kα

4 kα

C is an arbitrary constant.

Reference : A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

3.2.3. Equations of the Form ∂ 2 w

2 = f (x) ∂ w 2 + g ∂t x, t, w, ∂w ∂x ∂x

This is a special case of equation 3.4.2.3 with f (w) = be λw .

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

w 1 = wC 1 x+C 2 , ✸ C 1 t+C 3 +

ln C 1 ,

where C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation.

2 ◦ . Traveling-wave solution:

1 exp( Ax + Aµt + B) − b

w=− ln

A(a − µ 2 )

where µ, A, and B are arbitrary constants.

3 ◦ . There is an exact solution of the form

1 x w(x, t) = F (z) − ln | t|, z= .

This is a special case of equation 3.2.3.5 with b = an.

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

w 1 = wC 2− n 1 x, ✻ C 1 t+C 2 +

ln C 1 ,

λ where C 1 and C 2 are arbitrary constants, are also solutions of the equation.

2 ◦ . Functional separable solution for n ≠ 2 and λ ≠ 0:

1 2 cλ(2 − n)

3 ◦ . Functional separable solution for n ≠ 2 (generalizes the solution of Item 2 ◦ ):

x 2− n

w = w(r),

C and k are arbitrary constants (k ≠ 0) and the function w(r) is determined by the ordinary differential equation

4 ◦ . There is an exact solution of the form

2 2 w(x, t) = F (z) − ln | t|, z = x|t| n−2 . λ

For n = 1 and n = 2, the equation describes the propagation of nonlinear waves with axial and central symmetry, respectively.

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

w 1 = wC 1 x, ✻

C 1 t+C 2 + ln C 1 ,

λ where C 1 and C 2 are arbitrary constants, are also solutions of the equation.

2 ◦ . Functional separable solution for n ≠ 0 and λ ≠ 0:

3 ◦ . Functional separable solution (generalizes the solution of Item 2 ◦ ):

w = w(r), 2 r k x a(t + C) ,

where

C and k are arbitrary constants (k ≠ 0) and the function w(r) is determined by the ordinary differential equation

4 ◦ . There is an exact solution of the form

2 x w(x, t) = F (z) − ln | t|, z= .

This is an equation of the propagation of nonlinear waves in an inhomogeneous medium. The substitution z = x + β leads to a special case of equation 3.2.3.5 with b = 0:

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

λ where C 1 and C 2 are arbitrary constants, are also solutions of the equation.

2 ◦ . Functional separable solution for n ≠ 2:

w = w(ξ),

ξ= 1 4 a(2 − n) 2 ( t + C) 2 − x 2− n .

Here,

C is an arbitrary constant, and the function w = w(ξ) is determined by the ordinary differential equation

A ≠ 1, an exact solution of equation (1) is given by

A = 1, which corresponds to b = 1

2 an, exact solutions of equation (1) are expressed as

cλξ(ln |ξ| + q) 2

cλξ cosh 2

( p ln |ξ| + q)

where p and q are arbitrary constants.

1) brings (1) to the generalized Emden–Fowler equation

For

A ≠ 1, the substitution ξ = kz 1− A ( k= ✼

In the special case A= 1 2 , which corresponds to b = a(n − 1), solutions of equation (2) are given by

1 − a(2 − n) 2

w(z) =

ln

2 kcλ(z + q) 2

1 ap 2 (2 − n) 2

w(z) =

ln

2 kcλ cosh 2 ( pz + q)

1 − ap 2 (2 − n) 2

w(z) =

ln

2 kcλ cos 2 ( pz + q)

where p and q are arbitrary constants.

w = w(y), y = At + B ln |x| + C,

where

A, B, and C are arbitrary constants, and the function w = w(y) is determined by the autonomous ordinary differential equation

aB 2 − 2 A ) w ′′

yy +( b − a)Bw ′ y + ce λw = 0.

√ Solution of equation (3) with A= ✽ B a:

1 cλ

w(y) = − ln

y+C 1 .

B(b − a)

Solutions of equation (3) with b = a:

1 2( A 2 − aB 2 )

w(y) = ln

cλ(y + q) 2

1 2 p 2 ( aB 2 − A 2 )

w(y) = ln

cλ cosh ( py + q)

1 2 p 2 ( A 2 − aB 2 )

w(y) = ln

cλ cos 2

( py + q)

where p and q are arbitrary constants.

This is a special case of equation 3.4.3.5 with f (w) = be λw . ∂ 2 w

This is a special case of equation 3.2.3.9 with b = aλ.

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

w 1 = w x− ln | C 1 |, ✽ C 1 t+C 2 +

ln | C 1 |,

where C 1 and C 2 are arbitrary constants, are also solutions of the equation.

2 ◦ . Functional separable solution for λ ≠ 0:

2 e λx

w = w(r),

where C 1 and k are arbitrary constants (k ≠ 0) and the function w(r) is determined by the autonomous ordinary differential equation

w rr ′′ + −1 ck e µw = 0.

Its general solution is expressed as

 1  cµ ln − sinh 2   −

  1 cµ  2 − ln cosh ( C

where C 2 and C 3 are arbitrary constants.

3 ◦ . There is an exact solution of the form

w(x, t) = F (z) − ln | t|, z=x+

ln | t|.

This is an equation of the propagation of nonlinear waves in an inhomogeneous medium. This is a special case of equation 3.2.3.9 with b = 0.

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

w 1 = w x− ln | C 1 |, ✾ C 1 t+C 2 +

ln | C 1 |,

where C 1 and C 2 are arbitrary constants, are also solutions of the equation.

2 ◦ . Functional separable solution:

k= ✾ 1, where

w = w(z), z= 4 ke − λx − akλ 2 ( t + C) 2 1 /2 ,

C is an arbitrary constant and the function w = w(z) is determined by the ordinary differential equation

A solution of equation (1) has the form

1 2 kλ(aλ − 2b)

w(z) =

ln

cµz 2

Note some other exact solutions of equation (1):

1 −2 akλ 2

w(z) =

cµ(z + B) 2

1 2 aA 2 kλ 2

w(z) =

cµ cosh ( Az + B)

1 −2 aA 2 kλ 2

w(z) =

cµ sinh ( Az + B)

1 −2 aA 2 kλ 2

w(z) =

w(z) =

ln

if b= aλ,

cµ(Az 2