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