Equations of the Form ∂w n = a∂ w n + F x, t, w, ∂w

11.1.5. Equations of the Form ∂w n = a∂ w n + F x, t, w, ∂w

∂t

∂x

∂x

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

1 = C n−2 1 w(C 1 x + 2bC 1 C 2 t+C 3 , C 1 n t+C 4 )+ C 2 x + bC 2 t+C 5 ,

where C 1 , ...,C 5 are arbitrary constants, is also a solution of the equation.

2 ◦ . Solution:

w(x, t) = C 1 t+C 2 +

θ(z) dz,

z = x + λt,

where C 1 , C 2 , and λ are arbitrary constants, and the function θ(z) is determined by the (n− 1)st-order autonomous ordinary differential equation

aθ ( n−1)

+ bθ − λθ − C 1 = 0.

To C 1 = 0 there corresponds a traveling-wave solution.

3 ◦ . Self-similar solution:

2− n

w(x, t) = t n u(ζ),

ζ = xt − n ,

where the function u(ζ) is determined by the ordinary differential equation

4 ◦ . There is a degenerate solution quadratic in x:

w(x, t) = ϕ(t)x 2 + ψ(t)x + χ(t).

connects the original equation with the generalized Burgers–Korteweg–de Vries equation 11.1.3.1:

If u = u(x, t) is a solution of equation (2), then the corresponding solution w = w(x, t) of the original equation can be found from the linear system of first-order equations (1).

+ f (t).

f (t) dt + Θ(z), z = x + λt, where C 1 , C 2 , and λ are arbitrary constants, and the function Θ(z) is determined by the autonomous

w(x, t) = C 1 t+C 2 +

ordinary differential equation

aΘ ( n) + bΘ ′ 2 z − λΘ ′ z z − C 1 = 0. Z

2 ◦ . The substitution w = U (x, t) +

f (t) dt leads to a simpler equation of the form 11.1.5.1:

+ cw + f (t).

∂t ∂x

∂x

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

w 1 = w(x + C 1 , t) + C 2 e ct , where C 1 and C 2 are arbitrary constants, is also a solution of the equation.

2 ◦ . Solution:

+ − e ct e ct f (t) dt + θ(z), z = x + λt, where

w(x, t) = Ae ct

A and λ are arbitrary constants, and the function θ(z) is determined by the autonomous ordinary differential equation

aθ ( n)

+ bθ ′ z − λθ ′ z + cθ = 0.

3 ◦ . There is a degenerate solution quadratic in x:

w(x, t) = ϕ(t)x 2 + ψ(t)x + χ(t). Z

4 − ◦ . The substitution w = U (x, t) + e ct e ct f (t) dt leads to the simpler equation

∂U

U 2 ∂U

+ b + cU .

∂t

∂x n

∂x

+ kw + f (t)w + g(t).

Generalized separable solution:

w(x, t) = ϕ(t) + ψ(t) exp(λx),

where 2 λ is a root of the quadratic equation bλ + cλ + k = 0, and the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations

2 t = kϕ + f (t)ϕ + g(t),

(2) The Riccati equation (1) is integrable by quadrature in some special cases, for example, (a) k = 0,

ψ t ′ = ( cλ + 2k)ϕ + f (t) + aλ n ψ.

f (t) = const, g(t) = const . See also Kamke (1977) and Polyanin and Zaitsev (2003). Whenever a solution of equation (1) is

(b) g(t) ≡ 0,

(c)

found, one can obtain the corresponding solution of the linear equation (2). ∂w

+ g(x) + h(t).

∂x n

∂x

1 ◦ . Additive separable solution:

w(x, t) = At + B +

h(t) dt + ϕ(x).

A and B are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation

Here,

aϕ ( n)

x + f (x) ϕ ′ x ) + g(x) − A = 0.

2 ◦ . The substitution w = U (x, t) + h(t) dt leads to the simpler equation

+ g(x).

+ bw + g(x) + h(t).

∂t ∂x

∂x

1 ◦ . Additive separable solution:

w(x, t) = ϕ(x) + Ae bt

+ − e bt e bt h(t) dt.

Here,

A is an arbitrary constant and the function ϕ(x) is determined by the ordinary differential equation

aϕ ( n)

x + f (x)(ϕ ′ 2 x ) + bϕ + g(x) = 0.

. The substitution − w = U (x, t) + e bt e bt h(t) dt leads to the simpler equation

∂U 2 ∂ n U ∂U

+ f (x)

+ bU + g(x).

∂t

∂x n

∂x

+ bf (t)w 2 + g(t)w + h(t).

∂t ∂x n

∂x

1 ◦ . Generalized separable solutions involving exponentials of x:

(1) where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential

w(x, t) = ϕ(t) + ψ(t) exp x − b ,

b < 0,

equations with variable coefficients

bf ϕ + gϕ + h,

(3) The arguments of the functions f , g, and h are not specified.

t = 2 bf ϕ + g + a(

− b) n ψ.

Equation (2) is a Riccati equation for ϕ = ϕ(t) and, hence, can be reduced to a second-order linear equation. The books by Kamke (1977) and Polyanin and Zaitsev (2003) present a large number of solutions to this equation for various f , g, and h.

Whenever a solution of equation (2) is known, the corresponding solution of equation (3) is computed by the formula

(4) where

ψ(t) = C exp a( − b) n t+ (2 bf ϕ + g) dt ,

C is an arbitrary constant. Note two special integrable cases of equation (2). Solution of equation (2) for h ≡ 0:

ϕ(t) = e G C 1 − b fe G dt , G=

g dt,

where C 1 is an arbitrary constant. If the functions f , g, and h are proportional,

the solution of equation (2) is expressed as

where C 2 is an arbitrary constant. On integrating the left-hand side of (5), one may obtain ϕ = ϕ(t) in explicit form.

2 ◦ . Generalized separable solution (generalizes the solutions of Item 1 ◦ ):

b < 0, (6) where the functions ϕ(t), ψ(t), and χ(t) are determined by the system of first-order ordinary

w(x, t) = ϕ(t) + ψ(t) exp x − b + χ(t) exp −x − b ,

differential equations with variable coefficients

(9) For equations of even order, with n = 2m, m = 1, 2, . . . , it follows from (8) and (9) that ψ(t) and

χ(t) are proportional. Then, by setting ψ(t) = Aθ(t) and χ(t) = Bθ(t), we can rewrite solution (6) in the form

w(x, t) = ϕ(t) + θ(t)

A exp x − b +

B exp −x − b ,

b < 0, (10) b < 0, (10)

(12) The function ϕ can be expressed from (12) via θ and then substituted into (11) to obtain a

θ ′ t = 2 bf ϕ + g + (−1) m ab m θ.

second-order nonlinear equation for θ. For f , g, h = const, this equation is autonomous and its order can be reduced.

Note two special cases where solution (10) is expressed in terms of hyperbolic functions: √

1 w(x, t) = ϕ(t) + θ(t) cosh x 1 − b if A=

2 , B= 2 ;

1 w(x, t) = ϕ(t) + θ(t) sinh x 1 − b if A=

2 , B=− 2 .

3 ◦ . Generalized separable solution involving trigonometric functions of x: √ √

b > 0, (13) where the functions ϕ(t), ψ(t), and χ(t) are determined by a system of ordinary differential equations (which is not written out here).

w(x, t) = ϕ(t) + ψ(t) cos x b + χ(t) sin x b ,

For equations of even order, with n = 2m, m = 1, 2, . . . , there are exact solutions of the form

(14) where c is an arbitrary constant and the functions ϕ(t) and θ(t) are determined by the system of first-order ordinary differential equations with variable coefficients

w(x, t) = ϕ(t) + θ(t) cos x b+c ,

(16) The function ϕ can be expressed from (16) via θ and then substituted into (15) to obtain a

θ ′ t = 2 bf ϕ + g + (−1) m ab m θ.

second-order nonlinear equation for θ. For f , g, h = const, this equation is autonomous and its order can be reduced. ✟✂✠

References : V. A. Galaktionov (1995), A. D. Polyanin and V. F. Zaitsev (2002).

+ xg(t) + h(t)

Passing to the new independent variables Z

g(t) dt , one arrives to the simpler equation

ϕ n ( t) dt, z = ϕ(t)x +

h(t)ϕ(t) dt, ϕ(t) = exp

which has a traveling-wave solution −1 w = u(kz + λτ ) and a self-similar solution w = v(zτ /n ). ∂w

+ g(t).

∂t

∂x

1 ◦ . Additive separable solution:

w(x, t) = At + B +

g(t) dt + ϕ(x).

Here,

A and B are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation

aϕ ( n) x + f x, ϕ ′ x −

A = 0.

2 ◦ . The substitution

w = U (x, t) +

g(t) dt

leads to the simpler equation

∂U

∂U

+ f x,

∂t

∂x n

∂x

10. =a

n +f x,

+ bw + g(t).

∂t ∂x

∂x

1 ◦ . Additive separable solution:

bt

bt

w(x, t) = ϕ(x) + Ae − + e e bt g(t) dt.

Here,

A is an arbitrary constant and the function ϕ(x) is determined by the ordinary differential equation

aϕ ( n) x + f x, ϕ ′ x + bϕ = 0.

2 ◦ . The substitution

w = U (x, t) + e − bt e bt g(t) dt

leads to the simpler equation

Multiplicative separable solution:

w(x, t) = A exp λx + aλ n t+

f (t, λ) dt ,

where

A and λ are arbitrary constants.

w n –1

11.1.6. Equations of the Form ∂w = a∂ n + F x, t, w, ∂w , . . ., ∂ w n

∂t

∂x

∂x

∂x –1

∂w i,j<n ∂ n w X ∂ i w ∂ j w X n–1

+ h(t).

i,j=0

Here, we adopt the notation: 0 ∂ w

∂x 0 ≡ w.

1 ◦ . In the general case, the equation has generalized separable solutions of the form

w(x, t) = ϕ(t) + ψ(t) exp(λx), i,j<n X

where λ is a root of the algebraic equation

b ij i+j λ = 0.

i,j=0

be zero for odd i + j. In this case, the original equation has also generalized separable solutions of the form

2 ◦ . Let n be an even number and let all coefficients b ij

w(x, t) = ϕ 1 ( t) + ψ 1 ( t)

A cosh(λx) + B sinh(λx) ,

w(x, t) = ϕ 2 ( t) + ψ 2 ( t)

A cos(λx) + B sin(λx) ,

where

A and B are arbitrary constants, the parameter λ is determined by solving algebraic equations, and the functions ϕ 1 ( t), ψ 1 ( t) and ϕ 2 ( t), ψ 2 ( t) are found from appropriate systems of first-order ordinary differential equations.

2. =a n +F x,

+ g(t).

1 ◦ . Additive separable solution:

w(x, t) = At + B +

g(t) dt + ϕ(x).

Here,

A and B are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation

aϕ ( n) x +

F x, ϕ ′ x , ...,ϕ ( n−1) x −

A = 0,

whose order can be reduced with the substitution U (x) = ϕ ′ x .

2 ◦ . The substitution w = u(x, t) + g(t) dt leads to the simpler equation

+ bw + g(t).

1 ◦ . Additive separable solution:

w(x, t) = ϕ(x) + Ae bt

+ − e bt e bt g(t) dt.

Here,

A is an arbitrary constant and the function ϕ(x) is determined by the ordinary differential equation

aϕ ( n) x +

F x, ϕ ′ x , ...,ϕ ( n−1) x + bϕ = 0.

. The substitution − w = u(x, t) + e bt e bt g(t) dt leads to the simpler equation

Multiplicative separable solution:

F (t, λ, . . . , λ n−1 ) dt , where

w(x, t) = A exp λx + aλ n t+

A and λ are arbitrary constants. ∂w

Multiplicative separable solutions:

2 w(x, t) = 2 A cosh(λx) + B sinh(λx) exp aλ n t+ F (t, λ , ...,λ n−2 ) dt , w(x, t) = 2 A cos(λx) + B sin(λx) exp (−1) n aλ n t + Φ(t) ,

2 Φ( 2 t) = F t, −λ , . . . , (−1) n−1 λ n−2 dt,

where

A, B, and λ are arbitrary constants.

n + f (x, t, w) ∂w + g(x, t, w)

∂t

∂x

∂x

∂w ∂ n w

1. = aw ∂t

+ f (t)w + g(t).

∂x n

1 ◦ . Degenerate solution:

n−1

g(t)

w(x, t) = F (t) A n−1 x

+ ···+A 1 x+A 0 +

where A 0 , A 1 , ...,A n−1 are arbitrary constants.

2 ◦ . Generalized separable solution:

w(x, t) = ϕ(t) x n

+ A n−1 x n−1 + ···+A 1 x+A 0 + ϕ(t)

g(t)

dt, ϕ(t)

f (t) dt , where A 0 , A 1 , ...,A n−1 , and

ϕ(t) = F (t)

C are arbitrary constants.

n + f (x)w +

∂t ∂x

k=0

Generalized separable solution:

X n−1

n−1

w(x, t) = t

where C 0 , C 1 , ...,C n−1 , and x 0 are arbitrary constants.

∂w ∂ n w

3. = aw ∂t

+ bw 2 + f (t)w + g(t).

∂x n

Generalized separable solution:

w(x, t) = ϕ(t)Θ(x) + ψ(t).

Here, the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations

2 t = Cϕ + bϕψ + f (t)ϕ, ψ 2 ′

t = Cϕψ + bψ + f (t)ψ + g(t),

where

C is an arbitrary constant and the function Θ(x) satisfies the nth-order linear ordinary differential equation

aΘ ( n) x + bΘ = C.

∂w

2 n – ak w + f (x)w + b 1 sinh(kx) + b 2 ∂t cosh(kx). ∂x Generalized separable solution linear in t:

4. = aw

w(x, t) = t b 1 sinh( kx) + b 2 cosh( kx) + ϕ(x).

Here, the function ϕ(x) is determined from the constant-coefficient linear nonhomogeneous ordinary differential equation

(2 aϕ 2 n)

− ak n ϕ + f (x) = 0.

5. = aw ∂x n

+ xf (t) + g(t)

+ h(t)w.

∂t

∂x

The transformation

F n ( t)H(t) dt, where the functions

w(x, t) = H(t)u(z, τ ), z = xF (t) +

g(t)F (t) dt, τ=

F (t) and H(t) are given by

h(t) dt , leads to the simpler equation

F (t) = exp

f (t) dt , H(t) = exp

which admits, for example, a traveling-wave solution u = u(kz + λτ ) and a self-similar solution of the form

u = u(ξ), ξ = zτ −1 /n . ∂w

∂w

6. = aw ∂t

+ f (x)w

+ g(t)w + h(t).

∂x n

∂x

Generalized separable solution:

w(x, t) = ϕ(t)Θ(x) + ψ(t), where the functions ϕ(t), ψ(t), and Θ(x) are determined by the ordinary differential equations

t = Cϕ + g(t)ϕ, ψ ′ t = Cϕ + g(t) ψ + h(t), aΘ ( n)

x + f (x)Θ ′ x = C,

where

C is an arbitrary constant. On integrating the first two equations successively, one obtains

ϕ(t) = G(t) A−C G(t) dt , G(t) = exp

g(t) dt ,

Z h(t)

ψ(t) = Bϕ(t) + ϕ(t)

A and B are arbitrary constants. ∂w

∂w

7. = aw ∂x n

+ f (x)w

+ g(x)w 2 + h(t)w.

∂t

∂x

Multiplicative separable solution:

h(t) dt , where

w(x, t) = ϕ(x)H(t) A+B

H(t) dt , H(t) = exp

A and B are arbitrary constants, and the function ϕ(x) is determined by the linear ordinary differential equation

aϕ ( n) x + f (x)ϕ ′ x + g(x)ϕ + B = 0.