Equations with Two Independent Variables, Nonlinear in Two or More Highest Derivatives
8.2.4. Equations with Two Independent Variables, Nonlinear in Two or More Highest Derivatives
f 2 =g 1 (x)g (y).
Generalized separable solution: w(x, y) = ϕ(x) + ψ(y) + C 1 xy + C 2 x+C 3 y+C 4 , where C 1 , ...,C 4 are arbitrary constants, and the functions ϕ = ϕ(x) and ψ = ψ(y) are determined
by the ordinary differential equations ( a is any)
f 1 ( ϕ ′′ xx )= ag 1 ( x),
af 2 ( ψ ′′ yy )= g 2 ( y).
The substitution u= ∂w ∂x leads to the first-order partial differential equation
For details about integration methods and exact solutions for such equations (with various
F ), see Kamke (1965) and Polyanin, Zaitsev, and Moussiaux (2002).
1 ◦ . Solution quadratic in both variables:
1 w(x, y) = 2
2 C 1 x C 2 xy + 2 1 F (C , C 2 ) y + C 3 x+C 4 y+C 5 ,
where C 1 , ...,C 5 are arbitrary constants.
2 ◦ . We differentiate the equation with respect to x, introduce the new variable
and then apply the Legendre transformation (for details, see Subsection S.2.3)
to obtain the second-order linear equation
where the subscripts
X and Y denote the corresponding partial derivatives.
Special case. Let
Solution: w = ϕ(z) + 1 ( A 2 A 3 − A 1 A 4 ) x 3 1 2 1 2 6 1 + 2 aA 1 A 3 x y+ 2 aA 2 A 3 xy + 6 ( a 2 A 1 A 3 + aA 2 A 4 ) y 3
+ 1 2 B 1 x 2 + B 2 xy + 1 2 B 3 y 2 + B 4 x+B 5 y+B 6 , z=A 1 x+A 2 y, where the A n and B m are arbitrary constants and the function ϕ(z) is determined by the ordinary differential equation
∂x∂y ∂y 2
1 ◦ . Solution quadratic in both variables:
2 w(x, y) = A 2
11 x + A 12 xy + A 22 y + B 1 x+B 2 y + C,
where A 11 , A 12 , A 22 , B 1 , B 2 , and
C are arbitrary constants constrained by F (2A 11 , A 12 ,2 A 22 ) = 0.
2 ◦ . Solving the equation for w yy (or w xx ), one arrives at an equation of the form 8.2.4.2.
∂x ∂x ∂x∂y
∂y ∂x∂y ∂y 2
Additive separable solution:
w(x, y) = ϕ(x) + ψ(y).
Here, ϕ(x) and ψ(y) are determined by the ordinary differential equations
C is an arbitrary constant.
Multiplicative separable solution:
w(x, y) = ϕ(x)ψ(y).
Here, the functions ϕ(x) and ψ(y) are determined by the ordinary differential equations
C is an arbitrary constant.
Multiplicative separable solution:
w(x, y) = ϕ(x)ψ(y).
Here, the functions ϕ(x) and ψ(y) are determined by the ordinary differential equations
1 x, ϕ ′ x /ϕ, ϕ ′′ xx /ϕ ψ k F 2 y, ψ ′ y /ψ, ψ ′′ yy /ψ
where
C is an arbitrary constant.
w = w(ξ), ξ = ax + by,
where the function w(ξ) is determined by the ordinary differential equation
The substitution u(x, y) = w(x, y) + kx + sy leads to an equation of the form 8.2.4.8:
∂u
∂u
F ax + by, u,
− k,
− s,
2 ∂y , ∂y 2 ,
∂x
∂x
∂x∂y
∂x ∂y ∂x 2 ∂x∂y ∂y 2
Traveling-wave solutions:
w = w(z),
z=a 2 x + (k − a 1 ) y,
where k is a root of the quadratic equation
k 2 −( a
1 + b 2 ) k+a 1 b 2 − a 2 b 1 = 0,
and the function w(z) is determined by the ordinary differential equation kzw ′ z =
, , . ∂x ∂y ∂x 2 ∂x∂y ∂y 2
Exact solutions are sought in the traveling-wave form
w = w(z),
z = Ax + By + C,
where the constants
A, B, and C are determined by solving the algebraic system
c 1 A k + c 2 B k = C. (3) Equations (1) and (2) are first solved for
A and B, and then C is evaluated from (3). The desired function w(z) is determined by the ordinary differential equation
∂x 2 ∂x∂y
, , , . ∂x ∂y ∂x 2 ∂x∂y ∂y 2
Traveling-wave solutions:
w = w(z),
z = Ax + By,
where the constants
A and B are determined by solving the algebraic system of equation
1 A + a 2 AB + a 3 B = A,
b 1 A 2 + b 2 AB + b 3 B 2 = B,
and the desired function w(z) is determined by the ordinary differential equation
Parts
» NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
» Equations of the Form ∂w = aw +f x, t, w, ∂w
» Equations of the Form ∂w m =a∂ w ∂w + bw k
» 1.13. Equations of the Form ∂w i =∂ f (w) ∂w + g(w)
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» Other Equations Explicitly Independent of x and t
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» Equations of the Form ∂w =a ∂ w ∂w +f (x, t) ∂w +g(x, t, w)
» 6.15. Equations of the Form ∂w i
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» Equations with Two Independent Variables, Nonlinear in Two or More Highest Derivatives
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» Solutions of Some Nonlinear Functional Equations and Their Applications
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