Equations Involving ∂ m w and ∂ w
11.4.2. Equations Involving ∂ m w and ∂ w
∂x m ∂y
1. a n +b ∂y n
= (ay n + bx n )f (w).
∂x Solution:
w = w(z),
z = xy,
where the function w(z) is determined by the autonomous ordinary differential equation
w ( n)
= f (w).
Remark. This remains true if the constants a and b in the equation are replaced by arbitrary functions a = a(x, y, w, w x , w y , . . .) and b = b(x, y, w, w x , w y , . . .).
Multiplicative separable solution:
w(x, y) = Ae λy ϕ(x),
where
A and λ are arbitrary constants, and the function ϕ(x) is determined by the nth-order ordinary differential equation
n ; w ∂y 2 w ,..., ∂x w ∂x w ∂y 2 m
1 ◦ . Multiplicative separable solution:
w(x, y) =
A cosh(λy) + B sinh(λy) ϕ(x),
where
A, B, and λ are arbitrary constants, and the function ϕ(x) is determined by the nth-order ordinary differential equation
2 F x, ϕ 2 ′ /ϕ, . . . , ϕ ( n) /ϕ; λ , ...,λ m
2 ◦ . Multiplicative separable solution:
w(x, y) =
A cos(λy) + B sin(λy) ϕ(x),
where
A, B, and λ are arbitrary constants, and the function ϕ(x) is determined by the nth-order ordinary differential equation
2 F x, ϕ 2
x /ϕ, . . . , ϕ x /ϕ; −λ , . . . , (−1) m λ m = 0.
( n)
Additive separable solution:
w(x, y) = ϕ(x) + ψ(y).
Here, the functions ϕ(x) and ψ(y) are determined by the ordinary differential equations
F 1 x, ϕ ′ x , ...,ϕ ( n) x − kϕ = C,
F 2 y, ψ ′ y , ...,ψ ( m) y − kψ = −C, where
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. ∂w
Additive separable solution:
w(x, y) = ϕ(x) + ψ(y).
Here, the functions ϕ(x) and ψ(y) are determined by the ordinary differential equations
e ( λϕ F 1 x, ϕ n)
x ′ , ...,ϕ x = C,
e λψ F 2 y, ψ ′ , ...,ψ ( m) y y =− C, where
C is an arbitrary constant.
w ∂y m 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
F 1 x, ϕ x ′ /ϕ, . . . , ϕ ( n) x /ϕ − k ln ϕ = C,
y ′ /ψ, . . . , ψ y /ψ − k ln ψ = −C, where
F ( 2 y, ψ m)
C is an arbitrary constant.
Solution: w = w(ξ), ξ = ax + by,
where the function w(ξ) is determined by the ordinary differential equation
F ξ, w, aw ′ ξ , ...,a n ( n) w , bw ξ ′ ξ , ...,b m ( m) w ξ = 0.
w = ϕ(ξ) + Cx, ξ = ax + by,
where
C is an arbitrary constant and the function ϕ(ξ) is determined by the ordinary differential equation
ξξ ...,a n ( , ϕ n)
ξ , bϕ
m) ξ ′ , ...,b ϕ ξ
10. n a 1 x+b 1 y + f (w)
a 2 x+b 2 y + g(w)
∂y m Solutions are sought in the traveling-wave form
w = w(z),
z = Ax + By,
where the constants
A and B are evaluated from the algebraic system of equations
a 1 A n+m + a 2 B n+m = A,
b 1 A n+m + b 2 B n+m = B.
The desired function w(z) is determined by the mth-order ordinary differential equation z+A n+m f (w) + B n+m g(w) w ( m)
= C 0 + C 1 z+···+C n−1 z , where C 0 , C 1 , ...,C n−1 are arbitrary constants.
∂y ∂y Generalized traveling-wave solution:
w = w(z),
z = Ax + By,
where the constants
A and B are evaluated from the algebraic system of equations
a 1 A n + a 2 B n = A,
b 1 A n + b 2 B n = B,
and the desired function w(z) is determined by the ordinary differential equation
, Bw z ′ , ...,B w z . Remark. If the right-hand side of the equation is also dependent on mixed derivatives, solutions
zw ( n) =
F w, Aw ′ , ...,A m w ( m)
( k)
are constructed likewise.
Supplements
Exact Methods for Solving Nonlinear Partial Differential Equations
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)
» Equations of the Form ∂w λw =a∂ e ∂w + f (w)
» Other Equations Explicitly Independent of x and t
» Equations of the Form ∂w 2 = f (x, t) ∂ w +g x, t, w, ∂w
» Equations of the Form ∂w = aw ∂ 2 + f (x, t, w) ∂w + g(x, t, w)
» Equations of the Form ∂w =a ∂ w ∂w +f (x, t) ∂w +g(x, t, w)
» 6.15. Equations of the Form ∂w i
» Equations of the Form ∂w 2 = f (x, t, w) ∂ w +g x, t, w, ∂w
» Equations with Cubic Nonlinearities Involving Arbitrary Functions
» Equations of General Form Involving Arbitrary Functions of a Single Argument
» Equations of General Form Involving Arbitrary Functions of Two Arguments
» Equations of the Form ∂w =a∂ w n ∂w w k +b∂ ∂w
» Heat and Mass Transfer Equations in Quiescent or Moving Media with Chemical Reactions
» Equations of Mass Transfer in Quiescent or Moving Media with Chemical Reactions
» Equations of Heat and Mass Transfer in Anisotropic Media
» Other Equations with Three Space Variables
» Equations with n Space Variables
» Three and n Dimensional Equations
» Equations of the Form ∂ 2 = a∂ w 2 + f (x, t, w)
» Sine Gordon Equation and Other Equations with Trigonometric Nonlinearities
» Equations of the Form ∂ + f (w) ∂w =∂ g(w) ∂w
» Equations Involving Arbitrary Parameters of the Form
» Equations Involving Power Law Nonlinearities
» Heat and Mass Transfer Equations of the Form
» Heat and Mass Transfer Equations of the Form h i h i h i
» Heat and Mass Transfer Equations with Complicating Factors
» Khokhlov–Zabolotskaya Equation
» Equation of Unsteady Transonic Gas Flows
» Equations of the Form ∂ 2 w ∂ 2 w
» Equations of the Form ∂w =F w, ∂w ,∂ w
» Equations of the Form ∂w =F t, w, ∂w ,∂ w
» Equations of the Form ∂w =F x, w, ∂w ,∂ w
» Equations of the Form ∂w =F x, t, w, ∂w ,∂ w
» Equations with Two Independent Variables, Nonlinear in Two or More Highest Derivatives
» Cylindrical, Spherical, and Modified Korteweg–de Vries Equations
» Equations of the Form ∂w +a∂ w 3 +f w, ∂w =0
» Burgers–Korteweg–de Vries Equation and Other Equations
» Steady Hydrodynamic Boundary Layer Equations for a Newtonian Fluid
» Steady Boundary Layer Equations for Non-Newtonian Fluids
» Unsteady Boundary Layer Equations for a Newtonian Fluid
» Unsteady Boundary Layer Equations for Non-Newtonian Fluids
» Equations Involving Second-Order Mixed Derivatives
» Equations Involving Third-Order Mixed Derivatives
» Equations of the Form ∂w =a∂ w 4 +F x, t, w, ∂w
» Boussinesq Equation and Its Modifications
» Kadomtsev–Petviashvili Equation
» Stationary Hydrodynamic Equations (Navier–Stokes
» Nonstationary Hydrodynamic Equations (Navier–Stokes equations)
» Equations of the Form ∂w n = a∂ w n + f (w) ∂w
» Equations of the Form ∂w n = a∂ w n + f (x, t, w) ∂w + g(x, t, w)
» Equations of the Form ∂w n = a∂ w n + F x, t, w, ∂w
» General Form Equations Involving the First
» Equations Involving ∂ m w and ∂ w
» Contact Transformations. Legendre and Euler Transformations
» B ¨acklund Transformations. Differential Substitutions
» Exponential Self Similar Solutions. Equations Invariant Under Combined Translation and Scaling
» Solution of Functional Differential Equations by Differentiation
» Solution of Functional Differential Equations by Splitting
» Simplified Scheme for Constructing Generalized Separable
» Special Functional Separable Solutions
» Splitting Method. Reduction to a Functional Equation with
» Solutions of Some Nonlinear Functional Equations and Their Applications
» Clarkson–Kruskal Direct Method: a Special Form for Similarity Reduction
» Some Modifications and Generalizations
» Group Analysis Methods 1. Classical Method for Symmetry Reductions
» First Order Differential Constraints
» Second and Higher Order Differential Constraints
» Connection Between the Differential Constraints Method and Other Methods
» Movable Singularities of Solutions of Ordinary Differential
» Solutions of Partial Differential Equations with a Movable
» Examples of the Painlev ´e Test Applications
» Method Based on Linear Integral Equations
» Conservation Laws 1. Basic Definitions and Examples
» Conservation Laws. Some Examples
» Characteristic Lines. Hyperbolic Systems. Riemann Invariants
» Self Similar Continuous Solutions. Rarefaction Waves
» Shock Waves. Rankine–Hugoniot Jump Conditions
» Evolutionary Shocks. Lax Condition (Various Formulations)
» Solutions for the Riemann Problem
» Examples of Nonstrict Hyperbolic Systems
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