Heat and Mass Transfer Equations of the Form
5.4.3. Heat and Mass Transfer Equations of the Form
f (x) ∂w +∂ g(y) ∂w = h(w) ∂x
∂x
∂y
∂y
◮ Equations of this form describe steady-state heat/mass transfer or combustion processes in inhomogeneous anisotropic media. Here, f = f (x) and g = g(y) are the principal thermal diffusivities
(diffusion coefficients) dependent on coordinates; h = h(w) is the kinetic function (source function), which defines the law of heat (substance) release of absorption. The simple solutions dependent on
a single coordinate, w = w(x) or w = w(y), are not considered in this subsection. ∂
ax n ∂w
by m ∂w = f (w).
1 ◦ . Functional separable solution for n ≠ 2 and m ≠ 2:
2 2− m 1 w = w(ξ), /2 ξ= b(2 − m) x + a(2 − n) y . Here, the function w(ξ) is determined by the ordinary differential equation
2 2− n
w ′′ ξξ +
w ξ ′ = Bf (w),
(2 − n)(2 − m)
ab(2 − n) 2 (2 − m) 2
For m = 4/n, a family of exact solutions to the original equation with arbitrary f = f (w) follows from (1). It is given by
ab(2 − n) where C 1 and C 2 are arbitrary constants.
w ζζ ′′ =
2 ζ 1− A f (w).
(1 − A)
A large number of exact solutions to equation (2) for various f = f (w) can be found in Polyanin and Zaitsev (2003). ❍☎■
Reference : V. F. Zaitsev and A. D. Polyanin (1996).
= f (w).
The transformation ζ = x + c, η = y + s leads to an equation of the form 5.4.3.1:
= f (w).
= f (w).
The transformation ζ = |x| + c, η = |y| + s leads to an equation of the form 5.4.3.1:
= f (w).
= f (w).
∂x ∂y
∂y
Functional separable solution for µ ≠ 0:
1 w = w(ξ), /2 ξ= bµ ( x+C
1 ) +4 ae µy
where C 1 is an arbitrary constant and the function w(ξ) is determined by the autonomous ordinary differential equation
w ′′
ξξ = abµ 2 f (w).
The general solution of this equation with arbitrary kinetic function f = f (w) is defined implicitly by
where C 2 and C 3 are arbitrary constants. ∂ 2 w
∂w
5. a 2 +
be µ|y|
= f (w).
∂x ∂y
∂y
The substitution ζ = |y| leads to an equation of the form 5.4.3.4. ∂
= f (w).
Functional separable solution for βµ ≠ 0:
w = w(ξ), 1 ξ = bµ e βx /2 + 2 aβ − e µy ,
where the function w(ξ) is determined by the ordinary differential equation
w ′′
ξξ − w ′ ξ ξ = Af (w),
The substitution ζ=ξ 2 brings (1) to the generalized Emden–Fowler equation
′′ = 1 w −1 ζζ
4 Aζ
f (w),
whose solutions with f (w) = (kw + s) −1 and f (w) = (kw + s) −2 ( k, s = const) can be found in Polyanin and Zaitsev (2003). ❍☎■
Reference : V. F. Zaitsev and A. D. Polyanin (1996).
7. ae β|x| ∂w
be µ|y| ∂w = f (w).
The transformation ζ = |x|, η = |y| leads to an equation of the form 5.4.3.6. ∂
= f (w).
Functional separable solution for n ≠ 2 and µ ≠ 0:
2 a(2 − n) − e µy 1 /2 , where the function w(ξ) is determined by the ordinary differential equation
2 w = w(ξ), 2− ξ= bµ x n +
2− n ξ ξ = abµ 2 (2 − n) 2 f (w). ∂
Multiplicative separable solution:
w(x, y) = ϕ(x)ψ(y),
where the functions ϕ(x) and ψ(y) are determined by the ordinary differential equations [ f (x)ϕ x ′ ] ′ x = kϕ ln ϕ + Cϕ,
[ g(y)ψ ′ y ] ′ y = kψ ln ψ − Cψ, and ▲☎▼
C is an arbitrary constant.
Reference : A. D. Polyanin and V. F. Zaitsev (2002).
This is a special case of equation 5.4.4.6 with k = a, h 1 ( x) = b, and h 2 ( y) = 0.
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
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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 Cubic Nonlinearities Involving Arbitrary Functions
» Equations of General Form Involving Arbitrary Functions of a Single Argument
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» Heat and Mass Transfer Equations in Quiescent or Moving Media with Chemical Reactions
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» 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
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» Equations Involving Second-Order Mixed Derivatives
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» Equations of the Form ∂w =a∂ w 4 +F x, t, w, ∂w
» Boussinesq Equation and Its Modifications
» Kadomtsev–Petviashvili Equation
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» Nonstationary Hydrodynamic Equations (Navier–Stokes equations)
» Equations of the Form ∂w n = a∂ w n + f (w) ∂w
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» 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
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