# type of partial differential equations?

• May 12th 2010, 10:23 AM
boysrule
type of partial differential equations?
> Given the second order non linear BVP
>
> (ƒ')^n = 1 + γ θ …..……………………… 1
>
> θ" + (λ +n+1/ 2n +1) ƒ θ' - n (2 λ +1/2n+ 1) * ƒ' θ = 0 ………… 2
>
> Prime in the above eqn’s describe partial differentiation with respect to η
>
> Boundary conditions are
> ƒ (0) = 0, θ'(0) = -1
> ƒ' (∞) = 0, θ(∞) = 0
>
> can anyone help me in deciding what type of partial differential equations are these
> i mean elliptic or parabolic or hyperbolic pde's and how to solve them(Headbang)(Headbang)
• May 13th 2010, 06:32 AM
Jester
Quote:

Originally Posted by boysrule
> Given the second order non linear BVP
>
> (ƒ')^n = 1 + γ θ …..……………………… 1
>
> θ" + (λ +n+1/ 2n +1) ƒ θ' - n (2 λ +1/2n+ 1) * ƒ' θ = 0 ………… 2
>
> Prime in the above eqn’s describe partial differentiation with respect to η
>
> Boundary conditions are
> ƒ (0) = 0, θ'(0) = -1
> ƒ' (∞) = 0, θ(∞) = 0
>
> can anyone help me in deciding what type of partial differential equations are these
> i mean elliptic or parabolic or hyperbolic pde's and how to solve them(Headbang)(Headbang)

Since you say that prime denote differentiation wrt $\eta$ then I'd say these are ODEs' not PDE's.
• May 13th 2010, 01:21 PM
boysrule
type of partial differential equations
Given the second order non linear BVP

(ƒ')^n = 1 + γ θ …..……………………… 1

θ" + (λ +n+1/ 2n +1) ƒ θ' - n (2 λ +1/2n+ 1) * ƒ' θ = 0 ………… 2

Prime in the above eqn’s describe partial differentiation with respect to η

Boundary conditions are
ƒ (0) = 0, θ'(0) = -1
ƒ' (∞) = 0, θ(∞) = 0

where η is a function of x & y given by
Similarity variable, η = x ^ (λ-n/2n+1) * y
Θ is a dimensionless temperature
ƒ is a dimensionless stream function given by
Stream function, ψ = x ^ (λ+n+1/2n+1) * ƒ(η)

how to solve the above system of eqn’s
i am sorry that last time i did not mention about waht is η
• May 13th 2010, 01:48 PM
Jester
Quote:

Originally Posted by boysrule
Given the second order non linear BVP

(ƒ')^n = 1 + γ θ …..……………………… 1

θ" + (λ +n+1/ 2n +1) ƒ θ' - n (2 λ +1/2n+ 1) * ƒ' θ = 0 ………… 2

Prime in the above eqn’s describe partial differentiation with respect to η

Boundary conditions are
ƒ (0) = 0, θ'(0) = -1
ƒ' (∞) = 0, θ(∞) = 0

where η is a function of x & y given by
Similarity variable, η = x ^ (λ-n/2n+1) * y
Θ is a dimensionless temperature
ƒ is a dimensionless stream function given by
Stream function, ψ = x ^ (λ+n+1/2n+1) * ƒ(η)

how to solve the above system of eqn’s
i am sorry that last time i did not mention about waht is η

What is the original set of equations (they look like the're from fluid mechanics, maybe for a non-Newtonian fluid). I might be able to give a reference.
• Jun 2nd 2010, 05:23 PM
boysrule
solving non linear PDE's in fluid mechanics
Given the second order non linear BVP

(ƒ')^n = 1 + γ θ …..……………………… 1

θ" + (λ +n+1/ 2n +1) ƒ θ' - n (2 λ +1/2n+ 1) * ƒ' θ = 0 ………… 2

Prime in the above eqn’s describe partial differentiation with respect to η

Boundary conditions are
ƒ (0) = 0, θ'(0) = -1
ƒ' (∞) = 0, θ(∞) = 0

where η is a function of x & y given by
Similarity variable, η = x ^ (λ-n/2n+1) * y
Θ is a dimensionless temperature
ƒ is a dimensionless stream function given by
Stream function, ψ = x ^ (λ+n+1/2n+1) * ƒ(η)

how to solve the above system of PD eqn’s
the above system of equations are related to mixed convection of non newtonian fluids
• Jun 22nd 2010, 04:50 PM
boysrule
fluid mechanics-boundary layer flow probem on vertical plate in porous medium
Given the second order non linear BVP

(ƒ')^n = 1 + γ θ …..……………………… 1

θ" + [(λ +n+1)/( 2n +1)] ƒ θ' - n [(2 λ +1)/(2n+ 1)] * ƒ' θ = 0 ………… 2

Prime in the above eqn’s describe partial differentiation with respect to η

Boundary conditions are
ƒ (0) = 0, θ'(0) = -1
ƒ' (∞) = 0, θ(∞) = 0

The above nonlinear coupled system of equations for ƒ and θ have been derived from conservation laws that govern the boundary layer flow on vertical plate in porous medium by introducing similarity variable η and stream function ψ

where
γ can be assumed as a Rayleigh number type i.e it represents the relative importance
of free to forced convection
η is a function of x & y given by
Similarity variable, η = x ^ (λ-n/2n+1) * y
n is the permeability
ƒ is a dimensionless stream function given by
Stream function, ψ = x ^ (λ+n+1/2n+1) * ƒ(η)
θ is a dimensionless temperature given by
T = x^ [{n(2 λ+1)/2n+1}]* θ(η) and
λ is a scalar obtained by assuming the surface heat flux Q(X)=x^ λ at y=0 which vary according to power laws

I would like to solve them for approximations of λ and n
I know that the above system of equations can be solved by finite difference methods using shooting technique.
I would like to know how to start the solution and would like to know what type of PDE’s are these i.e. parabolic, elliptic, and hyperbolic since MATLAB can solve elliptic nonlinear PDE’s as far as I know and also I am not familiar with MATLAB.
So can any one suggest me the right path.