type of partial differential equations?

May 2010
9
0
> 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)
 

Jester

MHF Helper
Dec 2008
2,470
1,255
Conway AR
> 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 \(\displaystyle \eta\) then I'd say these are ODEs' not PDE's.
 
May 2010
9
0
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
 

Jester

MHF Helper
Dec 2008
2,470
1,255
Conway AR
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.
 
May 2010
9
0
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
 
May 2010
9
0
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.