1 D steady state diffusion equation in the atmosphere

[Juliousceasor]

Hello,

I have a 1-D steady state (dc/dt=0) differential equation in the atmosphere. It looks like follows,

K*C'' + (K'+K/H)*C' + (1/H*K'- (K/H^2)*H'- (L+Si))C + S = 0

where,
C = concentration of the contaminant in the atmosphere at different heights z
K = vertical diffusion coefficient
H = scale height
L = decay constant
Si= constant
S = source term

C'' = double derivative of C w.r.t. z
C',K',H'= derivative of C,K,H w.r.t. z

K,H,Si,S,L are all known values.

I am trying to solve the above differential equation numerically by means of finite differences of 1st order with boundary conditions,

At the top boundary: C = S/L
at the bottom boundary k*dc/dx = 0



Can anyone tell me how to write this routine in matlab?

help would be greatly appricieted! :)

 

[NateD]

Thanks in advance!The first step is to convert the equation into a system of linear equations. To do this, you will need to replace each of the derivatives with finite differences. For example, if you have a grid of points z_i for i=1,...,n then the derivative of C with respect to z can be approximated by the finite difference C'(z_i) ~ (C_(i+1)-C_(i-1))/(2*dz). Here, dz is the spacing between two consecutive grid points. Using this approach, you can rewrite the differential equation as a system of linear equations. Once you have the system of linear equations, you can solve it using the MATLAB function "solve" or any other numerical method. You can also use MATLAB's built-in numerical solvers, such as ode45, to solve the differential equation directly. This is generally more accurate than using finite differences.