How can I solve equation with no explicit solution?

I am trying to solve for aerosol optical thickness t, using radiative transfer equation. I have simplified the code but i need to get the final solution for t. The code is below:

%syms L k S R P u t phi alpha

PI = 3.142; k = 28.72155; L = 31.51617; S = 0.79574; R = 41.76734;

P = 0.05039; alpha = 98.0708; phi = 2.29082; u = 0.774689;

equation = (alpha -k -L == S*R*P*(1/PI)*exp(-(t/u)-(1/(6*u*t)))- L*exp(-phi*t))

solution = vpasolve(equation, t)

[Bjorn Gustavsson]

What exactly are you trying to solve for? (In a physics sense). You'll be lucky to get a solution in simple closed form. Since the aerosol-density will vary from any simple functional form in reality you will have to turn to numerical solutions sooner or later - so you might just as well turn to such methods now and get ahead in building tools and experience sooner rather than later.

Answer 1

[Alan Stevens]

Are you sure your equations are correct? They seem to result in a negative value for t:

% First plot graph to see where solution might lie
tg = [-1:0.01:-0.04 NaN 0.01:0.1:2];
z = fn(tg);
plot(tg,z),grid
% Looks like a solution between 0 and -0.1 
t0 = -0.1;  % Initial guess
t = fzero(@fn,t0);
disp(t)

 function z = fn(t)
     k = 28.72155; L = 31.51617; S = 0.79574; R = 41.76734;
     P = 0.05039; alpha = 98.0708; phi = 2.29082; u = 0.774689;
     LHS = alpha-k-L;
     RHS = S*R*P*(1/pi)*exp(-(t/u)-(1./(6*u*t)))- L*exp(-phi*t);
     z = LHS - RHS;
 end