As Walter has pointed out, it is surely not possible for 'solve' to find an explicit solution for t in terms of the other variables, since your expression involves the 'erfc' function. The best you can hope for is a numerical solution. To accomplish this I would suggest you make the following substitutions in G to simplify matters. Let x = sqrt(d*t)/h and k = 4*h^2/a^2 so that
G = g(x) = ((1-2*x^2)*exp(x^2)*erfc(x)+2*x/sqrt(pi))/(1+k*x^2)
Then
dG/dd = dg/dx*dx/dd = g'(x)*sqrt(t/d)/(2*h)
and
d(dG/dd)/dt = g"(x)*dx/dt*sqrt(t/d)/(2*h)+g'(x)/sqrt(d*t)/(4*h)
= g"(x)*(sqrt(d/t)/(2*h))*sqrt(t/d)/(2*h)+g'(x)/sqrt(d*t)/(4*h)
= g"(x)/(4*h^2)+g'(x)/(4*h^2)/x
= (g"(x)*x+g'(x))/(4*h^2*x)
Hence the equation you want to solve is
For this purpose you can use matlab's 'fzero'. For any given value of k=4*h^2/a^2 you can find the x root or roots of this equation. Having found it or them, you can immediately use the solution(s) to solve for t as
You can use your symbolic toolbox to evaluate and simplify g'(x) and g"(x) in setting up the expression g"(x)*x+g'(x) for 'fzero' to solve.
Best Answer