In my heat transfer classes, we encounter the Gaussian error function when dealing with unsteady heat conduction of a semi-infinite surface. The equation I have to solve is usually of the form:
$$\text{erf}(x)=c$$
where $c$ is a number and $x$ is the variable.
For these questions, the method I use involves guessing a bunch of $x$'s and getting them as close to $c$ as possible. I can compute error function in my scientific calculator using:
$$\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}{e^{-u^2}du}$$
The other method I know involves quadratic interpolation about three nearby points, but obtaining the polynomial and solving it is too time-consuming, so that's ruled out as well.
I would appreciate any method to compute error function equations which can be done reasonably by a scientific calculator. I knew the error function had an inverse, but I couldnt find a suitable form which I can compute via calculator.
It is possible to use the Newton-Raphson iteration method to solve it:
$$x_n=x_{n-1}-\frac{c-\int_{0}^{x_{n-1}}{e^{-u^2}du} }{-e^{-x_{n-1}^2}}$$
A suitable choice of $x_0$ is required for a speedy convergence.