Numerical and analytical solutions of an ODE: Strengths and weaknesses of the analytical series solution

Relative distance between two scalar fields. Application to mathematical modelling approximation

“…When we can obtain analytical solutions of these approximated PDEs, we can compare them with the numerical solution of the original PDE by using the distance D . The principal advantage of using D instead of (1), (2) or (5) is that the value given by D has a very easy interpretation because of (9); that is, a value close to 0 means relatively near, but a value close to 1 means relatively infinitely far.…”

“…Mathematically speaking, we want to compare two scalar fields f and g within a certain connected open set D Â R n . In the literature, we may find a graphical method for comparing numerical solutions with analytical approximations, which yields a qualitative information [2]. However, a quantitative method of comparison is demanded.…”

“…Nevertheless, the evaluation of (1) or (2) does not give us a value, which provides an idea whether the scalar fields f and g are close or far from each other. In order to solve this problem, we may define the relative error between f and g as…”