Equations and Solutions

The art of fitting ordinary differential equations models to experimental results

“…As for any differential equation, the solutions depend on the initial conditions and, therefore, we need to know the values z i0 at t = t 0 . For any time t t 0 , we can use equations (10) and (11) to iterate in time steps h until step n is reached, satisfying the condition t n t t n+1 . Then, z i (t) can be obtained from a linear interpolation.…”

“…It is interesting to note that it was the need to solve differential equations that gave birth both to analog computers [9] and to the first digital computer [10]; differential equations describe a very large fraction of physics [11]; simple sets of simultaneous differential equations, like the Lotka-Volterra equations [12] or the Brusselator [13], may present several dynamic regimes; additional fitting parameters are needed for solving differential equations given that initial conditions must be defined (but these can be estimated from experimental data); differential equations of motion ease the modeling of forces and/or torques (see for instance [14]); the minimization of the χ 2 automatically avoids singularities in the solutions of differential equations; in systems lacking analytical descriptions, differential equations remain the main modeling tool (for instance, in the case of space probe trajectories [15]); sophisticated modeling is actually achieved by means of software implemented analog computers (this is case, for instance, of [16]); fitting models to data may be considered an element of 'data science' [17].…”