zt ln ln /dt exp /dt which for constant results in rt note that exp tis the same as as will be described laterr must in more realistic models depend on the rn method of separation of variables then requires to integrate = /dn which quickly becomes non-trivial for many choices of the only generally applicable solution approach is therefore numerical method numerical solution there is huge collection of numerical methods for problems like ( )and in general any equation of the form ut/where tis the unknown function in the problemand is some known formula of and optionally for examplef utd ru in ( we will first present simple finite difference method solving utthe idea is four-fold introduce mesh in time with points tn we seek the unknown at the mesh points tn and introduce un as the numerical approximation to tn /see fig assume that the differential equation is valid at the mesh points approximate derivatives by finite differencessee fig formulate computational algorithm that can compute new value un based on previously computed values ui an example will illustrate the steps firstwe introduce the meshand very often the mesh is uniformmeaning that the spacing between points tn and tnc is constant this property implies that tn ntn secondthe differential equation is supposed to hold at the mesh points note that this is an approximationbecause the differential equation is originally valid at all real values of we can express this property mathematically as tn un tn / for examplewith our model equation ruwe have the special case tn run or tn tn /un if depends explicitly on