Picard succession approximation method

We can also simplify the solution given in (15.93) using the Picard successive approximation method when we know the initial conditions. If t0 and z0(t0) are the initial conditions and the differential equation is written as z (t) =fit, z), the equation can be written as... 

Thus, Equations 15.62 and 15.63 represent the rough approximation of Ad>(y) and y,(v) and can be plotted as a function of y in Figure 15.4 for the same data as in Figure 15.1. Both A(y) and j,(y) dependences are expected but the latter shows a quasi-limiting behavior not predicted by the condition of quasi-primary current distribution. The variation in jt(y) has to be zero if we consider all the components given by the Picard s method of successive approximation. Thus, the plots in Figure 15.4 have to be taken with care. 

The existence of solution to Eq. (4.1.15) is generally established by a procedure that guarantees the convergence of the method of successive approximations (also called Picard s iteration). This method consists in substituting into the right-hand side of (4.1.15), the nth approximation for the population density denoted by in order to calculate the (n + l)st approximant. Thus we have... 

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