A CONSISTENT NUMERICAL METHOD MONITORED BY RESIDUALS FOR SOLVING THE PARTIAL OR ORDINARY DIFFERENTIAL EQUATIONS OF FIRST ORDER

May 8, 2015

There are numerous methods used for the numerical integration of Ordinary Differential Equations (ODE) or Partial Differential Equations (PDE). The method presented within this book is completely different from the other methods considered "classics" by the scientific and engineering world, because:

It starts by effectively integrating the Differential Equations that leads to Integral Equations. The integration procedure, which is carefully avoided by some "classical" methods such as Euler, Heun, Runge-Kutta, Finite Differences, is based on the use of different degrees polynomials – called Concordant Functions (CF) – as functions that replace the original unknown function. The major part of the computation is dedicated to obtain the CF, which is determined – on one side – by the known START parameters given by the Boundary Conditions and – on the other side – by the unknown TARGET parameters. Consequently, the method is implicit, therefore stable. The results of the procedure are the TARGET parameters.

Any computation procedure, based a coherent algorithm, leads to certain numerical result. This is of a little use – or even useless – if it not accompanied by a reasonable method for the evaluation of the accuracy. Some "classical methods" use sophisticated and time-consuming approaches for this purpose or ignore them completely. The method developed here replaces the Target unknowns in the given ODE/PDE and – at the "cost" of two/three multiplication and a sum – obtains a RESIDUAL, which gives a sure and reliable evaluation of the global accuracy. Thus, the integration is monitored for all the elements. The method being consistent can be applied similarly for both ODEs and PDEs, with constant or variable coefficients, linear or nonlinear, explicit or implicit. Numerous examples illustrate its various possibilities.