Linear integrator Runge Kutta

In Fig. 10, the transients exhibit quite different behavior from opal A to opal CT. In particular, a bi-exponential decay (Eq. 2) failed to reproduce the kinetics of opal CT. In this material, the emission is red-shifted towards 2.6 eV and the PL is strongly quenched at shorter time delays, with an unusual, non-linear kinetics in semi-log scale, indicating a complex decay channel either involving multi-exponential relaxation or exciton-exciton annihilations. Runge-Kutta integration of Eq. 5 seems to confirm the latter assumption with satisfactory reproduction of the observed decays. The lifetimes and annihilation rates are Tct = 9.3 ns, ta = 13.5 ns, 7ct o = 650 ps-1 and 7 0 = 241 ps-1, for opal CT and opal A, respectively. 

In ref 174 a new embedded pair of explicit exponentially fitted Runge-Kutta-Nystrom methods is developed. The methods integrate exactly any linear combinations of the functions from the set (exp(/iO,exp(—/it) (ji e ifl or /i e i9I). The new methods have the following characteristics ... 

In 28 the author have developed a family of trigonometrically fitted Runge-Kutta methods for the numerical integration of the radial Schrodinger equation. The developed method is based on the Runge-Kutta Zonneveld method. More specifically the new methods are developed in order to integrate exactly any linear combination of the functions ... 

Using the boundary conditions (equations (5.54) and (5.55)) the boundary values uo and un+i can be eliminated. Hence, the method of lines technique reduces the nonlinear parabolic PDE (equation (5.48)) to a nonlinear system of N coupled first order ODEs (equation (5.52)). This nonlinear system of ODEs is integrated numerically in time using Maple s numerical ODE solver (Runge-Kutta, Gear, and Rosenbrock for stiff ODEs see chapter 2.2.5). The procedure for using Maple to solve nonlinear parabolic partial differential equations with linear boundary conditions can be summarized as follows ... 

A combination of the Runge Kutta method and methods of non-linear regression allows a parameter identification from the time-course data. This technique starts with a given set of parameters, performs the numeric integration of the rate equation... 

Linear statistical examinations require only the solution of a system of linear equations. Dynamically nonlinear problems, on the other hand, require the application of highly developed integration methods, most of which are based on further developments of the Runge-Kutta method. 

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