19.1 Introduction

In this section we evaluate the complex system of ordinary differential equations (ODE) by the Runge-Kutta method. This method can be applied to various linear and nonlinear problems [see e.g. Jian Zou, 2001a; Salasnich, 2003a; Burlak et al., 2002; Kartashov et al., 2003a,b]. The following program is a translation into C++ from [Press et al., 1993] and is contained in class CRunge2. Since in C++ the complex type is available (see STL [Nicolai, 1999]) it is allowed to rewrite the mentioned code for complex type ODE. Section 19.2 shows the program code (with GUI), which allows one to evaluate the solution of ODE dy _i/ dx = f _i( x, y ₁, , y _N ), where y _i and f _i are complex numbers. Readers can construct their own function (see TDiffEqsSystem2, void MySystem2 ets) to their own calculation. The next test program for the calculation for ODE is considered

The exact solution is given by [Kamke, 1983]

Figure 19.1 shows the initial data and output. In this GUI one can write down the initial conditions x ₀, y ₀₁, y ₀₂, y ₀₃, the order of ODE, accuracy of calculations ? and the value of the independent variable x to find the solution. In this case ? =10 _?8 and x = 3. From Fig. 19.1 one can see that the calculation accuracy is...