**35.2****
****Solving Differential Equations** (.NET, C#, CSharp, VB, Visual Basic, F#)

Class **RungeKuttaSolver**
solves first order initial value differential equations by the Runge-Kutta
method. The solver computes the unknown function *y*
as set of tabulated values such that .

**Constructing RungeKuttaSolver
Instances**

Instances of **RungeKuttaSolver**
are constructed from the number of tabulated points and a nonzero value
*delta*. From the number of points and
the delta, the set is determined as .

For instance:

Code Example – C# ordinary differential equations (ODE)

int n = 2000;

double delta = .001;

var solver = new RungeKuttaSolver( n, delta );

Code Example – VB ordinary differential equations (ODE)

Dim N As Integer = 2000

Dim Delta As Double = 0.001

Dim Solver As New RungeKuttaSolver(N, Delta)

Optionally, the order of the Runge-Kutta method may also be specified, using the enum RungeKutterSolver.SolverOrder.

Code Example – C# ordinary differential equations (ODE)

var solver = new RungeKuttaSolver( n, delta, SolverOrder.First );

Code Example – VB ordinary differential equations (ODE)

Dim Solver As New RungeKuttaSolver(N, Delta, SolverOrder.First)

By default, the fourth order Runge-Kutta method (RK4) is used.

**Solving First Order Initial
Value Problems**

The Solve() method on
**RungeKuttaSolver** solves a given
**FirstOrderInitialValueProblem**
by the common Runge-Kutta method. The Solve()
method computes the unknown function *y*
as set of tabulated values such that . The tabulated
values are returned either as a KeyValuePair<double[],
double[]>, or as a CenterSpace.NMath.Core.TabulatedFunction
passed by reference. (See Section 13.5
for more information on tabulated functions.)

For example:

Code Example – C# ordinary differential equations (ODE)

KeyValuePair<double[], double[]> tabulatedValues =

solver.Solve( prob );

Code Example – VB ordinary differential equations (ODE)

Dim TabulatedValues As KeyValuePair(Of Double(), Double()) =

Solver.Solve(Prob)

or

Code Example – C# ordinary differential equations (ODE)

TabulatedFunction ftab = null;

solver.Solve( prob, ref ftab );

Code Example – VB ordinary differential equations (ODE)

Dim FTab As TabulatedFunction = Nothing

Solver.Solve(Prob, FTab)

Similarly, in the following code, results are returned
as a **LinearSpline**, a subclass of
**TabulatedFunction** that provides
linear interpolation between tabulated values:

Code Example – C# ordinary differential equations (ODE)

var spline = new LinearSpline();

solver.Solve( prob, ref spline );

Code Example – VB ordinary differential equations (ODE)

Dim Spline As New LinearSpline()

Solver.Solve(Prob, Spline)