REQUEST
A polluted reservoir exacerbated by large-scale earth disturbances and drainage overrun from a nearby mining company was found to have an unacceptable concentration of bacteria and other acidic pollutants. Government regulations prompted the company to initiate a complete study of the situation. We were called to study the particulars and asked to find acceptable levels of concentration of the pollutants as fresh water would replenish the reservoir. We were given starting conditions, such as, the differential equation that governs the concentration of the pollutant as a function of time (in weeks) and the initial concentration of pollutant in parts per volume.

Our task was to find concentration levels of the pollutant for specific time ranges (after 4 weeks, after 8 weeks, and so forth).

Note:
This exact problem was originally solved using Euler's Method. By using the Second-Order Runge-Kutta Method we basically reinforced our first findings and wanted to show how we can obtain better approximations. For a visual comparison of approximations (check IMPLEMENTATION result tables under Euler's Method and Second-Order Runga-Kutta).

SOLUTION
To numerically solve the problem we applied the first-order differential equation using Second-Order Runge-Kutta Method (illustrated as sample model of execution under SECOND-ORDER RUNGE-KUTTA METHOD and IMPLEMENTATION) to solve the problem . We set the initial step size of 2.5 weeks and found approximate concentration of pollutant for different time periods. A report listing levels of pollutant concentrations was printed as wells as a graphical representation of the findings.

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           static void TestRungeKutta2();
              {
                 double h = 0.001;
                 double x0 = 0;
                 double y0 = 1.0;
                 double result = y0;
                 double trct = 0;
                 for (int i = 0; i < 11; i++)
                 {
                    double x = 0.1 * i;
                    result = ODE.RungeKutta2(f, x0, result, h, x);
                    double exact = 2.0 * Math.Exp(x) - x - 1.0;
                    double error = ((exact - result) / exact) * 100000;
                    int r = (int)(error * 10000);
                    trct = r / 10000.0;
                    listBox1.Items.Add(" " + x );
                    listBox2.Items.Add(" " + result);
                    listBox3.Items.Add(" " + exact);
                    listBox4.Items.Add(" " + trct);
                    x0 = x;
                 }
                 this.Text = "Second-Order Runge-Kutta Method / KeystoneMiningPost";               }