ODE Euler Method - Keystone Mining Post

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Differential Equations

~ RungeKutta-Fehlberg

EULER METHOD

OUR SOLUTIONS

SOLUTIONS
Differential Equations

Apply differential equations to solve case problems depending on specific requirements. Use differential equations numerical methods available such as:

Second-Order Runga-Kutta traditionally associated with a class of methods for the numerical integration of ordinary differential equations.
Fourth-Order Runga-Kutta to achieve great accuracy with less computational effort.
High-Order Differential Equations most problems in engineering governed by differential equations are either high-order equations or coupled differential equation systems.

We include our real-world solution to a Reservoir Pollution problem request.

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).

SOLUTION
To numerically solve the problem we applied the first-order differential equation using Euler's Method (illustrated as sample model of execution under EULER'S 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.

EULER'S TEST

Testing Euler's Method

In order to test Euler Method as defined before a new method TestEuler() has been added and executed:

►

           static void TestEuler();
              {
                 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.Euler(f, x0, result, h, x);
                    double exact = 2.0 * Math.Exp(x) - x - 1.0;
                    double error = ((exact - result) / exact) * 100;
                    int r = (int)(error * 1000000);
                    trct = r / 1000000.0;
                    listBox1.Items.Add(" " + x );
                    listBox2.Items.Add(" " + result);
                    listBox3.Items.Add(" " + exact);
                    listBox4.Items.Add(" " + trct);
                    x0 = x;
                 }
                 this.Text = "Euler Method Approximation / KeystoneMiningPost";
              }

Results are shown below.

METHOD
Euler Method

The simplest method for the numerical integration of a first-order ordinary differential equation is Euler's method. Consider the problem:

dx/dy = f (x, y) = g (x)

This method assume that for a small distance ∆x along the x-axis from some initial point x₀, the function g(x) is a constant equal to g(x₀).

dx/dy |_x=x₀ = f (x₀, y₀) = g(x₀)

Assuming that g(x) = g(x₀) for all values of x between x₀ and x₁ = x + ∆x, then the change in y corresponding to the small change ∆x in x is given approximately by:

∆x/∆y = f (x₀, y₀)

If y₁ is used to denote y₀ + ∆y, then the above equation becomes:

y₁ = y₀ + ∆x f (x₀, y₀) = y₀ + h f (x₀, y₀)

where h = ∆x; y₁ can then be calculated from the above equation. This completes the solution process for one step along the x-axis. The process can be repeated by using the previous solution as the starting values for the current step, yielding a general solution:

y₂ = y₁ + h f (x₁, y₁)
y₂ = y₂ + h f (x₂, y₂)
·
·
·
y_{n + 1} = y_n + h f (x_n, y_n)

This is known as Euler's method. The calculation process, step by step, along the x-axis from the initial point x₀ to the required finishing point.

One of our main algorithm implementation mechanisms include writing algorithms using differential equations. The next tab shows the solution and numerical implementation of a first-order differential equation.