Introduction and History
Leonhard Euler, one of the most prolific mathematicians of the 18th Century, invented the Euler’s method. To those unaware, he was a very eminent mathematician, and his influence on the world of mathematics is something one cannot oversee.
Above quote is rightly describes the importance of how great the work of Euler was in the field of mathematics. He has written about 500 books and papers during his lifetime. Euler’s method is not the just one concept named after him. There are many other concepts named after him; Euler’s constant, Euler line of triangle, Euler’s equation of motion, Eulerian graphs, to name a few. He has worked on a variety of area including astronomy, logarithms, calculus, the motion of the moon and plenty more. It is not surprising that he is such a high figure of influence in the world of mathematics. His concepts made mathematical applications so understandable for everyone today. Euler’s method is one such concept which we are going to implement using MATLAB in this blog!
A Quick Understanding of Euler’s Method
Euler’s method is a technique to solve first order initial value problems (IVP), numerically. The standard form of equation for Euler’s method is given as
where y(x0) = y0 is the initial value. We need to find the value of y at point ‘n’ i.e. y(xn).
Right now, we know only one point (x0, y0). The blue graph below is the ‘Exact’ (Solution Curve in Diagram) and the red / tangent line is the Euler’s method. Our aim is to find the set of points along with this blue graph. Let us see the steps to find the solution.
- We use the tangent line (red line) as an approximation at the only known point to us, (x0, y0).
- We move a short distance ahead on this red line which is close to our true solution (x1, y1). This short distance is the step size and denoted by ‘h’.
- For better approximate values, keep the value of h as small as possible.
- We now use the point (x1, y1) to plot the point (x2, y2) in the similar fashion.
We will use this method to find the solution to a differential equation of a basic RC circuit. Let us see how!
We’re going to approximate the charge on the capacitor in the circuit given below and plot the graph using MATLAB. Then compare the Exact and the Euler’s graphs.
Differential Equation of the Given RC Circuit
In the circuit, R represents resistance in Ohms, C is capacitance in Farads, i(t) is current through the resistor measured in Amperes and the capacitor has the voltage drop Q/C. Q(t) is the charge on capacitor at time t. When the switch is closed at time t = 0, by applying Kirchhoff’s law we get the equation:
A few math steps will give us the equation in the form of Euler’s method as below:
We’re going to use equation (2) to write the code and plot the approximation graph in MATLAB
- E = 200 V. R = 1500 Ω. C = 0.0025F.
- Initial Condition: Q = 0 at t = 0
- Find Q at t = 1.
Steps to MATLAB Code
- Defining the equation 2 in MATLAB
- Set the initial and final time values a = 0, b = 1 and y0 = 0;
- Number of steps N = 10
- Therefore: Step size h = (b-a)/N
- Run the for loop till N number of steps
- (increment t by h)
- Plot (t, y) – value of Q against time t
- Repeat the same steps for True value with the following equation. This is the first order differential through mathematical analysis.
You may also plot a table to display value of Q at time ‘t’ for N number of steps. To do so you may use the function table in MATLAB. We’ve shown the values for both the methods below!
Exact & EulerValues
Charge at Capacitor – Exact
Charge at Capacitor - Euler
The graph and tables above show how accurately we could calculate the charge at the capacitor in above RC circuit. Hence, we could safely use Euler’s method to solve the first order differential equations even in the problems where we do not know the exact solution.
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eulers method by matlab has advantage for research work…