Modify the Euler’s method MATLAB code to plot and compare the approximate solution using the modified Euler method, for a step size of 0.1 and 0.01
Added by Elizabeth G.
Step 1
First, we need to modify the Euler's method code to implement the modified Euler method. The modified Euler method is also known as the improved Euler method or the Heun's method. It is a second-order numerical method that uses a midpoint approximation to improve Show more…
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Define the total error as: Total Error = ∑ (step size * Absolute Error(i)) Here N is the total number of steps taken when approximating Euler's method using a specified step size. This means that if are to approximate the solution for t ∈ [0, 5] using a step size of 1 then the total number of steps would be N = 5. Compute the total error for the three different step sizes, h = 1, 0.1, 0.01 Using a smaller time step size takes more computational time and sometimes the increase in accuracy is not worth the increase in computational cost. Make a log-log plot (in matlab loglog(x,y)) of the total error as a function of the number of steps, N when N = 1, 2, 3, ..., 10000 and determine the best number of steps to take. Justify your answer.
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Use Euler's Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use $n$ steps of size $h$. $$y^{\prime}=e^{x y}, \quad y(0)=1, \quad n=10, \quad h=0.1$$
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Apply Euler's method with $h=0.1$ to the initial value problem $y^{\prime}=y^{2}-1, y(0)=3$ and estimate $y(0.5) .$ Repeat with $h=0.05$ and $h=0.01 .$ In general, Euler's method is more accurate with smaller $h$ -values. Conjecture how the exact solution behaves in this example. (This is explored further in exercises $34-36 .)$
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