Posted: August 25th, 2021
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Introduction to Differential Equations
The generalized of the problem to be solved:
At t = 0, y = yo …. (Boundary condition)
How does y change with t?
Approximation solution using Euler’s Method:
… Add on h to progress one-step in t.
… Use the formula to progress one-step in y [Euler’s Method].
The Initial Value Problem (IVP) with y as the dependent variable and t an independent variable is described as , with and. However, the analytic solution for this IVP is:
Part 1 A: Approximation of IVP solutions using Euler’s method based on the following conditions:
Initial condition:
Euler’s Method: | |||
i | t | y | f (t,y) |
0.0 | 0.0 | 0.5 | 0.5 |
1.0 | 0.1 | 0.5 | 0.5 |
2.0 | 0.1 | 1.5 | 1.5 |
3.0 | 0.2 | 1.6 | 1.6 |
4.0 | 0.3 | 2.5 | 2.5 |
5.0 | 0.3 | 2.7 | 2.6 |
6.0 | 0.4 | 3.5 | 3.3 |
7.0 | 0.4 | 3.7 | 3.5 |
8.0 | 0.5 | 4.5 | 4.1 |
9.0 | 0.6 | 4.8 | 4.2 |
10.0 | 0.6 | 5.5 | 4.7 |
11.0 | 0.7 | 5.8 | 4.8 |
12.0 | 0.8 | 6.5 | 5.2 |
13.0 | 0.8 | 6.8 | 5.3 |
14.0 | 0.9 | 7.5 | 5.5 |
15.0 | 0.9 | 7.8 | 5.6 |
16.0 | 1.0 | 8.5 | 5.7 |
17.0 | 1.1 | 8.9 | 5.7 |
18.0 | 1.1 | 9.5 | 5.8 |
19.0 | 1.2 | 9.9 | 5.7 |
20.0 | 1.3 | 10.5 | 5.8 |
Table 1: IVP Solutions using Euler’s Method (See Excel Calculations)
Figure 1: Euler’s Method Approximation
Figure 2: Absolute error semi-log plot
Part 1 B: Approximation of IVP solutions using Improved Euler’s method based on the following conditions:
Using the trapezoidal rule, approximate the integral to obtain the following:
Since the formula relies on the knowledge that, which was being estimated in the first case, it is essential to first get the estimate for the set by applying Euler’s formula as follows;
Afterward, the prediction is used in the above formula to obtain a new estimate of referred to as the correction.
Hence, the improved Euler’s Method. The following table shows the results of improved Euler’s approximation;
Conditions;
i | t | Improved Approx. y | f (t,y) | Absolute Error |
0.0 | 0.0 | 0.5 | 0.5 | 0.00 |
1.0 | 0.1 | 0.5 | 0.5 | 0.00 |
2.0 | 0.1 | 0.6 | 0.6 | 0.01 |
3.0 | 0.2 | 0.6 | 0.6 | 0.01 |
4.0 | 0.3 | 0.6 | 0.7 | 0.02 |
5.0 | 0.3 | 0.7 | 0.7 | 0.02 |
6.0 | 0.4 | 0.7 | 0.8 | 0.02 |
7.0 | 0.4 | 0.8 | 0.8 | 0.02 |
8.0 | 0.5 | 0.8 | 0.9 | 0.02 |
9.0 | 0.6 | 0.9 | 0.9 | 0.02 |
10.0 | 0.6 | 1.0 | 1.0 | 0.01 |
11.0 | 0.7 | 1.0 | 1.0 | 0.00 |
12.0 | 0.8 | 1.1 | 1.1 | 0.02 |
13.0 | 0.8 | 1.1 | 1.1 | 0.04 |
14.0 | 0.9 | 1.2 | 1.2 | 0.06 |
15.0 | 0.9 | 1.3 | 1.2 | 0.08 |
16.0 | 1.0 | 1.4 | 1.2 | 0.11 |
17.0 | 1.1 | 1.4 | 1.3 | 0.14 |
18.0 | 1.1 | 1.5 | 1.3 | 0.18 |
19.0 | 1.2 | 1.6 | 1.3 | 0.21 |
20.0 | 1.3 | 1.7 | 1.4 | 0.26 |
Figure 3: Improved Euler’s Approximation for values of Y
Figure 4: Absolute error semi-log plot
Part 1 C: Approximate the solution to the IVP using the RK4 method with the following conditions
i | t | RK4 Approximation | f (t,y) | Absolute Error | k1 | k2 | k3 | k4 |
0.0 | 0.0 | 0.5 | 0.5 | 0.00 | 0.250 | 0.259 | 0.259 | 0.320 |
1.0 | 0.1 | 0.5 | 0.5 | 0.00 | 0.265 | 0.279 | 0.279 | 0.345 |
2.0 | 0.1 | 0.5 | 0.5 | 0.01 | 0.290 | 0.308 | 0.309 | 0.379 |
3.0 | 0.2 | 0.5 | 0.5 | 0.01 | 0.323 | 0.347 | 0.348 | 0.422 |
4.0 | 0.3 | 0.6 | 0.6 | 0.02 | 0.367 | 0.396 | 0.398 | 0.475 |
5.0 | 0.3 | 0.6 | 0.6 | 0.03 | 0.421 | 0.457 | 0.458 | 0.539 |
6.0 | 0.4 | 0.6 | 0.6 | 0.04 | 0.487 | 0.529 | 0.531 | 0.615 |
7.0 | 0.4 | 0.6 | 0.6 | 0.05 | 0.565 | 0.615 | 0.617 | 0.704 |
8.0 | 0.5 | 0.6 | 0.7 | 0.06 | 0.656 | 0.715 | 0.717 | 0.806 |
9.0 | 0.6 | 0.7 | 0.7 | 0.07 | 0.762 | 0.830 | 0.833 | 0.923 |
10.0 | 0.6 | 0.7 | 0.8 | 0.07 | 0.884 | 0.963 | 0.967 | 1.057 |
11.0 | 0.7 | 0.7 | 0.8 | 0.07 | 1.024 | 1.116 | 1.121 | 1.210 |
12.0 | 0.8 | 0.8 | 0.8 | 0.06 | 1.185 | 1.293 | 1.298 | 1.385 |
13.0 | 0.8 | 0.8 | 0.9 | 0.05 | 1.370 | 1.496 | 1.503 | 1.585 |
14.0 | 0.9 | 0.9 | 0.9 | 0.03 | 1.584 | 1.732 | 1.741 | 1.814 |
15.0 | 0.9 | 1.0 | 1.0 | 0.01 | 1.832 | 2.007 | 2.018 | 2.079 |
16.0 | 1.0 | 1.1 | 1.0 | 0.02 | 2.121 | 2.330 | 2.344 | 2.387 |
17.0 | 1.1 | 1.2 | 1.1 | 0.06 | 2.462 | 2.713 | 2.732 | 2.747 |
18.0 | 1.1 | 1.3 | 1.2 | 0.10 | 2.867 | 3.174 | 3.200 | 3.174 |
19.0 | 1.2 | 1.4 | 1.2 | 0.15 | 3.356 | 3.735 | 3.770 | 3.687 |
20.0 | 1.3 | 1.5 | 1.3 | 0.21 | 3.954 | 4.431 | 4.481 | 4.311 |
Table 2: RK4 Method Approximation
Figure 7: RK4 method approximation plot
Figure 8: Absolute error RK4 approximation semi-log plot
Given the IVP where y (t) is an independent variable in a function;
Part A: Approximate solution to IVP using Improved Euler’s method. Conditions given are;
Initial condition,
Time steps, h and
Figure 9: Improved Euler’s Approximation, h = 1/8
Figure 10: Improved Euler’s Approximation, h = 1/32
Figure 11: Improved Euler’s Approximation, h = 1/64
Figure 12: Improved Euler’s Approximation, h = 1/16
From the analysis, with every increase in the time steps, the dependent value is affected. Further, the accuracy of the estimation increases when the improved Euler’s approximation method is used as it helps reduce the outliers in estimation.
Part 2B: Approximate the solution to the IVP using the RK4 method with the following conditions;
Conditions given are;
Initial condition,
Time steps, h and
Figure 13: RK4 approximation plot, h = 1/8
Figure 14: RK4 approximation plot, h = 1/16
Figure 15: RK4 approximation plot, h = 1/32
Figure 16: RK4 approximation plot, h = 1/64
Unlike in the Improved Euler’s method of approximation, the RK4 case exhibits a unique trend in the behavior of dependent variables. According to the curves, as the time steps reduce, the curve almost adopts a uniform shape with limited changes in the trend over time.
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