Introduction to Differential Equations

Posted: August 25th, 2021

Student’s Name

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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].

Problem 1

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:

1. Based on the above information, the estimated IVP solutions are as shown in the table below:

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)

ii.                  Plot Euler’s Method Approximation

Figure 1: Euler’s Method Approximation

1. Plot Absolute Error between Approximation and Actual Value on Semi log Plot

Figure 2: Absolute error semi-log plot

Part 1 B: Approximation of IVP solutions using Improved Euler’s method based on the following conditions:

1. Improved Euler Method

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

1. Plot Improved Euler’s Approximation

                               Figure 3: Improved Euler’s Approximation for values of Y

1. Plot Absolute Error Using Semi-Log Plot

Figure 4: Absolute error semi-log plot

Part 1 C: Approximate the solution to the IVP using the RK4 method with the following conditions

1. The Following Table Is The RK4 Approximation

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

1. Plot the RK4 Method Approximation

Figure 7: RK4 method approximation plot

1. Absolute Error RK4 Method Approximation Semi-Log Plot

Figure 8: Absolute error RK4 approximation semi-log plot

Problem 2

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

1. Solution (See excel).
2. Plot all Improved Euler’s Method Approximations

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

iii.                Discussion

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

1. Solutions (See Excel worksheet)

1. Plot the RK4 Approximations for all 4 times steps

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

1. Discussion

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.