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The aim of the assignment is that the student/group studies and applies numerical methods such as Euler's method, the Improved Euler's method and the Runge-Kutta method to solve first-order differential equations numerically. The main reference is Section 1.7 of the textbook "Notes on diffy Qs differential equations for engineers" by Lebl. The improved Euler's method is explained after Exercise 1.7.103 and the Runge-Kutta method is explained after Exercise 1.7.6. The assignment has a value of 53 points, and it is worth 15% of the total marks of the course. It consists of two parts: a powerpoint presentation and two exercises to be solved. The assignment is meant to completed by groups or individually. Students are encouraged to
join a group.

PowerPoint presentation

The student/group should create a PowerPoint presentation of up to eleven slides explaining the main ideas behind Euler's method, the Improved Euler's method and the Runge-Kutta method. The presentation should be clear enough so that any student from the class can understand its elements. It should explain the following topics for each method.

(1) Geometric reason behind the corresponding method. A graph should be presented (for each method).

(2) A clear explanation of the method, which should include one example. Use the same example for the three methods.

(3) A table comparing the numerical values of the three methods, as in the table below.

xn

yn

Euler

Improved Euler

Runge-Kutta

Exercises

The marks are computed as follows: 5 points for the exact solution, 3 5 points for the formulas and workings for h = 0.1, 5 points for the h = 0.1 table, and 5 points for the Excel spreadsheet.

Each student/group will solve two of the following four exercises. The solution of each ex- ercise should again be clear enough so that any student from the class can understand its basic elements. The assignment could be handwritten. Just make sure that your writing is clear.

Statement. Use Euler's method, the Improved Euler's method and the Runge-Kutta method to obtain a four-decimal approximation of the indicated value for two of the exercises below.

(1) For each method use h = 0.1.

(2) Provide the exact solution of each exercise.

(3) For each method, you need to provide the relevant formulas and workings to obtain the five xn and yn for h = 0.1. This part should be done manually.

(4) After running each method, provide a table as the one below.
Recall that the error is the difference between the actual solution and the approximate solution.

(5) The numerical solutions of each exercise should be implemented in Excel but for h = 0.01. Attach a copy of one Excel spreadsheet which follows the format of the table.

Table 1. Comparison of numerical methods with h = 0.1

xn

yn

Euler

Improved Euler

Runge-Kutta

Exact Value

Error

Ex 1.: yj = 2x 3y + 1, y(1) = 5; y(1.5).
Ex 2.: yj = e-y, y(0) = 0; y(0.5). (This is a separable equation)
Ex 3.: yj = 2xy, y(1) = 1; y(1.5). (This is a separable equation)
Ex 4.: yj = y - y2, y(0) = 0.5; y(0.5). (This is a separable equation) The method for selecting the exercises is the following.

(1) In a group, select the person whose surname comes first in alphabetical order, out of the surnames of the group members. If there is only one student, select his/her surname.

(2) Obtain the position of the first letter of the surname in the English alphabet. For instance, a surname starting with D has position 4, while a surname starting with F has position 6.

(3) To select the two exercises from the four exercises given above, obtain the remainder r of the integer division of the position of the surname and 3. For instance, 4 divided by 3 has remainder r = 1 and 5 has remainder r = 2. Solve the exercises r + 1 and r + 2. That is, if the remainder is r = 2, you should solve the exercises 3 and 4.

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