MA2552 Introduction to Computing (DLI) 2023/24

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Computational Project

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Aims and Intended Learning Outcomes

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The aims of the Project are to describe methods for solving given computational problems, develop and test Matlab code implementing the methods, and demonstrate application

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of the code to solving a specific computational problem. In this Project, you be will be required to demonstrate

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• ability to investigate a topic through guided independent research, using resources

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available on the internet and/or in the library;

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• understanding of the researched material;

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• implementation of the described methods in Matlab;

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• use of the implemented methods on test examples;

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• ability to present the studied topic and your computations in a written Project Report.

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Plagiarism and Declaration

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• This report should be your independent work. You should not seek help from other

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students or provide such help to other students. All sources you used in preparing your

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report should be listed in the References section at the end of your report and referred

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to as necessary throughout the report.

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• Your Project Report must contain the following Declaration (after the title page):

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DECLARATION

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All sentences or passages quoted in this Project Report from other people’s work have

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been specifically acknowledged by clear and specific cross referencing to author, work and

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page(s), or website link. I understand that failure to do so amounts to plagiarism and

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will be considered grounds for failure in this module and the degree as a whole.

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Name:

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Signed: (name, if submitted electronically)

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Date:

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Project Report

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The report should be about 6-8 pages long, written in Word or Latex. Equations should

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be properly formatted and cross-referenced, if necessary. All the code should be included in

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the report. Copy and paste from MATLAB Editor or Command Window and choose ‘Courier

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New’ or another fixed-width font. The Report should be submitted via Blackboard in a single

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file (Word document or Adobe PDF) and contain answers to the following questions:

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1

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MA2552 Introduction to Computing (DLI) 2023/24

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Part 0: Context

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Let f(x) be a periodic function. The goal of this project is to implement a numerical method

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for solving the following family of ordinary differential equations (O.D.E):

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an

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d

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nu(x)

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dxn

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+ an−1

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d

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n−1u(x)

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dxn−1

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+ . . . + a0u(x) = f(x), (1)

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where ak, k = 0, · · · , n, are real-valued constants. The differential equation is complemented

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with periodic boundary conditions:

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d

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ku(−π)

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dxk

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=

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d

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ku(π)

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dxk

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for k = 0, · · · , n − 1.

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We aim to solve this problem using a trigonometric function expansion.

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Part 1: Basis of trigonometric functions

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Let u(x) be a periodic function with period 2π. There exist coefficients α0, α1, α2, . . ., and

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β1, β2, . . . such that

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u(x) = X∞

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k=0

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αk cos(kx) +X∞

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1

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βk sin(kx).

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The coefficients αk and βk can be found using the following orthogonality properties:

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Z π

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−π

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cos(kx) sin(nx) dx = 0, for any k, n

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Z π

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−π

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cos(kx) cos(nx) dx =

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

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

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

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0 if k ̸= n

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π if k = n ̸= 0

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2π if k = n = 0.

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Z π

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−π

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sin(kx) sin(nx) dx =

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(

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0 if k ̸= n

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π if k = n ̸= 0.

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1. Implement a function that takes as an input two function handles f and g, and an

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array x, and outputs the integral

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1

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π

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Z π

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−π

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f(x)g(x) dx,

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using your own implementation of the Simpson’s rule scheme. Corroborate numerically

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the orthogonality properties above for different values of k and n.

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2. Show that

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αk =

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(

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1

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π

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R π

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−π

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u(x) cos(kx) dx if k ̸= 0

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1

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2π

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R π

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−π

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u(x) dx if k = 0

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βk =

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1

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π

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Z π

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π

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u(x) sin(kx) dx.

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2

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MA2552 Introduction to Computing (DLI) 2023/24

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3. Using question 1 and 2, write a function that given a function handle u and an integer

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m, outputs the array [α0, α1 . . . , αm, β1, . . . , βm].

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4. Write a function that given an array [α0, α1 . . . , αm, β1, . . . , βm], outputs (in the form

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of an array) the truncated series

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um(x) := Xm

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k=0

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αk cos(kx) +Xm

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k=1

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βk sin(kx), (2)

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where x is a linspace array on the interval [−π, π].

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5. Using the function from question 3, compute the truncated series um(x) of the following

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functions:

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• u(x) = sin3

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(x)

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• u(x) = |x|

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• u(x) = (

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x + π, for x ∈ [−π, 0]

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x − π, for x ∈ [0, π]

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,

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and using question 4, plot u(x) and um(x) for different values of m.

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6. Carry out a study of the error between u(x) and um(x) for ∥u(x)−um(x)∥p with p = 2

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and then with p = ∞. What do you observe?

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Part 2: Solving the O.D.E

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Any given periodic function u(x) can be well approximated by its truncate series expansion (2) if m is large enough. Thus, to solve the ordinary differential equation (1)

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one can approximate u(x) by um(x):

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u(x) ≈

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Xm

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k=0

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αk cos(kx) +Xm

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k=1

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βk sin(kx),

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Since um(x) is completely determined by its coefficients [α0, α1 . . . , αm, β1, . . . , βm],

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to solve (1) numerically, one could build a system of equations for determining these

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coefficients.

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7. Explain why under the above approximation, the boundary conditions of (1) are automatically satisfied.

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8. We have that

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dum(x)

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dx =

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Xm

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k=0

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γk cos(kx) +Xm

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k=1

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ηk sin(kx)

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Write a function that takes as input the integer m, and outputs a square matrix D that

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maps the coefficients [α0, . . . , αm, β1, . . . , βm] to the coefficients [γ0, . . . , γm, η1, . . . , ηm].

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3

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MA2552 Introduction to Computing (DLI) 2023/24

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9. Write a function that given a function handler f and the constants ak, solves the

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O.D.E. (1). Note that some systems might have an infinite number of solutions. In

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that case your function should be able identify such cases.

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10. u(x) = cos(sin(x)) is the exact solution for f(x) = sin(x) sin(sin(x))−cos(sin(x)) (cos2

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(x) + 1),

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with a2 = 1, a0 = −1 and ak = 0 otherwise. Plot the p = 2 error between your numerical solution and u(x) for m = 1, 2, . . .. Use a log-scale for the y-axis. At what rate

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does your numerical solution converge to the exact solution?

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11. Show your numerical solution for different f(x) and different ak of your choice.

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