MATH 535 | Course Introduction and Application Information - Applied Mathematics and Statistics – With Thesis

Course Name

Wavelet Based Methods for Ordinary Differential Equations With MATLAB

Code	Semester	Theory (hour/week)	Application/Lab (hour/week)	Local Credits	ECTS
MATH 535	Fall/Spring	3	0	3	7.5

Prerequisites	None
Course Language	English
Course Type	Elective
Course Level	Second Cycle
Mode of Delivery	-
Teaching Methods and Techniques of the Course	Problem Solving Lecture / Presentation
National Occupation Classification	-
Course Coordinator	Dr.Öğretim Üyesi Demet ERSOY ÖZDEK
Course Lecturer(s)	Dr.Öğretim Üyesi Demet ERSOY ÖZDEK
Assistant(s)	Arş.Gör. Merve KAHRAMAN ARİMAN

Course Objectives	This course aims to provide a solid understanding of orthogonal polynomials and wavelet-based numerical methods for solving ordinary differential equations. It covers the use of these numerical techniques for a wide variety of problems in science, engineering, and other fields, and emphasizes the evaluation of their accuracy and convergence.
Learning Outcomes	The students who succeeded in this course; Explain the theory and properties of orthogonal polynomials. Find orthonormal bases of orthogonal polynomials. Solve differential equations using orthogonal polynomials for. Apply wavelet-based approaches in numerical solutions. Analyze the accuracy and convergence of polynomial and wavelet-based approximations. Solve ordinary differential equations using a computer program.
Course Description	This course focuses on the theory and applications of orthogonal polynomials and wavelet-based numerical methods for solving ordinary differential equations. The course covers the use of orthogonal polynomials and wavelet-based approximation techniques in solving ordinary differential equations and error and convergence analysis.
Related Sustainable Development Goals

Course Category	Core Courses
	Major Area Courses
	Supportive Courses
	Media and Management Skills Courses
	Transferable Skill Courses

Week	Subjects	Related Preparation
1	Introduction to Orthogonal Polynomials and Their Computational Applications	Arfken, G. B. and Weber H.J. (2013). Mathematical Methods for Physicists. 7th ed. Amsterdam: Elsevier. (Chapter 10)
2	Taylor Polynomials, Their Properties and Numerical Approximation; MATLAB-based Computation of Taylor Expansions and Error Analysis	Keskin, A. Ü. (2019). Ordinary differential equations for engineers: Problems with MATLAB solutions. Springer. https://doi.org/10.1007/978-3-319-95243-7 (Chapter 4)
3	Legendre Differential Equations and Legendre Polynomials; Generation and Visualization of Legendre Polynomials via MATLAB	Arfken, G. B. and Weber H.J. (2013). Mathematical Methods for Physicists. 7th ed. Amsterdam: Elsevier. (Chapter 12)
4	Associated Legendre Polynomials; Numerical Evaluation of Associated Legendre Functions Using MATLAB	Arfken, G. B. and Weber H.J. (2013). Mathematical Methods for Physicists. 7th ed. Amsterdam: Elsevier. (Chapter 12)
5	Chebyshev Polynomials; Numerical generation of Chebyshev Nodes Using MATLAB	Keskin, A. Ü. (2019). Ordinary differential equations for engineers: Problems with MATLAB solutions. Springer. https://doi.org/10.1007/978-3-319-95243-7 (Chapter 5)
6	Laguerre and Hermite Polynomials; Computational Experiments With Laguerre and Hermite polynomials	Arfken, G. B. and Weber H.J. (2013). Mathematical Methods for Physicists. 7th ed. Amsterdam: Elsevier. (Chapter 13)
7	Gram–Schmidt Orthogonalization and Weighted Inner Products; Numerical Orthogonalization procedures in MATLAB	Arfken, G. B. and Weber H.J. (2013). Mathematical Methods for Physicists. 7th ed. Amsterdam: Elsevier. (Chapter 10)
8	Collocation Methods for Solving Ordinary Differential Equations; Implementation of Collocation Schemes in MATLAB	Frazier, M. W. (1999). An Introduction to Wavelets Through Linear Algebra. Springer-Verlag. (Chapter 5)
9	Collocation Methods for Solving Ordinary Differential Equations; Implementation of Collocation Schemes in MATLAB	Frazier, M. W. (1999). An Introduction to Wavelets Through Linear Algebra. Springer-Verlag. (Chapter 5)
10	Error and Convergence Analysis; Numerical Error Norms and Convergence Rates	Keskin, A. Ü. (2019). Ordinary differential equations for engineers: Problems with MATLAB solutions. Springer. https://doi.org/10.1007/978-3-319-95243-7 (Chapter 8)
11	Introduction to Wavelets, Legendre Wavelets and Operational Matrices With Their MATLAB-based Representations	Frazier, M. W. (1999). An Introduction to Wavelets Through Linear Algebra. Springer-Verlag. (Chapter 5)
12	Chebyshev Wavelets and Operational Matrices with Their MATLAB-based Representations	Frazier, M. W. (1999). An Introduction to Wavelets Through Linear Algebra. Springer-Verlag. (Chapter 5)
13	Laguerre Wavelets and Operational Matrices with Their MATLAB-based Representations	Mason, J. C. and Handscomb, D. C. (2002). Chebyshev Polynomials 1st ed. Chapman & Hall/CRC. (Chapter 10)
14	Solving Ordinary Differential Equations with Wavelet-Based Numerical Methods; MATLAB Implementations and Numerical Experiments	Mason, J. C. and Handscomb, D. C. (2002). Chebyshev Polynomials 1st ed. Chapman & Hall/CRC. (Chapter 10)
15	Semester Review
16	Final Exam

Course Notes/Textbooks

Arfken, G. B. and Weber H.J. (2013). Mathematical Methods for Physicists. 7th ed. Amsterdam: Elsevier. ISBN-13: 978-0-12-384654-9

Keskin, A. Ü. (2019). Ordinary differential equations for engineers: Problems with MATLAB solutions. Springer. ISBN: 978-3-319-95242-0. https://doi.org/10.1007/978-3-319-95243-7

Frazier, M. W. (1999). An Introduction to Wavelets Through Linear Algebra. Springer-Verlag. ISBN-13: 978-0-387-98639-5.

Mason, J. C. and Handscomb, D. C. (2002). Chebyshev Polynomials 1st ed. Chapman & Hall/CRC. ISBN 978-0-420-03611-4.

Suggested Readings/Materials

Burden, R. L. and Faires, J. D. (2010). Numerical Analysis, 9th ed. Brooks & Cole, Cengage Learning. ISBN-13: 978-0-538-73351-9.

Timothy S. (2012). Numerical Analysis, 2nd ed., Pearson.

ISBN-13: 978-0-321-78367-7

Kreyszig, E. O. (2007). Introductory Functional Analysis with Applications. Wiley India. ISBN 978-8126511914.

Ahmadi, M. R. and Adibi, H. (2007). The Chebyshev Tau Technique for the Solution of Laplace's Equation. Applied Mathematics and Computation, 184(2), 895–900.

Ibraheem, G. H. and Al-Jawary, M. A. (2020). The operational matrix of Legendre polynomials for solving thin film flow problems. Alexandria Engineering Journal, 59(5), 4027–4033.

Baishya, C. and Veeresha, P. (2021). Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 477(2254), 20210438. https://doi.org/10.1098/rspa.2021.0438

Gupta, A. K., Saha Ray, S. S. (2014). Wavelet methods for solving fractional order differential equations. Mathematical Problems in Engineering, 2014, Article ID 140453. https://doi.org/10.1155/2014/140453

Semester Activities	Number	Weigthing
Participation
Laboratory / Application
Field Work
Quizzes / Studio Critiques
Portfolio
Homework / Assignments	1	10
Presentation / Jury	1	20
Project	1	20
Seminar / Workshop
Oral Exams
Midterm
Final Exam	1	50
Total

Weighting of Semester Activities on the Final Grade	3	50
Weighting of End-of-Semester Activities on the Final Grade	1	50
Total

Semester Activities	Number	Duration (Hours)	Workload
Theoretical Course Hours (Including exam week: 16 x total hours)	16	3	48
Laboratory / Application Hours (Including exam week: '.16.' x total hours)	16		0
Study Hours Out of Class	14	5	70
Field Work			0
Quizzes / Studio Critiques			0
Portfolio			0
Homework / Assignments	1	10	10
Presentation / Jury	1	20	20
Project	1	33	33
Seminar / Workshop			0
Oral Exam			0
Midterms			0
Final Exam	1	44	44
		Total	225

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Program Competencies/Outcomes

* Contribution Level

1

2

3

4

5

1

To be able to demonstrate independent and critical thinking in Applied Mathematics and Statistics.

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X

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2

To be able to define problems in Applied Mathematics/Statistics and verify whether they are mathematically/statistically consistent.

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3

To be able to analyse and solve real life problems using applied methods and interdisciplinary approach of Mathematics/Statistics.

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X

4

To be able to independently conduct, conclude, and report on specialized research in Applied Mathematics and Statistics.

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X

-

5

To be able to efficiently use national and international resources, for staying updated in the field, communicating with colleagues, and following the related literature.

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6

To be able to develop proficiency in using computer software widely utilized in the fields of Applied Mathematics and Statistics.

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X

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7

To be able to evaluate solution processes efficiently using mathematical reasoning and modeling in order to contribute to the solutions of social and scientific problems.

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8

To be able to synthesize theoretical frameworks with practical applications through mathematical and statistical methods.

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9

To be able to develop strategies, policies and plans for problems and research areas in Applied Mathematics/Statistics in order to interpret the results and translate them into practice.

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10

To be able to translate key topics, events, and phenomena in Applied Mathematics and Statistics into the context of other scientific disciplines.

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11

To be able to engage in lifelong learning by continuously updating and improving knowledge and skills in Applied Mathematics and Statistics.

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MATH 535 | Course Introduction and Application Information

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