| CODE | MAT3717 | ||||||
| TITLE | Partial Differencial Equations & Fourier Analysis | ||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||
| EQF/MQF LEVEL | Not Applicable | ||||||
| ECTS CREDITS | 5 | ||||||
| DEPARTMENT | Mathematics | ||||||
| DESCRIPTION | The following outlines the course content, but it is expected that not all of the later topics in the list will be taught in each year. 1. First order PDEs and the Method of Characteristics • Applications to Second Order Equations 2. The One Dimensional Wave Equation and d'Alembert's Solution 3. The Laplacian, Laplace's Equation and Harmonic Functions 4. Fourier Transforms • Schwartz Space, Fourier Inversion and Plancherel's Theorem 5. Fourier Series • Pointwise, Cesàro, and Abel convergence, application to the Dirichlet problem on the unit disk 6. The Heat Equation 7. The Method of Separation of Variables • Special Functions such as Bessel Functions, Spherical Harmonics, Legendre functions 8. An elementary introduction to distributions 9. Green's Functions Study-Unit Aims: Partial differential equations lie at the heart of most mathematical models of real-world systems, e.g. planetary motion, electrostatics, heat conduction and wave propagation. In this study-unit, students will be introduced to the theory of partial differential equations and will learn different techniques to solve some categories of these equations. Topics such as Fourier Transforms, Fourier Series, Distributions and various types of Special Functions shall be discussed, especially in relation to their applications to solving PDEs. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - Discuss the application of the method of characteristics in the study of differential equations; - Describe basic properties of the Laplacian operator and harmonic functions; - State and prove properties of some of the special functions that arise as solutions of important differential equations of mathematical physics; - State and prove important theorems about Fourier transforms such as the Fourier Inversion Theorem and the Plancherel Theorem; - Describe the notion of Green's functions and how they are related to solutions of PDEs. 2. Skills: By the end of the study-unit the student will be able to: - Use the method of characteristics to solve some categories of first-order and second-order PDEs; - Use the method of characteristics to solve ordinary differential equations; - State and prove basic properties of harmonic functions, such as the mean value property; - Apply the notion of Abel convergence of Fourier series to solve the Dirichlet problem in the unit disk; - Apply the method of separation variables to PDEs where the solution involves special functions such as Bessel functions, spherical harmonics and Legendre functions; - Use Green's functions to solve PDEs. Main Text/s and any supplementary readings: View reading list |
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| ADDITIONAL NOTES | Pre-requisite Study-unit: MAT2213 Co-requisite Study-unit: MAT3211 |
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| STUDY-UNIT TYPE | Lecture and Tutorial | ||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | James L. Borg |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2026/7. It may be subject to change in subsequent years. |
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