| CODE | MAT3712 | ||||||||
| TITLE | Partial Differential Equations and Calculus of Variations | ||||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 6 | ||||||||
| ECTS CREDITS | 5 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | - Partial differential equations: - First order quasi-linear equations - Method of characteristics - One dimensional wave equation, d’Alembert’s solution. - Elliptic equations: - Gravitation - Laplace’s equation - Poisson’s equation - Harmonic functions. - Calculus of variations: - Motivating examples - Continuous and piecewise differentiable solution - The Euler-Lagrange equation - Problems with fixed and non-fixed endpoints - Problems with Constraints - Necessary conditions for a minimum and corner conditions. Study Unit Aims: Linear partial differential equations are ubiquitous in mathematical physics; for example, classical gravitation, electrostatics, heat, diffusion, and waves all obey such equations. This study unit starts with quasi-linear first order partial differential equations and the powerful method of characteristics that can be used to solve them. Second order equations are studied next. Some can be reduced to a couple of first order equations. Among the irreducible equations, the potential equation is considered, of which the Laplace equation is a special case. Its solutions are the harmonic functions, which are generalizations of the real and imaginary parts of holomorphic functions. The second part of the study-unit is concerned with finding an optimal function that maximizes or minimizes a given integral constraint, for example, finding the shape of a roller-coaster which minimizes the time spent in falling. The remaining chapters treat variations on this basic problem, such as when there are constraints and/or variable endpoints, for example, to find that function with maximum area and a given length. Learning Outcomes: By the end of the study-unit, students will be able to: - Solve a quasi-linear first order partial differential equations and reducible second order equations, such as the one-dimensional wave equation, using the method of characteristics; - Use this method to solve general first order ordinary differential equations; - Understand the main properties of harmonic functions, including the mean value theorem; - Derive the Euler-Lagrange equation of an optimization problem; - Solve such equations with integral constraints (such as areas or arclengths), or non-fixed endpoints. Suggested Reading - Sneddon I., Elements of Partial Differential Equations, McGraw-Hill, 1957. - Haberman R., Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, Prentice Hall, New Jersey, 3rd Edition, 1998. - Farbon S., Partial Differential Equations for scientists and Engineers, Dover Publications, 1993. - Clegg J.C., Calculus of Variations, Oliver and Boyd, 1968. - Pars L.A., An Introduction to the Calculus of Variations, Heinemann, 1962. - Pinch E.R., Optimal Control and the Calculus of Variations, Oxford University Press, 1993. |
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| ADDITIONAL NOTES | Follows from: MAT3711 Leads to: MAT5713, MAT5616 |
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| STUDY-UNIT TYPE | Lecture | ||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | Joseph Muscat |
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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 2024/5. It may be subject to change in subsequent years. |
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