FirstHalf

# FirstHalf - MATH 529 – Mathematical Methods for Physical...

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Unformatted text preview: MATH 529 – Mathematical Methods for Physical Sciences II Christoph Kirsch Contents 0 Overview of the lecture 1 0.1 Partial differential equations . . . . . . . . . . . . . . . . . . . . . 2 0.2 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Review of Fourier analysis 4 1.1 Fourier transform: definition . . . . . . . . . . . . . . . . . . . . 5 1.2 Fourier transform: properties . . . . . . . . . . . . . . . . . . . . 5 1.3 Periodic functions: Fourier series . . . . . . . . . . . . . . . . . . 6 1.4 Application: solution of ODEs . . . . . . . . . . . . . . . . . . . 7 12 Partial Differential Equations (PDEs) 8 12.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 12.2 1D Wave Equation: Vibrating String . . . . . . . . . . . . . . . . 12 12.3 1D Wave Equation: Separation of Variables, Use of Fourier Series 13 12.4 D’Alembert’s Solution of the Wave Equation. Characteristics. . . 20 12.5 Heat Equation: Solution by Fourier Series . . . . . . . . . . . . . 24 12.6 1D Heat Equation: Solution by Fourier Integrals and Transforms 30 12.7 2D Wave Equation: Vibrating Membrane . . . . . . . . . . . . . 36 12.8 Rectangular Membrane. Double Fourier Series . . . . . . . . . . 38 12.9 Laplacian in Polar Coordinates. Circular Membrane. Fourier- Bessel Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 12.10Laplace’s Equation in Cylindrical and Spherical Coordinates. Po- tential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 12.11Solution of PDEs by Laplace Transforms . . . . . . . . . . . . . . 58 12.12Application: hydrogen-like atomic orbitals . . . . . . . . . . . . . 59 12.13Application: Unbounded domains and artificial boundaries. Dirichlet- to-Neumann map. . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Overview of the lecture This is the second half of a one year course on mathematical methods for phys- ical sciences. The course web site is http://www.unc.edu/ ~ ckirsch/MATH529 , 1 which contains the course syllabus, course material such as these lecture notes, as well as the homework assignments. Please check this web site regularly for updates. The two topics covered in the lecture are • partial differential equations • complex analysis The course is based on the textbook Erwin Kreyszig: Advanced Engineering Mathematics. 9th Edition; Wiley, 2006 We will basically go through chapters 12–18 of this textbook. It is presumed that students are familiar with multivariable calculus. Fourier series, integrals and transforms will be reviewed as needed. This lecture will connect naturally to the first half, MATH 528. Students who have not attended that lecture may notice a larger gap in the first few weeks....
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FirstHalf - MATH 529 – Mathematical Methods for Physical...

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