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# 401-0363-10L  Analysis III

 Semester Autumn Semester 2015 Lecturers A. Iozzi Periodicity yearly recurring course Language of instruction English (lecture), German (exercise)

 Abstract Introduction to partial differential equations. Differential equations which are important in applications are classified and solved. Elliptic, parabolic and hyperbolic differential equations are treated. The following mathematical tools are introduced: Laplace transforms, Fourier series, separation of variables, methods of characteristics. Objective Mathematical treatment of problems in science and engineering. To understand the properties of the different types of partlial differentail equations. Content Laplace Transforms:- Laplace Transform, Inverse Laplace Transform, Linearity, s-Shifting - Transforms of Derivatives and Integrals, ODEs- Unit Step Function, t-Shifting- Short Impulses, Dirac's Delta Function, Partial Fractions- Convolution, Integral Equations- Differentiation and Integration of TransformsFourier Series, Integrals and Transforms:- Fourier Series- Functions of Any Period p=2L- Even and Odd Functions, Half-Range Expansions- Forced Oscillations- Approximation by Trigonometric Polynomials- Fourier Integral- Fourier Cosine and Sine TransformPartial Differential Equations:- Basic Concepts- Modeling: Vibrating String, Wave Equation- Solution by separation of variables; use of Fourier series- D'Alembert Solution of Wave Equation, Characteristics- Heat Equation: Solution by Fourier Series- Heat Equation: Solutions by Fourier Integrals and Transforms- Modeling Membrane: Two Dimensional Wave Equation- Laplacian in Polar Coordinates: Circular Membrane, Fourier-Bessel Series- Solution of PDEs by Laplace Transform Literature E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 9. Auflage, 2011C. R. Wylie & L. Barrett, Advanced Engineering Mathematics, McGraw-Hill, 6th ed.G. Felder, Partielle Differenzialgleichungen für Ingenieurinnen und Ingenieure, hypertextuelle Notizen zur Vorlesung Analysis III im WS 2002/2003.Y. Pinchover, J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press, 2005For reference/complement of the Analysis I/II courses:Christian Blatter: Ingenieur-Analysis (Download PDF) Prerequisites / Notice Up-to-date information about this course can be found at:http://www.math.ethz.ch/education/bachelor/lectures/hs2013/other/analysis3_itet