A Course in Analysis Vol. IV: Fourier Analysis, Ordinary Differential Equations, Calculus of Variations
Authors : Niels Jacob (Swansea University, UK) and Kristian P Evans (Swansea University, UK)
Publisher : World Scientific
ISBN : 978-981-3273-51-1 (Hardcover), 978-981-3274-52-5 (Softcover)
Our treatment of ordinary differential equations starts with a discussion of some classical methods to obtain explicit integrals, followed by the existence theorems of Picard–Lindelöf and Peano which are proved by fixed point arguments. Linear systems are treated in great detail and we start a first discussion on boundary value problems. In particular, we look at Sturm–Liouville problems and orthogonal expansions. We also handle the hypergeometric differential equations (using complex methods) and their relations to special functions in mathematical physics. Some qualitative aspects are treated too, e.g. stability results (Ljapunov functions), phase diagrams, or flows.
Our introduction to the calculus of variations includes a discussion of the Euler–Lagrange equations, the Legendre theory of necessary and sufficient conditions, and aspects of the Hamilton–Jacobi theory. Related first order partial differential equations are treated in more detail.
The text serves as a companion to lecture courses, and it is also suitable for self-study. The text is complemented by ca. 260 problems with detailed solutions.
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