Introduction to Partial Differential Equations and Hilbert Space Methods
This text covers all the major concepts and techniques in an elementary and clear style and explores some fields usually dealt with only in the advanced literature, also in a simplified manner. This edition includes new discussions on the three basic numerical methods, an introduction to computational fluid dynamics, and an introduction to Lie group methods for partial differential equations. All topics are treated as brief descriptions of needed results, not as condensed short courses. The text's practical approach will motivate students to pursue many areas of study, such as Fourier series, complex analysis, Lebesgue integration and functional analysis.
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applications approximation assume Banach spaces bifurcation theory boundary conditions bounded calculus called characteristic closed coefficients complete consider constant continuous convergence corresponding depending derivatives difference Dirichlet problem discussion domain eigenfunctions eigenvalue energy example exists fact Figure finite formally formula Fourier series fundamental further given Green's function Hilbert space identity illustrate important inequality initial value problem integral interval latter Lebesgue limit linear mathematical means method namely nonlinear norm Note obtain operator ordinary differential equations partial differential equations physical plane pointwise positive principle proof quantum mechanics question recall regular resulting satisfies scattering Section seen sense separation of variables shown solution solve square sums surface theorem theory transform un(x uniqueness usually variational vector wave zero