Introduction to Partial Differential Equations [Herman]

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Russell L. Herman’s Introduction to Partial Differential Equations is a broad, application-oriented textbook that develops the main analytical and numerical techniques used to study PDEs in mathematics, physics, and engineering. It begins with first-order equations and the method of characteristics, including conservation laws, shock and rarefaction waves, then introduces the classical second-order heat, wave, and Laplace equations through boundary-value problems, separation of variables, and d’Alembert’s solution. 

The book progressively builds the mathematical machinery needed for more advanced problems, covering Fourier series, Sturm–Liouville theory, Legendre and Bessel functions, higher-dimensional equations, spherical harmonics, Green’s functions, complex-variable methods, Fourier and Laplace transforms, and finally finite-difference numerical methods with discussions of stability and error. Throughout, theory is tied to concrete models such as vibrating strings and membranes, heat flow, fluid flow, traffic flow, quantum mechanics, and diffusion, making the text essentially a comprehensive two-semester introduction to both the theory and practical solution of partial differential equations. 

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