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This book is addressed to mathematics and physics students who want to develop an interdisciplinary view of mathematics, from the age of Riemann, Poincaré and Darboux to basic tools of modern mathematics. It enables them to acquire the sensibility necessary for the formulation and solution of difficult problems, with an emphasis on concepts, rigour and creativity. It consists of eight self-contained parts: ordinary differential equations; linear elliptic equations; calculus of variations; linear and non-linear hyperbolic equations; parabolic equations; Fuchsian functions and non-linear equations; the functional equations of number theory; pseudo-differential operators and pseudo-differential equations. The author leads readers through the original papers and introduces new concepts, with a selection of topics and examples that are of high pedagogical value.

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1 Ordinary differential equations.- 2 Linear elliptic equations.- 3 Calculus of variations.- 4 Linear and non-linear hyperbolic equations.- 5 Parabolic equations.- 6 Fuchsian functions.- 7 The Riemann -function.- 8 A window on modern theory.- 9 References.

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"This is a comprehensive textbook, making it well suited for beginner to intermediate graduate-level courses in partial differential equations. ... This book would benefit from the addition of exercise problems, but this fact does not detract from its many merits. In the end, this text is well organized and offers a clear conceptual framework to approach proofs. Summing Up: Recommended. Graduate students, researchers, and faculty." (V. K. Chellamuthu, Choice, Vol. 55 (9), May, 2018)

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Prof. Giampiero Esposito (1962) obtained an honours (cum laude) degree in Physics from Naples University in 1986, and was a St. John´s Benefactor´s Scholar at DAMTP in Cambridge (UK) from 1987 to 1991, where he received the J.T. Knight Prize Essay award in 1989 and obtained his Ph.D. degree. He was elected to INFN and ICTP post-doctoral positions at Naples and Trieste, respectively, and has been an INFN Research Fellow at Naples (position with tenure) since 1993, and INFN Primo Ricercatore since 2007.
His original contributions are mainly devoted to quantum gravity and quantum field theory on manifolds with boundary (one-loop conformal anomalies, mixed and diff-invariant boundary conditions for Euclidean quantum gravity, heat-kernel asymptotics, Casimir effect and measurement of variations of zero-point energy), spontaneous symmetry breaking in the early universe, accelerated expansion of the universe, singularity avoidance in quantum cosmology, and scattering from singular potentials in quantum mechanics.

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