This book is an introduction to the study of ordinary differential equations and partial differential equations, ranging from elementary techniques to advanced tools. The presentation focusses on initial value problems, boundary value problems, equations with delayed argument and analysis of periodic solutions: main goal is the analysis of diffusion equation, wave equation Laplace equation and signals. The study of relevant examples of differential models highlights the notion of well-posed problem. An expanded tutorial chapter collects the topics from basic undergraduate calculus that are used in subsequent chapters. A wide exposition concerning classical methods for solving problems related to differential equations is available: mainly separation of variables and Fourier series, with basic worked exercises. A whole chapter deals with the analytic functions of complex variable. An introduction to function spaces, distributions and basic notions of functional analysis is present. Several chapters are devoted to Fourier and Laplace transforms methods to solve boundary value problems and initial value problems for differential equations. Tools for the analysis appear gradually: first in function spaces, then in the more general framework of distributions, where a powerful arsenal of techniques allows dealing with impulsive signals and singularities in both data and solutions of differential problems.

Franco Tomarelli received a Degree in Mathematics at the Scuola Normale Superiore di Pisa, graduated at Pisa University in 1978, pursued his studies at the Universite¿ Pierre et Marie Curie in Paris and the Minneapolis School of Mathematics of the University of Minnesota. Associate professor at Pavia University from 1988 to 1990. Full professor of Mathematical Analysis at the Politecnico di Milano since 1990. Award Premio Bruno Finzi 1997. Socio Corrispondente of the Istituto Lombardo Accademia di Scienze e Lettere since 2006. Head of Seminario Matematico e Fisico di Milano from 2009 to 2015. Managing Editor of the Milan Journal of Mathematics (Springer) since 2016. His main research achievements concern topics in calculus of variations, differential problems in continuum mechanics and variational models for segmentation and inpainting of digital images.