Analytical Methods in Applied Mathematics

This book compiles an extensive list of solved and proposed problems in mathematical topics in analysis, aimed at students of mathematics, applied mathematics, physics, and engineering.

The book begins with an exploration of simple linear and nonlinear ordinary differential equations in Chapter 1, advancing through topics such as power series and the Frobenius method for solving differential equations in Chapter 2. In subsequent chapters, the discussion expands to include functions of complex variables, special functions constructed through the hypergeometric function, and series solutions including Fourier, Fourier-Bessel, and Fourier-Legendre series. Problems in integral transforms, Sturm-Liouville systems, Green's function, linear partial differential equations are also included. The work finishes with a special chapter on fractional calculus and practical applications of the topics presented.

With solved examples and step-by-step exercises, this book can be of value to undergraduate and graduate students seeking a hands-on approach on the listed topics, and as a bibliographical complement to STEM courses as well.

Preface.- Ordinary Differential Equations.- Power Series and the Frobenius Method.- Laurent Series and Residues.- Special Functions.- Fourier, Fourier-Bessel and Fourier-Legendre Series.- Laplace and Fourier Transforms.- Sturm-Liouville Systems.- Partial Differential Equations.- Separation of Variables.- Fractional Calculus.- Applications.- Answers and Hints.- Index.

Edmundo Capelas de Oliveira is a Professor and Researcher at the Department of Applied Mathematics at the Institute of Mathematics, Statistics and Scientific Computing, State University of Campinas (Unicamp), Brazil. He got his PhD in Physics (1982) from the same university, and did post-doc studies (in 1991 and 1995) at the Università degli Studi di Perugia, Italy. In 2012, he was distinguished by Unicamp with the First Award for Undergraduate Teaching Incentive for outstanding teaching skills at the undergraduate level. He authored "Solved Exercises in Fractional Calculus" (2019) and co-authored "Mastering Calculus through Practice" (2021) and "The Many Faces of Maxwell, Dirac and Einstein Equations: A Clifford Bundle Approach" (2016), all published by Springer. His research interests lie in fractional calculus, differential equations, and special functions.

José Emílio Maiorino is a public officer at Universidade Estadual de Campinas (Unicamp), Brazil. He holds a Master's degree in Logic and Philosophy of Science and a PhD in Physics from the same university.