Format: Paperback / softback, 677 pages, height x width: 235x155 mm,
weight: 1289 g, 1 Illustrations, black and white; XVIII, 677 p. 1 illus.,
Series: UNITEXT 147
Pub. Date: 16-Jan-2023
This work is an updated version of a book evolved from courses offered on partial differential equations (PDEs) over the last several years at the Politecnico di Milano. These courses had a twofold purpose: on the one hand, to teach students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other to provide them with a solid theoretical background for numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first part, chapters 2 to 5, is more elementary in nature and focuses on developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. In the second part, chapters 6 to 10 concentrate on the development of Hilbert spaces methods for the variational formulation and the analysis of (mainly) linear boundary and initial-boundary value problems, while Chapter 11 deals with vector-valued conservation laws, extending the theory developed in Chapter 4. The main differences with respect to the previous editions are: a new section on reaction diffusion models for population dynamics in a heterogeneous environment; several new exercises in almost all chapters; a general restyling and a reordering of the last chapters. The book is intended as an advanced undergraduate or first-year graduate course for students from various disciplines, including applied mathematics, physics and engineering.
1 Introduction.- 2 Diffusion.- 3 The Laplace Equation.- 4 Scalar Conservation Laws and First Order Equations.- 5 Waves and Vibration.- 6 Elements of Functional Analysis.- 7 Distributions and Sobolev Spaces.- 8 Variational Formulation of Elliptic Problems.- 9 Weak Formulation of Evolution Problems.- 10 More Advanced Topics.- 11 Systems of Conservation Laws.- Appendix A: Measures and Integrals.- Appendix B: Identities and Formulas.
Format: Paperback / softback, 81 pages, height x width: 235x155 mm, 2 Illustrations, black and white; IX, 81 p. 2 illus.
Series: SpringerBriefs in Mathematics
Pub. Date: 05-Feb-2023
ISBN-13: 9789811978760
This book introduces the theory of enveloping semigroups-an important tool in the field of topological dynamics-introduced by Robert Ellis. The book deals with the basic theory of topological dynamics and touches on the advanced concepts of the dynamics of induced systems and their enveloping semigroups. All the chapters in the book are well organized and systematically dealing with introductory topics through advanced research topics. The basic concepts give the motivation to begin with, then the theory, and finally the new research-oriented topics. The results are presented with detailed proof, plenty of examples and several open questions are put forward to motivate for future research. Some of the results, related to the enveloping semigroup, are new to the existing literature. The enveloping semigroups of the induced systems is considered for the first time in the literature, and some new results are obtained. The book has a research-oriented flavour in the field of topological dynamics.
Introduction.- Basic Definitions and Elementary Properties.- Dynamics of Induced Systems.- Dynamical Properties of Enveloping Semigroups.- Enveloping Semigroup of the Induced Systems.- More Investigations.
Format: Hardback, 311 pages, height x width: 235x155 mm, weight: 693 g, 1 Illustrations, black and white; XV, 311 p. 1 illus
Series: RSME Springer Series 7
Pub. Date: 12-Jan-2023
ISBN-13: 9783031211348
The book is devoted to the qualitative study of differential equations defined by piecewise linear (PWL) vector fields, mainly continuous, and presenting two or three regions of linearity. The study focuses on the more common bifurcations that PWL differential systems can undergo, with emphasis on those leading to limit cycles. Similarities and differences with respect to their smooth counterparts are considered and highlighted. Regarding the dimensionality of the addressed problems, some general results in arbitrary dimensions are included. The manuscript mainly addresses specific aspects in PWL differential systems of dimensions 2 and 3, which are sufficinet for the analysis of basic electronic oscillators. The work is divided into three parts. The first part motivates the study of PWL differential systems as the natural next step towards dynamic complexity when starting from linear differential systems. The nomenclature and some general results for PWL systems in arbitrary dimensions are introduced. In particular, a minimal representation of PWL systems, called canonical form, is presented, as well as the closing equations, which are fundamental tools for the subsequent study of periodic orbits. The second part contains some results on PWL systems in dimension 2, both continuous and discontinuous, and both with two or three regions of linearity. In particular, the focus-center-limit cycle bifurcation and the Hopf-like bifurcation are completely described. The results obtained are then applied to the study of different electronic devices. In the third part, several results on PWL differential systems in dimension 3 are presented. In particular, the focus-center-limit cycle bifurcation is studied in systems with two and three linear regions, in the latter case with symmetry. Finally, the piecewise linear version of the Hopf-pitchfork bifurcation is introduced. The analysis also includes the study of degenerate situations. Again, the above results are applied to the study of different electronic oscillators.
Part I: Introduction1 From linear to piecewise linear differential systems 1.1 Some caveats about the notation used in the book 1.2 A short review on linear systems in R2 1.2.1 Real and distinct eigenvalues 1.2.2 Complex eigenvalues 1.2.3 Nonzero double eigenvalues 1.2.4 A canonical form for affine systems 1.2.5 The case of vanishing determinant 1.3 Degenerate bifurcations in planar affine systems 1.4 The one-parameter Lienard form 1.5 Computing the exponential matrix 1.6 Passing from linear to piecewise linear systems 1.7 Limit cycles in a continuous piecewise linear worked example 1.7.1 The right half-return map 1.7.2 The left half-return map 1.7.3 A bifurcation analysis 2 Preliminary results 2.1 A unified Lienard form for continuous planar piecewise linear systems 2.2 Canonical forms for Lure systems in higher dimension 2.3 Some generic results about equilibria 2.3.1 Observable continuous piecewise linear systems with two zones 2.3.2 Observable symmetric continuous PWL systems with three zones 2.4 Analysis of periodic orbits through their closing equations 2.4.1 Observable continuous piecewise linear systems with two zones 2.4.2 Symmetric continuous PWL systems with three zones 2.5 Periodic orbits and Poincare maps in piecewise linear systems 2.5.1 Derivatives of transition maps 2.5.2 Continuous piecewise linear systems with two zones 2.5.3 Continuous piecewise linear systems with three zones Part II: Planar piecewise linear differential systems 3 Continuousplanar systemswithtwo zones 3.1 Equilibria in continuous planar piecewise linear systems with two zones 3.2 Some preliminary results on limit cycles 3.3 The Massera's method for uniqueness of limit cycles 3.4 General results about limit cycles 3.5 Refracting Systems 3.6 The bizonal focus-center-limit cycle bifurcation 4 Continuousplanar systemswiththree zones 4.1 Limit cycle existence and uniqueness 4.2 The focus-center-limit cycle bifurcation for symmetric systems 5 Boundary equilibriumbifurcations and limit cycles 5.1 Boundary Equilibrium Bifurcations in systems with two zones 5.2 Boundary Equilibrium Bifurcations in systems with three zones 5.3 Analysis of Wien bridge oscillators 6 Algebraically computable continuousPWLnodal oscillators 6.1 Preliminary results 6.2 Analysis of equilibria and periodic orbits 6.3 Application to a piecewise linear van der Pol oscillator 7 The focus-saddle boundary bifurcation 7.1 Preliminary results 7.2 Main results 7.3 Application to the analysis of memristor oscillators Part III: Three-dimensional piecewise linear differential systems 8 The FCLC bifurcation in 3D systems with 2 zones 8.1 The generic focus-center-limit cycle bifurcation 8.2 The onset of asymmetric oscillations in Chua's circuit 8.3 The degenerate FCLC bifurcation in systems with 2 zones 8.4 The degenerate FCLC bifurcation in Chua's circuit 9 The FCLC bifurcation in 3D symmetric PWL systems 9.1 The symmetric focus-center-limit cycle bifurcation 9.2 The degeneration of the FCLC bifurcation in symmetric systems 10 The analogue of Hopf-pitchfork bifurcation 10.1 A one-parameter bifurcation analysis 10.2 The degenerate PWL Hopf-pitchfork bifurcation 10.3 The Hopf-pitchfork bifurcation in Chua's circuit 10.4 An extended Bonhoeffer-van der Pol oscillator 11 Afterword Appendices A The piecewise linear characteristics of Chua's diode B The Chua's oscillator C Auxiliary results Bibliography
Format: Hardback, 502 pages, height x width: 235x155 mm, VIII, 502 p.; VIII, 502 p.,
Series: Problem Books in Mathematics
Pub. Date: 25-Jan-2023
ISBN-13: 9783031212611
This book, the much-anticipated sequel to (Almost) Impossible, Integrals, Sums, and Series, presents a whole new collection of challenging problems and solutions that are not commonly found in classical textbooks. As in the author's previous book, these fascinating mathematical problems are shown in new and engaging ways, and illustrate the connections between integrals, sums, and series, many of which involve zeta functions, harmonic series, polylogarithms, and various other special functions and constants. Throughout the book, the reader will find both classical and new problems, with numerous original problems and solutions coming from the personal research of the author. Classical problems are shown in a fresh light, with new, surprising or unconventional ways of obtaining the desired results devised by the author. This book is accessible to readers with a good knowledge of calculus, from undergraduate students to researchers. It will appeal to all mathematical puzzlers who love a good integral or series and aren't afraid of a challenge.
Chapter
1. Integrals.
Chapter
2. Hints.
Chapter
3. Solutions.
Chapter
4. Sums and Series.
Chapter
5. Hints.
Chapter
6. Solutions.
Format: Hardback, 156 pages, height x width: 235x155 mm, 22 Illustrations, color; 14 Illustrations, black and white; X, 156 p. 36 illus., 22 illus. in color.
Series: Springer INdAM Series 50
Pub. Date: 09-Feb-2023
ISBN-13: 9789811977152
The content of the book collects some contributions related to the talks presented during the INdAM Workshop "Fractional Differential Equations: Modelling, Discretization, and Numerical Solvers", held in Rome, Italy, on July 12-14, 2021. All contributions are original and not published elsewhere. The main topic of the book is fractional calculus, a topic that addresses the study and application of integrals and derivatives of noninteger order. These operators, unlike the classic operators of integer order, are nonlocal operators and are better suited to describe phenomena with memory (with respect to time and/or space). Although the basic ideas of fractional calculus go back over three centuries, only in recent decades there has been a rapid increase in interest in this field of research due not only to the increasing use of fractional calculus in applications in biology, physics, engineering, probability, etc., but also thanks to the availability of new and more powerful numerical tools that allow for an efficient solution of problems that until a few years ago appeared unsolvable. The analytical solution of fractional differential equations (FDEs) appears even more difficult than in the integer case. Hence, numerical analysis plays a decisive role since practically every type of application of fractional calculus requires adequate numerical tools. The aim of this book is therefore to collect and spread ideas mainly coming from the two communities of numerical analysts operating in this field - the one working on methods for the solution of differential problems and the one working on the numerical linear algebra side - to share knowledge and create synergies. At the same time, the book intends to realize a direct bridge between researchers working on applications and numerical analysts. Indeed, the book collects papers on applications, numerical methods for differential problems of fractional order, and related aspects in numerical linear algebra.The target audience of the book is scholars interested in recent advancements in fractional calculus.
Chapter
1. A New Diffusive Representation for Fractional Derivatives, Part I: Construction, Implementation and Numerical Examples.
Chapter
2. Exact solutions for the fractional nonlinear Boussinesq equation.
Chapter
3. A numerical procedure for fractional-time-space differential equations with the spectral fractional Laplacian.
Chapter
4. Spectral Analysis of Matrices in B-Spline Galerkin Methods for Riesz Fractional Equations.
Chapter 5.Do the Mittag-Leffler functions preserve the properties of their matrix arguments?.
Chapter
6. On the solutions of the fractional generalized Gierer-Meinhardt Model.
Chapter
7. A convolution-based method for an integro-differential equation in mechanics.
Chapter
8. A MATLAB code for fractional differential equations based on two-step spline collocation methods.
Format: Hardback, 370 pages, height x width: 235x155 mm, 2 Illustrations, color; XII, 370 p. 2 illus. in color.
Series: Applied Mathematical Sciences 214
Pub. Date: 13-Feb-2023
ISBN-13: 9783031210495
This book discusses numerical methods for solving time-fractional evolution equations. The approach is based on first discretizing in the spatial variables by the Galerkin finite element method, using piecewise linear trial functions, and then applying suitable time stepping schemes, of the type either convolution quadrature or finite difference. The main concern is on stability and error analysis of approximate solutions, efficient implementation and qualitative properties, under various regularity assumptions on the problem data, using tools from semigroup theory and Laplace transform. The book provides a comprehensive survey on the present ideas and methods of analysis, and it covers most important topics in this active area of research. It is recommended for graduate students and researchers in applied and computational mathematics, particularly numerical analysis.
Existence, Uniqueness, and Regularity of Solutions.- Semidiscrete Discretization.- Convolution Quadrature.- Finite Difference Methods: Construction and Implementation.- Finite Difference Methods on Uniform Meshes.- Finite Difference Methods on Graded Meshes.- Nonnegativity Preservation.- Discrete Fractional Maximal Regularity.- Subdiffusion with time-dependent coefficients.- Semilinear Subdiffusion Equations.- Time-Space Formulation and Finite Element Approximation.- A Spectral Petrov-Galerkin Method.- Incomplete Iterative Solution at the Time Levels.- Optimal Control with Subdiffusion Constraint.- Backward Subdiffusion Problems.- Appendix: Mathematical Preliminaries.