Linear Partial Differential Equations & Fourier theory by Marcus Pivato
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About this book :-
Linear Partial Differential Equations & Fourier theory written by
This is a textbook for an introductory course on linear partial differential equations (PDEs) and initial/boundary value problems (I/BVPs). It also provides a mathematically rigorous introduction to Fourier analysis (Chapters 7, 8, 9, 10, and 19), which is the main tool used to solve linear PDEs in Cartesian coordinates.
Finally, it introduces basic functional analysis (Chapter 6) and complex analysis (Chapter 18). The first is necessary to characterize rigorously the convergence of Fourier series, and also to discuss eigenfunctions for linear differential operators. The second provides powerful techniques to transform domains and compute integrals, and also offers additional insight into Fourier series.
This book is not intended to be comprehensive or encyclopaedic. It is designed for a one-semester course (i.e. 36–40 hours of lectures), and it is therefore strictly limited in scope. First, it deals mainly with linear PDEs with constant coefficients. Thus, there is no discussion of characteristics, conservation laws, shocks, variational techniques, or perturbation methods, which would be germane to other types of PDEs. Second, the book focuses mainly on concrete solutionmethods to specific PDEs (e.g. the Laplace, Poisson, heat,wave, and Schr¨odinger equations) on specific domains (e.g. line segments, boxes, disks, annuli, spheres), and spends rather little time on qualitative results about entire classes of PDEs (e.g. elliptic, parabolic, hyperbolic) on general domains. Only after a thorough exposition of these special cases does the book sketch the general theory; experience shows that this is far more pedagogically effective than presenting the general theory first. Finally, the book does not deal at all with numerical solutions or Galerkin methods.
One slightly unusual feature of this book is that, from the very beginning, it emphasizes the central role of eigenfunctions (of the Laplacian) in the solution methods for linear PDEs. Fourier series and Fourier–Bessel expansions are introduced as the orthogonal eigenfunction expansions which are most suitable in certain domains or coordinate systems.
Book Detail :-
Title: Linear Partial Differential Equations & Fourier theory
Author(s): Marcus Pivato
Publisher: Cambridge University Press
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About Author :-
The author Marcus Pivato is B.Sc. in Mathematics, University of Alberta (1994) and Ph.D. in Mathematics, University of Toronto (2001).
He is Professeur, Universit´e de Cergy-Pontoise, UFR d’Economie et Gestion. He is Member of THEMA, Editorial Board for Journal of Cellular Automata (Formerly Associate Professor, Dept. of Mathematics, Trent University). His research field is social choice theory, social welfare, decision theory and game theory.
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Book Contents :-
Linear Partial Differential Equations & Fourier theory written by
cover the following topics.
1. Heat and diffusion
Fourier’s law (i) ...in one dimension and (ii) ...in many dimensions, The heat equation (i) ...in one dimension (ii) ...in many dimensions, Laplace’s equation, The Poisson equation, Properties of harmonic functions, Transport and diffusion, Reaction and diffusion, Practice problems
2. Waves and signals
The Laplacian and spherical means, The wave equation (i) ...in one dimension: the string, (ii) ...in two dimensions: the drum, (iii) ...in higher dimensions, The telegraph equation, Practice problems
3. Quantum mechanics
Basic framework, The Schr¨odinger equation, Stationary Schr¨odinger equation, Practice problems, General theory
4. Linear partial differential equations
Functions and vectors, Linear operators (on finite dimensional vector spaces, on C∞, Kernels), Eigenvalues, eigenvectors, and eigenfunctions, Homogeneous vs. nonhomogeneous, Practice problems
5. Classification of PDEs and problem types
Evolution vs. nonevolution equations, Initial value problems, Boundary value problems (Dirichlet boundary conditions, Neumann boundary conditions, Mixed (or Robin) boundary conditions, Periodic boundary conditions), Uniqueness of solutions (Uniqueness for the Laplace and Poisson equations, Uniqueness for the heat equation, Uniqueness for the wave equation), Classification of second order linear PDEs (in two dimensions, with constant coefficients, in general), Practice problems
6. Some functional analysis
Inner products, L2, More about L2, Complex L2, Riemann vs. Lebesgue integrals, Orthogonality, Convergence concepts (L2, Pointwise convergence, Uniform convergence, Convergence of function series), Orthogonal and orthonormal Bases, Practice problems
7. Fourier sine series and cosine series
Fourier (co)sine series on [0, π] (Sine series on [0, π], Cosine series on [0, π]), Fourier (co)sine series on [0, L] (Sine series on [0, L], Cosine series on [0, L]) Computing Fourier (co)sine coefficients (Integration by parts, Polynomials, Step functions, Piecewise linear functions, Differentiating Fourier (co)sine series), Practice problems
8. Real Fourier series and complex Fourier series
Real Fourier series on [−π, π], Computing real Fourier coefficients (Polynomials, Step functions, Piecewise linear functions, Differentiating real Fourier series), Relation between (co)sine series and real series, Complex Fourier series
9. Multidimensional Fourier series
in two dimensions, in many dimensions, Practice problems
10. Proofs of the Fourier convergence theorems
Bessel, Riemann and Lebesgue, Pointwise convergence, Uniform convergence, convergence (Integrable functions and step functions in L2[−π, π], Convolutions and mollifiers, Proof of Theorems), BVP solutions via eigenfunction expansions
11. Boundary value problems on a line segment
The heat equation on a line segment, The wave equation on a line (the vibrating string), The Poisson problem on a line segment, Practice problems
12. Boundary value problems on a square
The Dirichlet problem on a square, The heat equation on a square (Homogeneous boundary conditions, Nonhomogeneous boundary conditions), The Poisson problem on a square (Homogeneous boundary conditions, Nonhomogeneous boundary conditions), The wave equation on a square (the square drum), Practice problems
13. Boundary value problems on a cube
The heat equation on a cube, The Dirichlet problem on a cube, The Poisson problem on a cube
14. Boundary value problems in polar coordinates
Introduction, The Laplace equation in polar coordinates (Polar harmonic functions, Boundary value problems on a disk, Boundary value problems on a codisk, Boundary value problems on an annulus, Poisson’s solution to Dirichlet problem on the disk), Bessel functions (Bessel’s equation; Eigenfunctions of 4 in Polar Coordinates, Boundary conditions; the roots of the Bessel function, Initial conditions; Fourier-Bessel expansions), The Poisson equation in polar coordinates, The heat equation in polar coordinates, The wave equation in polar coordinates, The power series for a Bessel function, Properties of Bessel functions, Practice problems
15. Eigenfunction methods on arbitrary domains
General solution to Poisson, heat and wave equation BVPs, General solution to Laplace equation BVPs, Eigenbases on Cartesian products, General method for solving I/BVPs, Eigenfunctions of self-adjoint operators
16. Separation of variables
in Cartesian coordinates on R, in Cartesian coordinates on R, in polar coordinates: Bessel’s equation, in spherical coordinates: Legendre’s equation, Separated vs. quasiseparated, The polynomial formalism, G Constraints (Boundedness, Boundary conditions)
17. Impulse-response methods
Introduction, Approximations of identity (in one dimension, in many dimensions), The Gaussian convolution solution (heat equation) (in one dimension, in many dimensions), d’Alembert’s solution (one-dimensional wave equation) (Unbounded domain, Bounded domain), Poisson’s solution (Dirichlet problem on half-plane), Poisson’s solution (Dirichlet problem on the disk), Properties of convolution, Practice problems
18. Applications of complex analysis
Holomorphic functions, Conformal maps, Contour integrals and Cauchy’s Theorem, Analyticity of holomorphic maps, Fourier series as Laurent series, Abel means and Poisson kernels, Linear Partial Differential Equations and Fourier Theory Marcus Pivato DRAFT, Poles and the residue theorem, Improper integrals and Fourier transforms, Homological extension of Cauchy’s theorem
19. Fourier transforms on unbounded domains
Fourier transforms, One-dimensional Fourier transforms, Properties of the (one-dimensional) Fourier transform, Parseval and Plancherel, Two-dimensional Fourier transforms, Three-dimensional Fourier transforms, Fourier (co)sine Transforms on the half-line, Momentum representation & Heisenberg uncertainty, Laplace transforms, Practice problems
20. Fourier transform solutions to PDEs
The heat equation (Fourier transform solution, The Gaussian convolution formula, revisited), The wave equation (Fourier transform solution, Poisson’s spherical mean solution; Huygen’s principle), The Dirichlet problem on a half-plane (Fourier solution, Impulse-response solution), PDEs on the half-line, General solution to PDEs using Fourier transforms, Practice problems
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