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Author Collins, Peter J
Title Differential and Integral Equations
Imprint Oxford : Oxford University Press, Incorporated, 2006
©2006
book jacket
Descript 1 online resource (387 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Note Intro -- Contents -- Preface -- How to use this book -- Prerequisites -- 0 Some Preliminaries -- 1 Integral Equations and Picard's Method -- 1.1 Integral equations and their relationship to differential equations -- 1.2 Picard's method -- 2 Existence and Uniqueness -- 2.1 First-order differential equations in a single independent variable -- 2.2 Two simultaneous equations in a single variable -- 2.3 A second-order equation -- 3 The Homogeneous Linear Equation and Wronskians -- 3.1 Some linear algebra -- 3.2 Wronskians and the linear independence of solutions of the second-order homogeneous linear equation -- 4 The Non-Homogeneous Linear Equation -- 4.1 The method of variation of parameters -- 4.2 Green's functions -- 5 First-Order Partial Differential Equations -- 5.1 Characteristics and some geometrical considerations -- 5.2 Solving characteristic equations -- 5.3 General solutions -- 5.4 Fitting boundary conditions to general solutions -- 5.5 Parametric solutions and domains of definition -- 5.6 A geometric interpretation of an analytic condition -- 6 Second-Order Partial Differential Equations -- 6.1 Characteristics -- 6.2 Reduction to canonical form -- 6.3 General solutions -- 6.4 Problems involving boundary conditions -- 6.5 Appendix: technique in the use of the chain rule -- 7 The Diffusion and Wave Equations and the Equation of Laplace -- 7.1 The equations to be considered -- 7.2 One-dimensional heat conduction -- 7.3 Transverse waves in a finite string -- 7.4 Separated solutions of Laplace's equation in polar co-ordinates and Legendre's equation -- 7.5 The Dirichlet problem and its solution for the disc -- 7.6 Radially symmetric solutions of the two-dimensional wave equation and Bessel's equation -- 7.7 Existence and uniqueness of solutions, well-posed problems -- 7.8 Appendix: proof of the Mean Value Theorem for harmonic functions
8 The Fredholm Alternative -- 8.1 A simple case -- 8.2 Some algebraic preliminaries -- 8.3 The Fredholm Alternative Theorem -- 8.4 A worked example -- 9 Hilbert-Schmidt Theory -- 9.1 Eigenvalues are real and eigenfunctions corresponding to distinct eigenvalues are orthogonal -- 9.2 Orthonormal families of functions and Bessel's inequality -- 9.3 Some results about eigenvalues deducible from Bessel's inequality -- 9.4 Description of the sets of all eigenvalues and all eigenfunctions -- 9.5 The Expansion Theorem -- 10 Iterative Methods and Neumann Series -- 10.1 An example of Picard's method -- 10.2 Powers of an integral operator -- 10.3 Iterated kernels -- 10.4 Neumann series -- 10.5 A remark on the convergence of iterative methods -- 11 The Calculus of Variations -- 11.1 The fundamental problem -- 11.2 Some classical examples from mechanics and geometry -- 11.3 The derivation of Euler's equation for the fundamental problem -- 11.4 The special case F = F(y, y') -- 11.5 When F contains higher derivatives of y -- 11.6 When F contains more dependent functions -- 11.7 When F contains more independent variables -- 11.8 Integral constraints -- 11.9 Non-integral constraints -- 11.10 Varying boundary conditions -- 12 The Sturm-Liouville Equation -- 12.1 Some elementary results on eigenfunctions and eigenvalues -- 12.2 The Sturm-Liouville Theorem -- 12.3 Derivation from a variational principle -- 12.4 Some singular equations -- 12.5 The Rayleigh-Ritz method -- 13 Series Solutions -- 13.1 Power series and analytic functions -- 13.2 Ordinary and regular singular points -- 13.3 Power series solutions near an ordinary point -- 13.4 Extended power series solutions near a regular singular point: theory -- 13.5 Extended power series solutions near a regular singular point: practice -- 13.6 The method of Frobenius -- 13.7 Summary
13.8 Appendix: the use of complex variables -- 14 Transform Methods -- 14.1 The Fourier transform -- 14.2 Applications of the Fourier transform -- 14.3 The Laplace transform -- 14.4 Applications of the Laplace transform -- 14.5 Applications involving complex analysis -- 14.6 Appendix: similarity solutions -- 15 Phase-Plane Analysis -- 15.1 The phase-plane and stability -- 15.2 Linear theory -- 15.3 Some non-linear systems -- 15.4 Linearisation -- Appendix: the solution of some elementary ordinary differential equations -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W
This clear, accessible textbook provides an introduction to both differential and integral equations. With numerous carefully worked examples and exercises, the text is ideal for any undergraduate with basic calculus to gain a thorough grounding in 'analysis for applications'
Description based on publisher supplied metadata and other sources
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
Link Print version: Collins, Peter J. Differential and Integral Equations Oxford : Oxford University Press, Incorporated,c2006 9780198533825
Subject Differential equations.;Integral equations
Electronic books
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