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1 online resource (387 pages) 

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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 Firstorder differential equations in a single independent variable  2.2 Two simultaneous equations in a single variable  2.3 A secondorder equation  3 The Homogeneous Linear Equation and Wronskians  3.1 Some linear algebra  3.2 Wronskians and the linear independence of solutions of the secondorder homogeneous linear equation  4 The NonHomogeneous Linear Equation  4.1 The method of variation of parameters  4.2 Green's functions  5 FirstOrder 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 SecondOrder 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 Onedimensional heat conduction  7.3 Transverse waves in a finite string  7.4 Separated solutions of Laplace's equation in polar coordinates and Legendre's equation  7.5 The Dirichlet problem and its solution for the disc  7.6 Radially symmetric solutions of the twodimensional wave equation and Bessel's equation  7.7 Existence and uniqueness of solutions, wellposed 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 HilbertSchmidt 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 Nonintegral constraints  11.10 Varying boundary conditions  12 The SturmLiouville Equation  12.1 Some elementary results on eigenfunctions and eigenvalues  12.2 The SturmLiouville Theorem  12.3 Derivation from a variational principle  12.4 Some singular equations  12.5 The RayleighRitz 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 PhasePlane Analysis  15.1 The phaseplane and stability  15.2 Linear theory  15.3 Some nonlinear 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

