Author | Evans, Gwynne A. author |
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Title | Analytic Methods for Partial Differential Equations [electronic resource] / by Gwynne A. Evans, Jonathan M. Blackledge, Peter D. Yardley |
Imprint | London : Springer London : Imprint: Springer, 1999 |
Connect to | http://dx.doi.org/10.1007/978-1-4471-0379-0 |
Descript | XII, 316 p. online resource |
1. Mathematical Preliminaries -- 1.1 Introduction -- 1.2 Characteristics and Classification -- 1.3 Orthogonal Functions -- 1.4 Sturm-Liouville Boundary Value Problems -- 1.5 Legendre Polynomials -- 1.6 Bessel Functions -- 1.7 Results from Complex Analysis -- 1.8 Generalised Functions and the Delta Function -- 2. Separation of the Variables -- 2.1 Introduction -- 2.2 The Wave Equation -- 2.3 The Heat Equation -- 2.4 Laplaceโs Equation -- 2.5 Homogeneous and Non-homogeneous Boundary Conditions -- 2.6 Separation of variables in other coordinate systems -- 3. First-order Equations and Hyperbolic Second-order Equations -- 3.1 Introduction -- 3.2 First-order equations -- 3.3 Introduction to dโAlembertโs Method -- 3.4 dโAlembertโs General Solution -- 3.5 Characteristics -- 3.6 Semi-infinite Strings -- 4. Integral Transforms -- 4.1 Introduction -- 4.2 Fourier Integrals -- 4.3 Application to the Heat Equation -- 4.4 Fourier Sine and Cosine Transforms -- 4.5 General Fourier Transforms -- 4.6 Laplace transform -- 4.7 Inverting Laplace Transforms -- 4.8 Standard Transforms -- 4.9 Use of Laplace Transforms to Solve Partial Differential Equations -- 5. Greenโs Functions -- 5.1 Introduction -- 5.2 Greenโs Functions for the Time-independent Wave Equation -- 5.3 Greenโs Function Solution to the Three-dimensional Inhomogeneous Wave Equation -- 5.4 Greenโs Function Solutions to the Inhomogeneous Helmholtz and Schrรถdinger Equations: An Introduction to Scattering Theory -- 5.5 Greenโs Function Solution to Maxwellโs Equations and Time-dependent Problems -- 5.6 Greenโs Functions and Optics: Kirchhoff Diffraction Theory -- 5.7 Approximation Methods and the Born Series -- 5.8 Greenโs Function Solution to the Diffusion Equation -- 5.9 Greenโs Function Solution to the Laplace and Poisson Equations -- 5.10 Discussion -- A. Solutions of Exercises