LEADER 05746nam 22006375 450 001 9910479882503321 005 20210913162831.0 010 $a1-4471-0379-3 024 7 $a10.1007/978-1-4471-0379-0 035 $a(CKB)3400000000088116 035 $a(SSID)ssj0000805105 035 $a(PQKBManifestationID)11456957 035 $a(PQKBTitleCode)TC0000805105 035 $a(PQKBWorkID)10822570 035 $a(PQKB)11515527 035 $a(DE-He213)978-1-4471-0379-0 035 $a(MiAaPQ)EBC3074316 035 $a(PPN)237992965 035 $a(EXLCZ)993400000000088116 100 $a20121227d1999 u| 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aAnalytic Methods for Partial Differential Equations$b[electronic resource] /$fby G. Evans, J. Blackledge, P. Yardley 205 $a1st ed. 1999. 210 1$aLondon :$cSpringer London :$cImprint: Springer,$d1999. 215 $a1 online resource (XII, 316 p.) 225 1 $aSpringer Undergraduate Mathematics Series,$x1615-2085 300 $a"With 25 Figures." 311 $a3-540-76124-1 320 $aIncludes bibliographical references and index. 327 $a1. 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 Laplace?s 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 d?Alembert?s Method -- 3.4 d?Alembert?s 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. Green?s Functions -- 5.1 Introduction -- 5.2 Green?s Functions for the Time-independent Wave Equation -- 5.3 Green?s Function Solution to the Three-dimensional Inhomogeneous Wave Equation -- 5.4 Green?s Function Solutions to the Inhomogeneous Helmholtz and Schrödinger Equations: An Introduction to Scattering Theory -- 5.5 Green?s Function Solution to Maxwell?s Equations and Time-dependent Problems -- 5.6 Green?s Functions and Optics: Kirchhoff Diffraction Theory -- 5.7 Approximation Methods and the Born Series -- 5.8 Green?s Function Solution to the Diffusion Equation -- 5.9 Green?s Function Solution to the Laplace and Poisson Equations -- 5.10 Discussion -- A. Solutions of Exercises. 330 $aThe subject of partial differential equations holds an exciting and special position in mathematics. Partial differential equations were not consciously created as a subject but emerged in the 18th century as ordinary differential equations failed to describe the physical principles being studied. The subject was originally developed by the major names of mathematics, in particular, Leonard Euler and Joseph-Louis Lagrange who studied waves on strings; Daniel Bernoulli and Euler who considered potential theory, with later developments by Adrien-Marie Legendre and Pierre-Simon Laplace; and Joseph Fourier's famous work on series expansions for the heat equation. Many of the greatest advances in modern science have been based on discovering the underlying partial differential equation for the process in question. J ames Clerk Maxwell, for example, put electricity and magnetism into a unified theory by estab­ lishing Maxwell's equations for electromagnetic theory, which gave solutions for problems in radio wave propagation, the diffraction of light and X-ray developments. Schrodinger's equation for quantum mechankal processes at the atomic level leads to experimentally verifiable results which have changed the face of atomic physics and chemistry in the 20th century. In fluid mechanics, the Navier-Stokes' equations form a basis for huge number-crunching activities associated with such widely disparate topics as weather forcasting and the design of supersonic aircraft. Inevitably the study of partial differential equations is a large undertaking, and falls into several areas of mathematics. 410 0$aSpringer Undergraduate Mathematics Series,$x1615-2085 606 $aMathematical analysis 606 $aAnalysis (Mathematics) 606 $aNumerical analysis 606 $aAnalysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12007 606 $aNumerical Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M14050 615 0$aMathematical analysis. 615 0$aAnalysis (Mathematics). 615 0$aNumerical analysis. 615 14$aAnalysis. 615 24$aNumerical Analysis. 676 $a515/.353 686 $a35C15$2msc 686 $a35A22$2msc 700 $aEvans$b G$4aut$4http://id.loc.gov/vocabulary/relators/aut$01034717 702 $aBlackledge$b J$4aut$4http://id.loc.gov/vocabulary/relators/aut 702 $aYardley$b P$4aut$4http://id.loc.gov/vocabulary/relators/aut 906 $aBOOK 912 $a9910479882503321 996 $aAnalytic Methods for Partial Differential Equations$92454015 997 $aUNINA