Partial Differential Equations

Partial Differential Equations : Analytical Solution Techniques

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This is a text for a two-semester or three-quarter sequence of courses in partial differential equations. It is assumed that the student has a good background in vector calculus and ordinary differential equations and has been introduced to such elementary aspects of partial differential equations as separation of variables, Fourier series, and eigenfunction expansions. Some familiarity is also assumed with the application of complex variable techniques, including conformal map ping, integration in the complex plane, and the use of integral transforms. Linear theory is developed in the first half of the book and quasilinear and nonlinear problems are covered in the second half, but the material is presented in a manner that allows flexibility in selecting and ordering topics. For example, it is possible to start with the scalar first-order equation in Chapter 5, to include or delete the nonlinear equation in Chapter 6, and then to move on to the second order equations, selecting and omitting topics as dictated by the course. At the University of Washington, the material in Chapters 1-4 is covered during the third quarter of a three-quarter sequence that is part of the required program for first-year graduate students in Applied Mathematics. We offer the material in Chapters 5-8 to more advanced students in a two-quarter sequence."show more

Product details

  • Paperback | 562 pages
  • 170.18 x 238.76 x 30.48mm | 1,020.58g
  • Springer
  • United States
  • English
  • Softcover reprint of the original 1st ed. 1990
  • 30 black & white illustrations, biography
  • 0534122167
  • 9780534122164

Table of contents

1 The Diffusion Equation.- 1.1 Heat Conduction.- 1.2 Fundamental Solution.- 1.2.1 Similarity (Invariance).- 1.2.2 Qualitative Behavior; Diffusion.- 1.2.3 Laplace Transforms.- 1.2.4 Fourier Transforms.- 1.3 Initial-Value Problem (Cauchy Problem) on the Infinite Domain; Superposition.- 1.4 Initial- and Boundary-Value Problems on the Semi-Infinite Domain; Green s Functions.- 1.4.1 Green s Function of the First Kind.- 1.4.2 Homogeneous Boundary-Value Problems.- 1.4.3 Inhomogeneous Boundary Condition u = g(t).- 1.4.4 Green s Function of the Second Kind.- 1.4.5 Homogeneous Boundary-Value Problems.- 1.4.6 Inhomogeneous Boundary Condition.- 1.4.7 The General Boundary-Value Problem.- 1.5 Initial- and Boundary-Value Problems on the Finite Domain; Green s Functions.- 1.5.1 Green s Function of the First Kind.- 1.5.2 Connection with Separation of Variables.- 1.5.3 Connection with Laplace Transform Solution.- 1.5.4 Uniqueness of Solutions.- 1.5.5 Inhomogeneous Boundary Conditions.- 1.5.6 Higher-Dimensional Problems.- 1.6 Burgers Equation.- 1.6.1 The Cole-Hopf Transformation.- 1.6.2 Initial-Value Problem on ? ? x ?.- 1.6.3 Boundary-Value Problem on 0 x ?.- Review Problems.- Problems.- References.- 2 Laplace s Equation.- 2.1 Applications.- 2.1.1 Incompressible Irrotational Flow.- 2.1.2 Two-Dimensional Incompressible Flow.- 2.2 The Two-Dimensional Problem; Conformai Mapping.- 2.2.1 Mapping of Harmonic Functions.- 2.2.2 Transformation of Boundary Conditions.- 2.2.3 Example, Solution in a Simpler Transformed Domain.- 2.3 Fundamental Solution; Dipole Potential.- 2.3.1 Point Source in Three Dimensions.- 2.3.2 Fundamental Solution in Two-Dimensions; Descent.- 2.3.3 Effect of Lower Derivative Terms.- 2.3.4 Potential Due to a Dipole.- 2.4 Potential Due to Volume, Surface, and Line Distribution of Sources and Dipoles.- 2.4.1 Volume Distribution of Sources.- 2.4.2 Surface and Line Distribution of Sources or Dipoles.- 2.4.3 An Example: Flow Over a Nonlifting Body of Revolution.- 2.4.4 Limiting Surface Values for Source and Dipole Distributions.- 2.5 Green s Formula and Applications.- 2.5.1 Gauss Integral Theorem.- 2.5.2 Energy Theorem and Corollaries.- 2.5.3 Uniqueness Theorems.- 2.5.4 Mean-Value Theorem.- 2.5.5 Surface Distribution of Sources and Dipoles.- 2.5.6 Potential Due to Dipole Distribution of Unit Strength.- 2.6 Green s and Neumann s Functions.- 2.6.1 Green s function.- 2.6.2 Neumann s function.- 2.7 Dirichlet s and Neumann s Problems.- 2.8 Examples of Green s and Neumann s Functions.- 2.8.1 Upper Half-Plane, y ? 0 (Two Dimensions).- 2.8.2 Upper Half-Space, z ? 0 (Three Dimensions).- 2.8.3 Interior (Exterior) of Unit Sphere or Circle.- 2.9 Estimates; Harnack s Inequality.- 2.10 Connection between Green s Function and Conformai Mapping (Two Dimensions); Dipole-Green s Functions.- 2.11 Series Representations; Connection with Separation of Variables.- 2.12 Solutions in Terms of Integral Equations.- 2.12.1 Dirichlet s Problem.- 2.12.2 Neumann s Problem.- Review Problems.- Problems.- References.- 3 The Wave Equation.- 3.1 The Vibrating String.- 3.2 Shallow-Water Waves.- 3.2.1 Assumptions.- 3.2.2 Hydrostatic Balance.- 3.2.3 Conservation of Mass.- 3.2.4 Conservation of Momentum in the X direction.- 3.2.5 Smooth Solutions.- 3.2.6 Energy Conservation.- 3.2.7 Initial-Value Problem.- 3.2.8 Signaling Problem.- 3.2.9 Small-Amplitude Theory.- 3.3 Compressible Flow.- 3.3.1 Conservation Laws.- 3.3.2 One-Dimensional Ideal Gas.- 3.3.3 Signaling Problem for One-Dimensional Flow.- 3.3.4 Inviscid, Non-Heat-Conducting Gas; Analogy with Shallow-Water Waves.- 3.3.5 Small-Disturbance Theory in One-Dimensional Flow (Signaling Problem).- 3.3.6 Small Disturbance Theory in Three Dimensional, Inviscid Non-Heat-Conducting Flow.- 3.4 The One-Dimensional Problem in the Infinite Domain.- 3.4.1 Fundamental Solution.- 3.4.2 General Initial- Value Problem on ? ? x ?.- 3.4.3 An Example.- 3.5 Initial- and Boundary-Value Problems on the Semi-Infinite Interval; Green s Functionshow more

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