None of the following textbooks is required for 18.306. The books are listed here only in case you are interested in further study. Optional reading from the recommended books will be assigned. A sessionwise schedule for readings is provided in the table below.
Recommended Textbooks in Applied Mathematics and Applications
Whitham, G. B. Linear and Nonlinear Waves. Canada: JohnWiley, 1999. ISBN: 0471359424.
Kevorkian, J. Partial Differential Equations: Analytical Solution Techniques. 2nd ed. New York: Springer Verlag, 2000. ISBN: 0387986057.
Levine, H. Partial Differential Equations. Providence, R.I.: American Mathematical Society: International Press, 1997. ISBN: 0821807757.
Hinch, E. J. Perturbation Methods. England: Cambridge University Press, 1991. ISBN: 0521378974.
Other Textbooks at a Similar Level
Debnath, L. Nonlinear Partial Differential Equations for Scientists and Engineers. Boston: Birkhauser, 1997. ISBN: 0817639020.
Carrier, G. F., and C. E. Pearson. Partial Differential Equations: Theory and Technique. 2nd ed. Boston: Academic Press, 1988. ISBN: 0121604519.
Barenblatt, G. I. Scaling, Self-similarity, and Intermediate Asymptotics. Cambridge, [England]; New York: Cambridge University Press, 1997. ISBN: 0521435226.
Drazin, P. G., and R. S. Johnson. Solitons: An Introduction. Cambridge, [England]; New York: Cambridge University Press, 1989. ISBN: 0521336554.
Books at a more Advanced Level, with Emphasis on the Rigorous Theory
The first 2 books have a theorem-proof type of exposition, for the brave ones!
Evans, L. C. Partial Differential Equations. Providence, R.I.: American Mathematical Society, 1998. ISBN: 0821807722.
DiBenedetto, E. Partial Differential Equations. Switzerland: Birkhauser, 1994. ISBN: 0817637087.
Garabedian, P. R. Partial Differential Equations. American Mathematical Society, 1998. ISBN: 0821813773.
Readings by Session
| 1 |
Introduction, Theme for the Course, Initial and Boundary Conditions, Well-posed and Ill-posed Problems |
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| 2 |
Conservation Laws in (1 + 1) Dimensions
Introduction to 1st-order PDEs: Linear and Homogeneous, and Linear, Non-Homogeneous PDEs |
Debnath - §§ 3.2-3.5 |
| 3 |
Theory of 1st-order PDEs (cont.): Quasilinear PDEs, and General Case, Charpit's Equations |
Debnath - §§ 4.2, 4.3 |
| 4 |
Theory of 1st-order PDEs (cont.): Examples, The Eikonal Equation, and the Monge Cone
Introduction to Traffic Flow |
Debnath - § 4.5
Garabedian - § 2.2
Whitham - § 3.1, pp. 68-71. |
| 5 |
Solutions for the Traffic-flow Problem, Hyperbolic Waves
Breaking of Waves, Introduction to Shocks, Shock Velocity
Weak Solutions |
Whitham - §§ 2.1-2.3, 2.7 |
| 6 |
Shock Structure (with a Foretaste of Boundary Layers), Introduction to Burgers' Equation
Introduction to PDE Systems, The Wave Equation |
Whitham - §§ 2.4-2.6, 5.1, 5.2 |
| 7 |
Systematic Theory, and Classification of PDE Systems |
Whitham - §§ 5.1, 5.2 |
| 8 |
PDE Systems (cont.): Example from Elementary Gas Dynamics, Riemann Invariants
More on the Wave Equation, The D'Alembert Solution |
Whitham - §§ 5.2, 5.3
Carrier and Pearson - §§ 3.1, 3.3 |
| 9 |
Remarks on the D'Alembert Solution
The Wave Equation in a Semi-infinite Interval
The Diffusion (or Heat) Equation in an Infinite Interval, Fourier Transform and Green's Function |
Carrier and Pearson - §§ 1.1, 1.3 |
| 10 |
Properties of Solutions to the Diffusion Equation (with a Foretaste of Similarity Solutions)
Conversion of Nonlinear PDEs to Linear PDEs: Simple Transformations, Parabolic PDE with Quadratic Nonlinearity, Viscous Burgers' Equation and the Cole-Hopf Transformation |
Evans - § 4.4.1 |
| 11 |
The Laplace Equation in a Finite Region, Separation of Variables in a Circular Disc
Conversion of Nonlinear PDEs to Linear PDEs: Potential Functions |
Evans - § 4.4.2 |
| 12 |
Generalities on Separation of Variables for Solving Linear PDEs, The Principle of Linear Superposition
Conversion of PDEs to ODEs, Traveling Waves, Fisher's Equation
Conversion of Nonlinear PDEs to linear PDEs: The Hodograph Transform
Quiz 1 |
Debnath - §§ 8.8, 8.9
Evans - § 4.4.3 |
| 13 |
Conversion of Nonlinear PDEs to Linear PDEs: The Legendre Transform
Natural Frequencies and Separation of Variables: Linear PDEs, Fourier Series, Example: Vibrating String
The Sturm-Liouville Problem
About the Question: Can One Hear the Shape of a Drum? |
Evans - § 4.4.4
Carrier and Pearson - §§ 3.5, 3.7, 11.1, 11.2 |
| 14 |
Natural Frequencies for Linear PDEs (cont.): Vibrating Circular Membrane, Bessel's Functions, Linear Schrödinger's Equation |
Garabedian - § 11.1 |
| 15 |
Vibrating Circular Membrane (cont.)
Natural Frequencies and Separation of Variables: Nonlinear PDEs, Example: Nonlinear Schrödinger's Equation, Elliptic Integrals and Functions |
|
| 16 |
Remarks on the Nonlinear Schrödinger Equation
General Eigenvalue Problem for Linear PDEs with Self-adjoint Operators
Classification of 2nd-order Quasilinear PDEs, Initial and Boundary Data |
Kevorkian - § 4.2 |
| 17 |
Introduction to Green's Functions, The Poisson Equation in 3D, Integral Equation for the "Nonlinear Poisson Equation"
Green's Functions for Nonlinear Problems |
|
| 18 |
Green's Functions for Nonlinear PDEs: Example: Infinite Vibrating String with Forcing, The Issue of (Classical) Causality, Formulation of the Integral Equation, Analytical Solution by Regular Perturbation |
|
| 19 |
Conversion of Self-adjoint Problems to Integral Equations
Introduction to Dispersive Waves, Dispersion Relations, Uniform Klein-Gordon Equation, Linear Superposition and the Fourier Transform, The Stationary-phase Method for Linear Dispersive Waves |
Whitham - §§ 11.1-11.3 |
| 20 |
Extra Lecture
Linear Dispersive Waves (cont.): Phase and Group Velocities, Energy Propagation, Theory of Caustics, Airy Function
Generalizations: Local Wave Number and Frequency, Slowly Varying Wave Amplitudes |
Whitham - §§ 11.4, 13.6 |
| 21 |
Asymptotic Expansions for Non-uniform PDEs, Example: Non-uniform Klein-Gordon Equation
Kinematic Derivation of Group Velocity |
Whitham - §§ 11.8, 11.5 |
| 22 |
Dimensional Analysis for Stationary-phase Method (Linear Dispersive Waves), Characteristic Length and Time of a Dispersive System
Introduction to Dimensional Analysis and Similarity for PDEs, Example: The Diffusion Equation |
Barenblatt - Chaps. 0, 1, 2, 3 |
| 23 |
Dimensional Analysis and Similarity (cont.): Idea of Stretching Transformations, Example: Nonlinear Diffusion |
Debnath - §§ 8.11-8.13 |
| 24 |
Extra Lecture
Dimensional Analysis and Similarity (cont.): More on Nonlinear Diffusion, Solutions of Compact Support |
Debnath - § 8.11 |
| 25 |
Comments on the Blasius Problem
Introduction to Perturbation Methods for PDEs: Regular Perturbation, Example |
Hinch - Chap. 4 |
| 26 |
Regular Perturbation for Linear Schrödinger Equation with a Potential
Perturbation Methods for PDEs: Singular Perturbation, Boundary Layers, Elementary Example |
Hinch - Chap. 5 |
| 27 |
Singular Perturbation for PDEs (cont.), More Advanced Examples
Quiz 2 |
Carrier and Pearson - §§ 16.2, 16.5.1 |
| 28 |
Boundary Layers (cont.): Anatomy of Inner and Outer Solutions
Introduction to Solitary Waves and Solitons, Water Waves, Solitary Waves for the KdV Equation, The Sine-Gordon Equation: Kink and Anti-kink Solutions |
Carrier and Pearson - § 16.5.1
Debnath - §§ 9.1, 9.2, 9.4, 11.7 |
| 29 |
Extra Lecture
(Heuristic) Definition of Soliton, Some Nonlinear Evolution PDEs with Soliton Solutions, Solutions to the Sine-Gordon Equation via Separation of Variables, Outline of the Inverse Scattering Transform Idea and Technique
Special Topics: The Painlevé Conjecture, The Painlevé Property, The Painlevé Equations |
Debnath - § 11.8
Drazin and Johnson - §§ 4.1-4.4, 7.1 |
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