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Advanced Partial Differential Equations with Applications >> Content Detail



Study Materials



Readings

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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

Amazon logo Whitham, G. B. Linear and Nonlinear Waves. Canada: JohnWiley, 1999. ISBN: 0471359424.

Amazon logo Kevorkian, J. Partial Differential Equations: Analytical Solution Techniques. 2nd ed. New York: Springer Verlag, 2000. ISBN: 0387986057.

Amazon logo Levine, H. Partial Differential Equations. Providence, R.I.: American Mathematical Society: International Press, 1997. ISBN: 0821807757.

Amazon logo Hinch, E. J. Perturbation Methods. England: Cambridge University Press, 1991. ISBN: 0521378974.

Other Textbooks at a Similar Level

Amazon logo Debnath, L. Nonlinear Partial Differential Equations for Scientists and Engineers. Boston: Birkhauser, 1997. ISBN: 0817639020.

Amazon logo Carrier, G. F., and C. E. Pearson. Partial Differential Equations: Theory and Technique. 2nd ed. Boston: Academic Press, 1988. ISBN: 0121604519.

Amazon logo Barenblatt, G. I. Scaling, Self-similarity, and Intermediate Asymptotics. Cambridge, [England]; New York: Cambridge University Press, 1997. ISBN: 0521435226.

Amazon logo 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!

Amazon logo Evans, L. C. Partial Differential Equations. Providence, R.I.: American Mathematical Society, 1998. ISBN: 0821807722.

Amazon logo DiBenedetto, E. Partial Differential Equations. Switzerland: Birkhauser, 1994. ISBN: 0817637087.

Amazon logo Garabedian, P. R. Partial Differential Equations. American Mathematical Society, 1998. ISBN: 0821813773.

Readings by Session

SES #TOPICSREFERENCES
1Introduction, Theme for the Course, Initial and Boundary Conditions, Well-posed and Ill-posed Problems
2Conservation Laws in (1 + 1) Dimensions

Introduction to 1st-order PDEs: Linear and Homogeneous, and Linear, Non-Homogeneous PDEs
Debnath - §§ 3.2-3.5
3Theory of 1st-order PDEs (cont.): Quasilinear PDEs, and General Case, Charpit's EquationsDebnath - §§ 4.2, 4.3
4Theory 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.
5Solutions for the Traffic-flow Problem, Hyperbolic Waves

Breaking of Waves, Introduction to Shocks, Shock Velocity

Weak Solutions
Whitham - §§ 2.1-2.3, 2.7
6Shock 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
7Systematic Theory, and Classification of PDE SystemsWhitham - §§ 5.1, 5.2
8PDE 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
9Remarks 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
10Properties 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
11The 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
12Generalities 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
13Conversion 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
14Natural Frequencies for Linear PDEs (cont.): Vibrating Circular Membrane, Bessel's Functions, Linear Schrödinger's EquationGarabedian - § 11.1
15Vibrating Circular Membrane (cont.)

Natural Frequencies and Separation of Variables: Nonlinear PDEs, Example: Nonlinear Schrödinger's Equation, Elliptic Integrals and Functions
16Remarks 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
17Introduction to Green's Functions, The Poisson Equation in 3D, Integral Equation for the "Nonlinear Poisson Equation"

Green's Functions for Nonlinear Problems
18Green'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
19Conversion 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
20Extra 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
21Asymptotic Expansions for Non-uniform PDEs, Example: Non-uniform Klein-Gordon Equation

Kinematic Derivation of Group Velocity
Whitham - §§ 11.8, 11.5
22Dimensional 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
23Dimensional Analysis and Similarity (cont.): Idea of Stretching Transformations, Example: Nonlinear DiffusionDebnath - §§ 8.11-8.13
24Extra Lecture

Dimensional Analysis and Similarity (cont.): More on Nonlinear Diffusion, Solutions of Compact Support
Debnath - § 8.11
25Comments on the Blasius Problem

Introduction to Perturbation Methods for PDEs: Regular Perturbation, Example
Hinch - Chap. 4
26Regular Perturbation for Linear Schrödinger Equation with a Potential

Perturbation Methods for PDEs: Singular Perturbation, Boundary Layers, Elementary Example
Hinch - Chap. 5
27Singular Perturbation for PDEs (cont.), More Advanced Examples

Quiz 2
Carrier and Pearson - §§ 16.2, 16.5.1
28Boundary 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
29Extra 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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