MA 523 - Introduction To Partial Differential Equations
Arshak Petrosyan
Spring 2004
Office hours: MWF 1:30-2:30pm, or by appointment (contact information)

Textbooks:
Primary: [J] F. John, Partial Differential Equations, 4th ed.
Recommended: [E] L.C. Evans, Partial Differential Equations
Reference for Calculus: W. Rudin, Principles of Mathematical Analysis
Final Exam
7:00-9:00pm Fri, May 7, 2004 BRNG 1245
[ Review Problems](Note: Not sample problems)
Topics for Final Exam
• Laplace's Equation, Green's Identities, Fundatmental Solution ([J] 4.1)
• Mean Value Property for Harmonic Functions, Sub- and Superharmonic Functions ([J] 4.1)
• Maximum Principle ([J] 4.2)
• The Dirichlet Problem, Green's Function, Poisson's Formula, Harnack's Inequality ([J] 4.3)
• The Dirichlet Principle ([E] 2.2.5), Hilbert-Space Methods ([J] 4.5)
• The Wave Equation in n-Dimensional Space, Method of Spherical Means, Kirchhoff's Formula ([J] 5.1a)
• Hadamard's Method of Descent, Nonhomogeneous Equations, Duhamel's Principle ([J] 5.1b,c)
• Energy Methods for the Wave Equation ([E] 2.4.3)
• Initial-Boundary-Value Problems, Separation of Variables, Eigenvalue Problem for Laplacian, Method of Reflection. ([J] 5.1d)
• The Heat Equation, Fundamental Solution, Initial-Value problem ([E] 2.3.1a,b)
• Nonhomogeneous Problem, Duhamel's Principle ([E] 2.3.1c)
• Maximum Principle, Uniqueness ([J] 7.1b)
• Mixed Problem, Separation of Variables, Methof of Reflection ([J] 7.1c)
• Energy Methods for the Heat Equation, Uniqueness and Backward Uniqueness ([E] 2.3.4)
• Fourier Transform ([E] 4.3.1a)
• Applications of Fourier Transform, Bessel Potentials, Schroedinger's Equation, the Heat Equation, the Wave equation ([E] 4.3.1b)
• Plane and Traveling Waves, Exponential solutions, Dispersive Equations, KdV equation, Solitons ([E] 4.2.1a,b)
• Hyperbolic Systems with Constant Coefficents ([E] 7.3.3)
• Midterm Exam
6:30-8:00pm Thu. Mar 4, 2004 RAWL 1086
[ Sample Problems](Note: Actual exam may vary in format)
Topics for Midterm Exam
• Partial Derivatives, Examples of PDE, Implicit Function Theorem
• Existence and Uniqueness of Solutions of ODE, Transport Equation ([J] 1.3), Quasi-Linear Equations ([J] 1.4)
• The Cauchy Problem for the Quasi-Linear Equation ([J] 1.5), Examples ([J] 1.6)
• Quasi-Linear Equations in Higher Dimensions ([J] 1.5), Burgers' Equation (Example 3 from [J] 1.6, also [E] 3.4.1 a), Rankine-Hugoniot Condition
• General First-Order Equations ([J] 1.7), The Cauchy Problem ([J] 1.8), The Eikonal Equation
• General First-Order Equations and the Cauchy Problem in Higher Dimensions ([J] 1.8), Envelope Solutions ([J] 1.9)
• Characteristics for Quasi-Linear Second-Order Equations ([J] 2.1), Linear Second-Order Hyperbolic Equations in the Plane ([J] 2.3)
• The One-Dimensional Wave Equation, D'Alambert's Formula ([J] 2.4)
• Weak Solutions of the Wave Equation, Initial-Boundary Value Problem, Solutions using Fourier Series ([J] 2.4)
• Multi-Index Notation ([J] 3.1), The Cauchy Problem, Compatability Conditions of the Cauchy Data, Noncharacteristic Hyper-Surfaces ([J] 3.2)
• Multiple Infinite Series ([J] 3.3(a)), Power-Series, Real Analytic Functions, Unique Continuation Property, Method of Majorants ([J] 3.3(b))
• The Cauchy-Kowalevski Theorem, Reduction to the Standard Form ([J] 3.3(d))
• Weak (Distributional) Solutions, Adjoint Operators ([J] 3.4, 3.6), Distributions (Generalized Functions), Dirac's Delta Function ([J] 3.6)
• Homework
• 1 p.18: 1, 6
• 2 p.19: 4, p.31: 1
• 3 p.40: 3, p.45: 3, 10
• 4 p.55: 1, 3, 4, 6, p.61: 2
• 5 p.69: 3, p.78: 1, 4
• 6 p.92: 1, 5
• 7 p.101: 2a, 5, p.105: 1
• 8 p.110: 3, 4, 7, p.125: 3
• 9 p.132: 1a,b, 2a, p.143: 2
• 10 [E] p.87: 10, 13
• 11 [E] p.234: 4
• Official Information