Chapters from Peter V. O’Neil textbook:
Thirteenth week (November 30, December 2, 4):
Background: (from comlex valued functions – optional) analytic functions, harmonic functions, mean value theorem
Lecture 32 – 33. (lecture notes)
Laplace Equation.
Lecture 34 (lecture notes)
Review.
Twelfth week (November 23, 25, 27):
Lecture 29 – 31. (lecture notes)
Maximum principle for the heat equation (infinite domain).
Eleventh week (November 16, 18, 20):
Background: (from vector calculus) Gradient, Laplacian, Divergence Theorem
Lecture 26. (lecture notes)
Solving PDEs on a disk.
Lecture 27-28. (lecture notes)
Maximum principle for the heat equation (finite domain).
Max (min) principle for the heat equation.
Tenth week (November 9, 11):
Background: (from vector calculus) Gradient, Laplacian, Divergence Theorem
Lecture 24-25. (lecture notes) + 4.11, 5.8
The Dirichlet Problem. (multidimensional)
Ninth week (November 4, 6):
Lecture 22. (lecture notes) + 4.11, 5.8
Separation of variables in two space dimensions. I
Lecture 23. (lecture notes) + 4.11, 5.8
Separation of variables in two space dimensions. II
Eighth week (October 26, 28, 30):
Background: (from calculus) complex valued functions, real and imaginary parts, integration by parts, double integrals, changing of the order of integration; (from ODE) initial value
problems for first and second order nonhomogeneous linear ODEs with constant coefficients;
(from linear algebra) basis, orthogonal basis, subspace, projections
Lecture 19. (lecture notes) + 4.10
Initial value problem by Fourier Integral.
Lecture 20. (lecture notes) + 3.6
Fourier transform.
Lecture 21. (lecture notes) + 4.10, 5.4
Application of Fourier transform.
Seventh week (October 19, 20, 21):
Background: (from calculus) odd/even functions, Riemann sum for a definite integral, double integrals, derivatives of integrals, integration by parts; (from ODE) initial value
problems for first and second order nonhomogeneous linear ODEs with constant coefficients;
(from linear algebra) basis, orthogonal basis, subspace, projections
Lecture 16. (lecture notes) + 3.3)
Convergence of Fourier Series. The best approximation.
Lecture 17. (lecture notes) + 4.9
Fourier method versus D’Alembert solution.
Lecture 18. (lecture notes) + 3.5
Fourier Integral.
Sixth week (October 14, 16):
Background: (from calculus) derivatives of integrals, integration by parts; (from ODE) initial value
problems for first and second order nonhomogeneous linear ODEs with constant coefficients;
(from linear algebra) basis, orthogonal basis, orthonormal basis, eigenvalues, eigenfunctions of symmetric (hermitian) matrixes, diagonalization
Lecture 14. (lecture notes) + 4.8, 4.9, 5.3, 8.1 )
Wave and heat equations. Separation of variables. Solutions by eigenfunction expansions.
Lecture 15. (lecture notes) + 3.3
Different types of convergence.
Example from the lectures 14-15.
Fifth week (October 7, 9):
Lecture 12. (lecture notes) + 3.1)
Wave and heat equations. Finite domain and boundary conditions. Concept of basis in functional spaces.
Lecture 13. (lecture notes) + 3.2, 3.4
Fourier series (sine, cosine, full).
Example from the lectures 12-13.
Fourth week (September 28, 30, October 2):
Lecture 9. (lecture notes + 4.3)
The characteristic triangle. Application of Green’s formula to the nonhomogeneous wave equation.
Lecture 10. (lecture notes) + 2.6
Wave equation. Approximate solutions by Taylor series.
Lecture 11. (lecture notes)
Wave equation. Approximate solutions by iterations. Conservation of energy.
Third week (September 21, 23, 25):
Lecture 6. (2.1 – 2.2 + lecture notes)
Linear second order PDEs. Classification. Hyperbolic canonical form.
Lecture 7. (lecture notes) + 4.1(partially)
Wave equation. Method of characteristics. D’Alembert solution.
Lecture 8. (lecture notes) + 4.1, 4.2 (partially)
Wave equation with lower order terms. Dissipative waves.
Example from the lecture 6. (See page 3)
Second week (September 14, 16, 18):
Lecture 3. (1.2-1.3 + lecture notes)
Method of characteristics for linear first-order PDEs. Initial Value Problem.
Lecture 4. (1.4 + lecture notes)
The Quasi-Linear Equations. Introduction.
Lecture 5. (1.4 + lecture notes)
The Quasi-Linear Equations. Solvability of IVPs.
Example from the lecture 5. Part I
Example from the lecture 5. Part II
First week (September 9, 11):
Lecture 1. (1.1 + lecture notes)
Introduction. Classifications of PDE. (linear, nonlinear, order of PDE, homogeneous, non-homogeneous)
Lecture 2. (1.2 + lecture notes)
Linear first-order equations. Geometric methods of solutions. ( directional derivative, geometric properties of a gradient )