Course APM346
Homeworks
Tests/Exam

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).

Example PDEs on the disk.

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.

Fourier Transform example.

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.

Some notes.

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.

Example from the lecture 9.

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.

Convergence

Third week (September 21, 23, 25):

Lecture 6. (2.1 - 2.2 + lecture notes)
Linear second order PDEs. Classification. Hyperbolic canonical form.

Example from the lecture 6.

Lecture 7. (lecture notes) + 4.1(partially)
Wave equation. Method of characteristics. D'Alembert solution.

Example from the lecture 6.

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.

Example from the lecture 3.

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 )

First Week Notes.