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Lecture 39:
Mon 15 Apr |
Review session
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Lecture 38:
Fri 12 Apr |
Field day: physical examples of vibrations with strings, plates, and fire
Chladni plate videos (nodal lines for a plate):
here,
here,
and here!
See nodal lines for design of a violin
and acoustic guitar.
Rubens tube video: here!
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Lecture 37:
Wed 10 Apr |
Finite elements for the wave equation: forward/backward Euler, trapezoid methods
Reading: Gockenbach, Section 7.3
MATLAB: wave_fem1.m: finite elements w/matrix exponential
MATLAB: wave_fem2.m: finite elements w/forward Euler (blow-up)
MATLAB: wave_fem3.m: finite elements w/backward Euler (decay)
MATLAB: wave_fem4.m: finite elements w/trapezoid rule (stable)
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Lecture 36:
Mon 8 Apr |
Spectral method for the wave equation
Reading: Gockenbach, Section 7.2
MATLAB: wave_fourier1.m: spectral method for wave equation
MATLAB: wave_fourier2.m: same initial position, different velocity
MATLAB: heat_fourier_comp.m: heat equation, same initial position
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Lecture 35:
Fri 5 Apr |
D'Alembert's solution of the wave equation on an unbounded domain
Reading: Gockenbach, Section 7.1
MATLAB: wave_inf_l.m: first order wave equation, left-moving
MATLAB: wave_inf_r.m: first order wave equation, right-moving
MATLAB: wave_inf.m: second order wave equation (left- and right-moving)
MATLAB: waveEqnDemo.m: Examples from Gockenbach
Jean le Rond d'Alembert on Wikipedia
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Lecture 34:
Wed 3 Apr |
Time-dependent heat equation: FEM implementation
Reading: Gockenbach, Section 6.4
MATLAB: invMK_eigs.m: eigenvalues of M-1K
Richard Courant
and his mathematical genealogy
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Lecture 33:
Mon 1 Apr |
Time-dependent heat equation: FEM implementation
Reading: Gockenbach, Section 6.4
MATLAB: heat_fem1.m: finite elements for heat equation (matrix exponential)
MATLAB: heat_fem2.m: finite elements for heat equation with forcing term
MATLAB: heat_fem3.m: finite elements for heat equation (forward Euler)
MATLAB: heat_fem4.m: finite elements for heat equation (backward Euler)
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Lecture 32:
Fri 29 Mar |
Time-dependent heat equation: Fourier series
Reading: Gockenbach, Section 6.2
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Lecture 31:
Wed 27 Mar |
Time-dependent heat equation: Fourier series
Reading: Gockenbach, Section 6.1
MATLAB: heat_fourier2.m: solution of heat equation,
inhomogeneous Dirichlet b.c.
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Lecture 30:
Mon 25 Mar |
Fourier series for the heat equation: Fourier series
Reading: Gockenbach, Section 6.1
MATLAB: heat_fourier1.m: solution of heat equation,
homogeneous Dirichlet b.c.
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Lecture 29:
Fri 22 Mar |
Dynamical systems: finite dimensional linear systems via the backward Euler method
Reading: Gockenbach, Section 4.4 and 4.5
MATLAB: bweulerdemo.m: Backward Euler approximation for
x'(t) = A x(t), symmetric A
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Lecture 28:
Wed 20 Mar |
Midterm return and discussion
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Lecture 27:
Mon 18 Mar |
Dynamical systems: finite dimensional linear systems via the forward Euler method
Reading: Gockenbach, Section 4.4 and 4.5
MATLAB: fweulerdemo.m: Forward Euler approximation for
x'(t) = A x(t), symmetric A
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Lecture 26:
Fri 15 Mar |
Dynamical systems: finite dimensional linear systems via the matrix exponential
Reading: Gockenbach, Section 4.3
MATLAB: expdemo.m: solution of
x'(t) = A x(t) for symmetric A
MATLAB: expdemo_nonsymm.m: solution of
x'(t) = A x(t) for nonsymmetric A
MATLAB: expdemo_norm.m: solution of
x'(t) = A x(t) showing decay w.r.t. time
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Lecture 25:
Wed 13 Mar |
Finite Element Method: convergence theory
Reading: lecture25.pdf
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Lecture 24:
Mon 11 Mar |
Finite Element Method: varying the boundary conditions
Reading: Gockenbach, Section 5.6
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Lecture 23:
Fri 8 Mar |
Finite Element Method: varying the boundary conditions
Reading: Gockenbach, Section 5.6
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Lecture 22:
Wed 6 Mar |
Review session
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Lecture 21:
Mon 4 Mar |
Finite Element Method
Reading: Gockenbach, Section 5.6
MATLAB: fem_demo1.m: finite element method for -u'' = f
MATLAB: hat.m: code for evaluating a hat function
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Lecture 20:
Fri 22 Feb |
The Galerkin Method
Reading: Gockenbach, Section 5.5
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Lecture 19:
Wed 20 Feb |
The weak form of the steady-state heat equation
Reading: Gockenbach, Section 5.4
Reading: lecture19.pdf
Lord Rayleigh (1842-1919) on Wikipedia
Walther Ritz (1878-1909) on Wikipedia and
Ritz's 1909 paper
Boris Galerkin (1871-1945) on Wikipedia
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Lecture 18:
Mon 18 Feb |
Spectral Method: varying the boundary conditions and the operator
Reading: Gockenbach, Section 5.3
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Lecture 17:
Fri 15 Feb |
Spectral Method: varying the boundary conditions and the operator
Reading: Gockenbach, Section 5.3
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Lecture 16:
Wed 13 Feb |
Spectral Method: varying the boundary conditions and the operator
Reading: Gockenbach, Section 5.3
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Lecture 15:
Mon 11 Feb |
Spectral Method for the Laplacian with Dirichlet boundary conditions
Reading: Gockenbach, Section 5.3
MATLAB: fone.m, uone.m: solve -u'' = 1,
u(0) = u(1) = 0
Helpful tip useful for computing trig integrals with Mathematica.
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Lecture 14:
Fri 8 Feb |
Eigenvalues and eigenfunctions of symmetric linear operators
Reading: Gockenbach, Sections 5.1 and 5.2
MATLAB: sindemo.m: eigenfunctions and boundary conditions
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Lecture 13:
Wed 6 Feb |
The spectral method for solving Ax=b; Symmetric linear operators
Reading: Gockenbach, Sections 3.5 and 5.1
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Lecture 12:
Mon 4 Feb |
Eigenvalues and eigenvectors of matrices; The Fredholm alternative
Reading: Gockenbach, Sections 3.5 and 3.2
Reading: lecture12.pdf
MATLAB: demo12: eigenvalue computations
Évariste Galois at Wikipedia
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Lecture 11:
Fri 1 Feb |
Best approximation and projections
Reading: Gockenbach, Section 3.4
MATLAB: proj_fun.m: best approximation from polynomials
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Lecture 10:
Wed 30 Jan |
Best approximation from a finite-dimensional subspace
Reading: Gockenbach, Section 3.4
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Lecture 9:
Mon 28 Jan |
Norms; Orthogonality; Best approximation from a one-dimensional subspace
Reading: Gockenbach, Section 3.4
MATLAB: proj1d_vec.m: best approximation from a 1d space (vectors)
MATLAB: proj1d_fun.m: best approximation from a 1d space (functions)
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Lecture 8:
Fri 25 Jan |
Inner products
Reading: Gockenbach, Section 3.4
MATLAB: ip_demo1.m: inner products of vectors and functions
MATLAB: ip_demo2.m: inner products of orthogonal functions
Euclid's proof of the Pythagorean Theorem, colorfully rendered by
Oliver Byrne
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Lecture 7:
Wed 23 Jan |
Span; Linear independence; Basis and dimension
Reading: Gockenbach, Section 3.3
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Lecture 6:
Fri 18 Jan |
Finite difference matrix approximations to the first and second derivative
Reading: lecture6.pdf
MATLAB: diff1.m: first differentiation matrix
MATLAB: diff1b.m: first differentiation matrix (inline functions)
MATLAB: diff2.m: second differentiation matrix
MATLAB: heateq.m: solve the steady-state heat equation
Richard Courant,
Kurt Friedrichs,
and
Hans Lewy at Wikipedia
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Lecture 5:
Wed 16 Jan |
Linear operators
Reading: Gockenbach, Section 3.1
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Lecture 4:
Mon 14 Jan |
Vector spaces and subspaces
Reading: Gockenbach, Section 3.1
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Lecture 3:
Fri 11 Jan |
Boundary and initial conditions; The heat equation at steady state
Reading: Gockenbach, Section 2.1
MATLAB: bar_bc_ss.m
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Lecture 2:
Wed 9 Jan |
Derivation of the heat equation
Reading: Gockenbach, Section 2.1
MATLAB: bar_bc.m
MATLAB: funplot.m: a simple code to plot functions in MATLAB.
MATLAB: matlabPrimer.pdf
heat capacity,
thermal conductivity at Wikipedia
Joseph Fourier (1768 - 1830) at Wikipedia
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Lecture 1:
Mon 7 Jan |
Overview and classification of differential equations
Reading: Gockenbach, Chapter 1
Syllabus: Syllabus
MATLAB: demo1.m, burgers.m
Nonlinear chemical waves:
Belousov-Zhabotisnsky
simulation and
in the petri dish
Ron Fedkiw's webpage
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