Inspired by Prof. Strang's CSE, beautiful Ocw courses: 85 and 86, videos: 85 (*), 86, and MIT homes: 85, 86.

HW:

1/18/2011
Review: matrices, differential equations, exponential growth, mass-spring                 HW: finish the problems started in class
Overview, motivation, linear <-> nonlinear, discrete <-> continuous.
Guessing and checking solutions.

Started up MATLAB and played with matrices.

1/20
Review DE and LA (matrices).                                       Work through Sect 1.1

Watch MIT Lect. 1. Four Special Matrices                        1.1 p9 / 1-5 try

1/25
Another look at the mass-spring problem: include gravity.
Linear equations: superposition principle
Homogeneous and non-homogeneous equations: x = x_h + x_p
This enables us to simply combine the solutions to pure oscillations without gravity and the static solution with gravity.
Played with MATLAB and the matrices: K and T and started p9/1.
                                                                                HW.  revisit  1.1 p9 / 1-5 try       review DE notes, for concepts we discussed today

1/27
Watch Linear Algebra (Vectors and Matrices) Review             
Continue with HW; come ready with questions to next class.

2/1
Practicing matrices and MATLAB.
Elimination, pivots, determinant, inverse matrix, formulas for the 2 by 2 case.
determinant = product of pivots
LU decomposition:  
>> [L, U] = lu(K)
Solving a system: Ax=b
>> A\b
HW:      practice for Test 1
If terribly bored, because you already know the material, watch the next video(s) in the series and continue through the book.
Advanced stuff may show up on the test as an extra credit problem.

2/3

Practice for Test 1


2/8 Test 1

2/15

Where do the matrices: Kn, Tn, Bn and Cn come from?

Systems of masses and springs.

Derive the 2 masses, 3 springs model.

Solve the static 2 masses - 3 springs model.

Review of the eigenvalue - eigenvector problem.

Practicing solving toy models of systems of differential equations.

1.5 Eigenvalues and Eigenvectors                                                      p.60 / 23-25  try        Skim Sect 1.2, 1.3

2/17

For another way to introduce Kn, Tn, Bn and Cn, watch:  Difference Equations

Compare with material in Sect 1.2.                                                      try 1.2 p.23 / 1-4,    1.3 p.33 / 1-4

2/22

Lessons learned on Test 1.

Logistic model as a refinement of the natural growth model.

Phase space analysis: fixed points, stable, unstable

Linear versus nonlinear: generally speaking, linear is easy and nonlinear is hard. Why? Because superposition principle holds for linear and not for nonlinear.

Reducing a second order ODE to a system of 1st order ODEs. This will help with p.60 / 24

Wave equ: u_tt = c^2 u_xx

Guessing solutions: u(x,t) = sin(x+t),  sin(x-t),    cos(x+t),     cos(x-t),     but not sin(xt) or sin(x/t)                      HW: check by plugging in

Delta function: a simple model of a point load

derivative of the step function is the delta function

Another look at Sect 1.3 Solving a linear system

2/24

1.4 Inverses and Delta Functions                                                             p.44 / 1,2

3/1

Solving the mass-spring oscillation model:  M d^2u/dt^2 = - K u + f

Simplest example: M=I, K = K_2 = [2 -1; -1 2]

Find evas: 1, 3   

Find evecs: [1 1]', [-1 1]'

With MATLAB,  >> [V, D] = eig(K_2)

Note: MATLAB packs the evas on the diagonal of the D matrix and evec in the columns of the V matrix; evecs are chosen to be length 1.

Plug into the formula:  u = C1 sin( sqrt(lambda1) t) v_lambda1 + C2 cos( sqrt(lanbda1) t ) v_lambda1 + same kind of stuff for lambda2, with C3 and C4

Cook some IC and use them to determine the constants: C1, C2, C3 and C4.

Note: if the IC is proportional to one of the eigenvectors, the solution is an normal mode of oscillation and it can be just written down.

In general, the solution is a superposition of normal modes, and it is not obvious - have to use brute force to solve for constants (but that's why we have computers).

Delta functions - useful for models.

Derivatives of step functions.

p23 / 1 the continuous case                                                                  Review and practice for Test 2

3/3

Practice for Test 2: 

Mostly solving systems of differential equations, in Sect 1.5, applied to mass-spring oscillations

1.2 Only easy delta function problems 

1.3 matrix problems with, most involving MATLAB

(1.4 deferred to Test 3)

Spring Break: Enjoy!

3/15 Test 2

Ch. 2 A framework for Applied Mathematics

2.3 Equilibrium and the Stiffness Matrix                      p.109 / 1,2,3

2.2. Oscillation by Newton's Law                                   skim

2.3 Lease Squares                                                    p.140 / 7-10, 12

2.4 Graph Models                                                    just started; skim   p.154 / 1 (try 2,3)

4/12 Test 3    What we covered in Ch 2 and some more eigenvalues / eigenvectors, so we don't forget them. Extra credits could of course be anything, perhaps illustrating transition from linear to nonlinear, or between discrete and continuous.

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Skim through the rest of the book - you may find ideas for projects that you can present at hrumc