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06-262 Math Methods Recap#

Taught you methods needed to solve steady-state and non-steady-state problems.
Concepts built on each other. Matrix algebra and the concept of eigenvalues and eigenvectors became important in solving systems of ODEs. Our understanding of $1^\circ$ ODEs helped us solve $2^\circ$ ODEs. Our understanding of ODEs helps solve PDEs. And homogeneous solutions help us solve non-homogeneous problems

Linear Algebra Recap#

(510)#\[\begin{align} \arr{A} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \rightarrow 2\times 2 \text{ matrix} \end{align}\]

Allows us to use shorthand in defining linear systems of equations

(511)#\[\begin{align} a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = b_1\\ a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = b_1\\ .\\.\\.\\ a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = b_m\\ \implies \arr{A}\vec{x} = \vec{b} \end{align}\]

Which is homogneous when $\vec{b} \equiv \vec{0}$ and non-homogeneous when $\vec{b} \neq \vec{0}$

We used Gauss Elimination to solve systems of equations.
E.g.the system

(512)#\[\begin{align} x_1 + 2x_2 - x_3 = 4\\ 4x_2 - 2x_3 = -2\\ x_1 - 2x_2 + 3x_3 = 0 \end{align}\]

can be represented by

(513)#\[\begin{align} \left[\begin{array}{rrr|r} 1 & 2 & -1 & 4\\ 0 & 4 & -2 & -2 \\ 1 & -2 & 3 & 0\end{array}\right] \rightarrow ^{\text{G.E.}} \left[\begin{array}{rrr|r} 1 & 0 & 0 & 5\\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 3\end{array}\right]\\ \implies x_1 = 5, \ x_2 = -2, \ x_3 = 3 \end{align}\]

Systems of equations can be
1. underdetermined (m<n)
2. determined (m=n)
3. overdetermined (m>n)
  And can have 3 solution scenarios:
4. infinite
5. unique
6. none
  Not dictated by how “determined” system is
Determinants $\rightarrow$ important property to calculate

(514)#\[\begin{equation} |\arr{A}| = \det(\arr{A})=\begin{cases} a_{11} & \text{if $n=1$}.\\ a_{11}A_{11} + a_{12}A_{12} + ... + a_{1n}A_{1n} & \text{if $n\neq 1$}. \end{cases} \end{equation}\]

(only for square matrices) Here $A_{11}, \ A_{12}, \ A_{1n} ...$ are cofactors
e.g. for a $2\times 2$ matrix, $\det(\arr{A}) = a_{11}a_{22} - a_{21}a_{12}$

A matrix is singular (non-invertible) when the det. is zero

Eigenvalues and eigenvectors
For an $n\times n$ matrix $\arr{A}$, when $ \arr{A}\vec{x}=\lambda\vec{x}$, $\lambda$ is an eigenvalue and $\vec{x}$ is the corresponding eigenvector
- $\lambda$’s found by solving $|\arr{A}-\lambda\arr{I}|=0$
- corresponding $\vec{x}$ found by solving $(\arr{A}-\lambda\arr{I})\vec{x}=0$
- the eigenspace of $\arr{A}$ is $\vec{x}=\vec{0}$ (always a solution) + all $\lambda,\vec{x}$ pairs
- An $n\times n$ matrix yields $n$ $\lambda,\vec{x}$ pairs.
Special matrices
- Diagonal
- Identity
- Triangular (upper/lower)
- Symmetric and skew-symmetric

ODE Recap#

We discussed several types of ODEs. Your ability to solve requires you to classify the ODE
- these 5 classifications, together, will determine your solution method

Order
Determined by the order of the highest derivative
A. first $\ y'=5$
B. second $\ y'' + y' + y = 5$
Seperable-ness

applies only to $1^\circ$ ODEs
can be written as $g(y)\cdot y'=f(x)$
A. separable $\ y^2\cdot y'= x + 2$
B. non-separable $\ y' + y = x$

Exactness

applies only to $1^\circ$ ODEs
can be written as $N(x,y)y'=M(x,y)$ with $\frac{\partial N}{\partial x} = \frac{\partial M}{\partial y}$
A. Exact $\ y\cdot y'=x^2$ (all separable ODEs are exact)
B. Non-exact $y\cdot y' = x^2 + y$

Linearity

can be written as $y''+p(x)y'+q(x)y=r(x)$ where $p,q$ and $r$ are continuous functions of $x$ only\
Constant coefficient is case when all p(x), q(x), etc are just constants A. Linear $\ y' + xy = 0$
B. Non-linear $\ y\cdot y' = x^2 + y$

Homogeneity

classification applies only to linear equations
occurs when $r(x) \equiv 0$
A. Homogeneous $y' + xy = 0$
B. Non-homogeneous $y'' + y = 5x$

Your Approach#

If it is first order:
A. separate if possible
B. check linearity. If linear, make exact with integrating factor $F(x) = \exp[\int p(x)dx]$ and use chain rule to obtain $y(x) = e^{-\int p(x)dx}[\int F(x)r(x)dx + c]$
C. Write in form $M(x,y)dx+N(x,y)dy=0$, check for exactness.
If not exact, try to find integrating factor $F(x)$ or $F(y)$

If $F(x)$, then $\frac{1}{F}\frac{dF}{dx} = \frac{1}{N}\left\{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}\right\} = R(x)$
and $F(x)=\exp[\int R(x)dx]$
If $F(y)$, then $\frac{1}{F}\frac{dF}{dy} = \frac{1}{M}\left\{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right\} = R(y)$
and $F(y)=\exp[\int R(y)dy]$
Once your ODE is exact, find $u(x,y)$ s.t. $\frac{\partial u}{\partial x} = M(x,y)$ and $\frac{\partial u}{\partial y} = N(x,y)$
Implicit solution is given by $u(x,y)=c$

If it is second order:
A. check homogeneity. If homogeneous, identify the two component solution basis, which depends on the roots of the characteristic equation $\lambda^2+a\lambda+b=0$
real, distinct roots
$y(x)=c_1e^{\lambda_1x}+c_2e^{\lambda_2x}$, where $\lambda=\frac{1}{2}(a\pm\sqrt{a^2-4b})$
repeated roots
$y(x)=c_1e^{\lambda x}+c_2xe^{\lambda x}$ where $\lambda=-\frac{1}{2}a$
complex roots
$y(x)=e^{-\frac{a}{2}x}[c_1\sin(\omega x)+c_2\cos(\omega x)]$ where $\omega=\sqrt{b-\frac{1}{4}a^2}$

B. if non-homogeneous, find homogeneous solution $y_H(x)$, then find a particular solution $y_P(x)$ using either

Method of undetermined coefficients * applicable only for non-homogeneous terms that return themselves as derivatives, e.g. $\sin x, e^x, x^n$ * assume $y_P(x)$ looks like $r(x)$
Variation of Parameters * works for any $r(x)$ * assume $y_P(x)=u(x)y_1(x)+v(x)y_2(x)$ * Find Wronskian, $W=y_1y_2'-y_2y_1'$, then,
$y(x)=y_H(x)+y_1(x)\int\frac{-r(x)y_2(x)}{W}dx + y_2(x)\int\frac{r(x)y_1(x)}{W}dx$

Boundary Value Problems
- IVPs always have a unique solution provided continuity is met
- BVPs have either no solution, unique or infinite number of solutions $\rightarrow$ you must think for these

Coupled ODEs#

e.g.\begin{align} y_1’-y_1+y_2=4\ y_2’+y_1=8x \end{align} especially when working with higher number of equations (like in chemical plant), best to solve simultaneously

Matrices come back:

(515)#\[\begin{align} \vec{y}'=\arr{A}\vec{y}+\vec{b}(x) \end{align}\]

where $\vec{b}(x)$ is the non-homogeneous term, $\vec{y}$ is an $n\times 1$ vector of unknown functions and $\arr{A}$ is an $n\times n$ coefficient matrix
For the example, $\arr{A}=\begin{bmatrix}1&-1 \\ -1&0 \end{bmatrix}$ and $\vec{b}=\begin{bmatrix} 4 \\ 8x \end{bmatrix}$

We proved that the eigenvalues and eigenvectors of $\arr{A}$ yield the homogenoeus solution:

(516)#\[\begin{align} \vec{y} = c_1\vec{x}^{(1)}e^{\lambda_1t} + c_2\vec{x}^{(2)}e^{\lambda_2t} + ... + c_n\vec{x}^{(n)}e^{\lambda_nt} \end{align}\]

for real, distinct eigenvalues

For real, repeated eigenvalues, we must use reduction of order to obtain linearly independent solutions
e.g. $c_1\vec{x}^{(1)}e^{\lambda_1t} + c_2\vec{x}^{(1)}te^{\lambda_1t}$ (remember there is also a term from the solution of the generalized eigenvalue problem!)
For complex eigenvalues, we must use Euler’s formula

(517)#\[\begin{align} e^{\omega it} = \cos(\omega t) + i\sin(\omega t) \end{align}\]

to change basis and obtain a real solution

(518)#\[\begin{align} \vec{y}=c_1 \begin{bmatrix} \cos\omega t \\ -sin\omega t \end{bmatrix} + c_2 \begin{bmatrix} \sin \omega t \\ \cos \omega t \end{bmatrix} \end{align}\]

Non-homogeneous systems are handled similarly to non-homogeneous single equations
- after finding homogeneous solution, find a particular solution using MUC or VoP
- In both cases, you assume a form of $y_P$
  MUC: $\vec{y}_P(t) = \vec{u}\sin t + \vec{v}\cos t$ (for, say, $5\cos t$)
  $\vec{y}_P(t) = \vec{u}t^2 + \vec{v}t + \vec{w}$ (for, say, $2t^2$)
  VoP: $\vec{y}_P(t) = \arr{Y}(t)\vec{u}(t)$
  where $\arr{Y}(t)$ is the fundamental matrix obtained from your two linearly independent solutions to homogeneous equation
For non-linear ODE’s where we can’t get the solution, we can still find steady states and do a non-linear stability analysis
- Steady states are straightforward $\vec{y}'=\vec{0}$, but you have to find every steady state!. Be careful.
- At each steady state, evaluate the jacobian and form a linearized ODE that holds near the steady state.
- Calculate the eigenvalues to understand the type (stability) of the steady state

PDEs#

2 or more independent variables
Order, linearity, homogeneity
solve using separation of variables to turn into multiple ODE’s, or if given special solution forms for specific PDE’s
If needed, use Fourier coefficients to transform initial conditions into the basis of solutions

Statistics#

At the very end, we covered some simple statistics that will help you next year.

Basic statistics
- Types of distributions (continuous, discrete) and how to evaluate them (PDF, CDF, sampling) in python
- Plotting distribution (NOT JUST HISTOGRAMS) and kernel density estimation.
- Fitting distributions based on the CDF
- Estimates for properties of a Gaussian distribution, and the errors on those estimates
Regression
- Linear regression [testable]
  - Form the augmented matrix $\arr{X}$ with your non-linear functions
  - Solve for the best parameters using the $\arr{X}^T$ trick
  - If you need uncertainty, use a package like statsmodels
- Non-linear regression
  - Turn a non-linear problem into a linear one, then solve like that
  - Use a tool like curvefit (easiest) or lmfit (if need uncertainty)

Mathematical Methods for Chemical Engineering

06-262 Math Methods Recap

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