Notes and References
The Taylor Series is a powerful tool for approximating functions using polynomials. It’s particularly useful in finding solutions to differential equations.
Example: Suppose we want to approximate the function $f(x) = \sin(x)$ around $x = 0$ using a Taylor series. The Taylor series expansion of $\sin(x)$ around $x = 0$ is:
$\sin(x) \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots$
$\dot x = \lambda x$
The rate of change of $x$ in time: $\dot x$ is linearly proportional to $x$, the state of the system or some variable that’s being kept track of.
Example: Let’s consider the simple ODE $\dot x = \lambda x$, where $\lambda$ is a constant. This equation models exponential growth or decay. For instance, if $\lambda = -0.5$, it represents exponential decay.
$\dot X = A X$
Instead of just one variable $x$, we keep track of lots of variables (usually in a matrix) $X$. $A$ is a matrix that tells how the system changes with time as a linear function of that system. Example: Consider a system of ODEs representing predator-prey dynamics, known as the Lotka-Volterra equations:
$\dot{x} = \alpha x - \beta xy$
$\dot{y} = \delta xy - \gamma y$
Where $x$ represents the prey population (e.g., rabbits) and $y$ represents the predator population (e.g., foxes).
$AT = TD$
The eigenvalues and eigenvectors(co-ordinate transformations) of the $A$ matrix are useful in solving the system of ODEs. they describe how systems behave under linear transformations.
Example: Suppose we have a matrix $A$ representing a linear transformation. Finding its eigenvalues and eigenvectors can help us understand how the transformation stretches or compresses space along certain directions. In mechanical systems, eigenvalues and eigenvectors determine natural frequencies and modes of vibration.
$\dot x = f(x)$
$f(x)$ is a nonlinear function. Unlike linear systems where the rate of change of $x$ = $\dot x$ is linearly proportional to the state $x$, in these systems, the rate of change is non-linear. Most real-world problems are non-linear. Chaotic systems can arise as solutions of non-linear differential equations.
Nonlinear systems exhibit behavior not explained by linear models.
Example: The logistic equation $\dot{x} = rx(1-x)$ models population growth with limited resources. Depending on the value of the parameter $r$, the system can exhibit stable equilibrium, periodic oscillations, or chaotic behavior.
Numerical methods provide approximations to solutions of differential equations and methods for simulating and plotting them.
Example: Using Python or MATLAB, we can numerically solve differential equations using methods like Euler’s method, Runge-Kutta methods, or finite difference methods. Finite element analysis uses numerical methods to simulate stress distribution in complex structures.
Formulas, derivations, symbolic techniques. Analytical techniques involve finding exact solutions using mathematical tools.
Example: Solving the differential equation $\frac{dy}{dx} = y$ yields $y = Ce^x$, where $C$ is a constant.
Computational methods involve using algorithms and computers to solve differential equations. We use the intuition derived from analytical methods to use programming to find solutions and create simulations.
Example: MATLAB’s ode45 function is commonly used to numerically solve ODEs of various complexities.
Data-driven approaches use empirical data to build models and make predictions.
Example: Neural networks can learn the dynamics of a system from data and predict future behavior.
$x = x + iy$
Solutions of differential equations involve complex numbers and complex functions.
Complex analysis deals with functions of complex variables.
Example: Residue theorem in complex analysis allows efficient evaluation of integrals along closed contours.
$\dfrac{d}{dx} \Longrightarrow \nabla = [\dfrac{d}{dx},\dfrac{d}{dy},\dfrac{d}{dz}]$
Example: Suppose we have a vector field representing the velocity of a fluid. Vector calculus helps us analyze properties such as divergence, curl, and line integrals in this field.
$\dfrac{\partial}{\partial t} u = \dfrac{\partial^2} {\partial x^2} u$
This generalizes Ordinary Differential Equations which are single derivatives in terms of the independent variables. Now we look at multiple derivatives of different independent variables. This is built on the language of Vector Calculus.
Vector Calculus and Partial Derivatives go together in the same way that Linear Algebra and Ordinary Differential Equations go together.
Example: The heat equation $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$ describes how the temperature $u(x, t)$ changes over time in a one-dimensional rod, where $\alpha$ is the thermal diffusivity.
$\mathcal{F}$ and $\mathcal{L}$
These are closely related to differential equations, especially partial differential equations. They are used to solve differential equations and are useful for applications like audio processing and image compression. This transitions from solving physics problems to solving data science problems.
Transform methods convert differential equations into algebraic equations.
Example: Laplace transform can be used to solve linear constant coefficient ODEs with initial conditions. Example: The Fourier transform can decompose a signal into its frequency components. For instance, it’s used in audio processing to analyze and manipulate sound waves.
$\text{SVD}(X) = U\Sigma V^T$, etc.
After Fourier Transforms and Laplace transforms, an understanding of linear algebra, it’s natural to introduce modern data-driven techniques, like data decomposition, machine learning, e.g., like the SVD - Singular Value Decomposition, which generalizes the Fourier transform to more complex and rich systems.
Example: Singular Value Decomposition (SVD) can be used for dimensionality reduction in datasets or for collaborative filtering in recommendation systems.
$\dot X = AX + Bu$
Control theory deals with influencing the behavior of dynamic systems. Differential equations are very applicable to control systems.
Example: In a cruise control system for a car, $x$ could represent the car’s velocity, $u$ the throttle position, and $\dot x$ the rate of change of velocity. The equation $\dot x = Ax + Bu$ describes how the car’s velocity changes over time under the influence of the throttle.