Course Summary: ... ---- # 1. Models | Defnitions | Theorems | Examples | | --------------------------------------------------------------------------------------------------------------------- | -------- | ---------------------------------------------------- | | 1.1 You should understand the purpose of mathematical modelling and the role of abstraction in the modelling process. | | | | | | 1.2 *[[SEIR Model for Covid-19 infections in Wales]] | | 1.3 *[[SQM Classification of Mathematical Models]] | | | | | | 1.3.4 [[RLC Circuit Model]] | | 1.4.1 [[Modelling Cycle]] | | | | | | 1.4.2 *[[Validation of Population Growth Model]] | | 1.4.3 *Limits of models | | | # 2. Steps | Defnitions | Theorems | Examples | | -------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------- | ----------------------------------------- | | 2.1 You should understand [[Recurrence Relation\|recurrence relations]]. Definition not given. | | 2.1.1 [[Fibonacci Sequence]] | | | | 2.1.2 [[Bessel Function]] | | | 2.2.1 [[Solution to Linear First Order Recurrence Relation with Constant Coefficients]] | | | 2.2 [[Stationary Point of First Order Autonomous Recurrence Relation]] & [[Stable Stationary Point of First Order Autonomous Recurrence Relation]] | 2.3 [[Condition for Stability of Stationary Point of First Order Autonomous Recurrence Relation]] | | | 2.5 *[[Periodic Point of First Order Autonomous Recurrence Relation]] | | 2.2.3 *[[Mixing Recurrence Relation]] | | | 2.3.1 [[Solution to Homogenous Second Order Linear Recurrence Relation with Constant Coefficients]] | 2.3.2 *[[Solution to Fibonacci Sequence]] | # 3. Change | Defnition | Theorems | Examples | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | ---------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------- | | 3.1 [[Scalar Ordinary Differential Equation]] | | | | 3.1.1 [[Order of a Differential Equation]] | | | | 3.1.2 [[Autonomous Scalar Ordinary Differential Equation]] | | | | 3.1.3 [[Linear Scalar Ordinary Differential Equation]] | | | | 3.1.4 [[Homogeneous Scalar Ordinary Differential Equation]] | | 3.1.5 [[Logistic Population Growth Model]] | | 3.2 [[Solution to Scalar Ordinary Differential Equation]]. You should understand explicit and implicit forms of solutions. | | | | 3.3 You should understand that IVPs must satisfy certain conditions to have a unique solution by theorem such as [[Picard–Lindelöf theorem]]. Knowledge of theorems not required. We can simply assume that given IVPs have unique solutions for the sake of the course. | | 3.3.1 [[Example of Initial Value Problem with Infinitely Many Solutions]] | | | 3.4 [[Fundamental theorem of calculus]] (proof not required) & [[Solution to Trivial Differential Equation Initial Value Problem]] | | | 3.5 [[Stationary Point of First Order Autonomous Scalar Ordinary Differential Equation]] & [[Stable Stationary Point of First Order Autonomous Scalar Ordinary Differential Equation]] | 3.6 [[Condition for Stability of Stationary Point of First Order Autonomous Scalar Ordinary Differential Equation]] | 3.7 [[Stationary Points of Population Growth Model]] | | 3.10 [[Separable Differential Equation]] | 3.8 [[Constant Solution to Initial Value Problem For Separable Equation]] | | | | 3.9 [[Implicit Solution to Initial Value Problem for Separable Equation]] | 3.10 [[Solution to Population Growth Model]]. | | | 3.12 [[Implicit Solution to First Order Linear Ordinary Differential Equation Initial Value Problem]] | 3.13 [[Estimating the time of death]] | | | 3.14 [[Solution to Bernoulli Equation]] | | # 4. Waves | Definitions | Theorems | Examples | | -------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------- | ---------------------------------------- | | 4.1 *[[Initial Value Problem for Second Order Linear Scalar Ordinary Differential Equation]] | | | | | 4.3 *[[Solution Space of Homogenous Second Order Linear Ordinary Differential Equation Forms a Vector Space]] | | | 4.4 *[[Linearly Independent Functions]] | 4.5 *[[Solution Space of Homogenous Second Order Linear Ordinary Differential Equation is Two-dimensional]] | | | | 4.8 *[[Superposition Principle for non-homogenous Second Order Linear Ordinary Differential Equation]] | | | | 4.2 [[Solution to homogenous 2nd order linear scalar ODE with real coefficients]] | | | | 4.2.4 [[Solution to Inhomogeneous Second Order Linear Scalar Ordinary Differential Equation with Real Coefficients]] | 4.2.5 *[[Solution to RLC Circuit Model]] | # 5. Dimension | Definitions | Theorems | Examples | | ---------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------ | | 5.1 [[Fundamental Dimensions]] & [[Derived Dimensions]] | 5.1.1 *[[Product Rule For Dimensions]] | | | 5.2 [[Dimensional Homogeneity]] & [[Dimensional Completeness]] | | | | 5.3 [[Scaling Procedure]] | | 5.6.2 [[Scaled Model for Projectile Launched Vertically From The Earth]] | | 5.5 [[Scaling Procedure for Scalar Ordinary Differential Equations]] | | | | 5.6 You should understand [[Perturbed Linear Ordinary Differential Equation]]. Definition not given. | 5.6.2 [[Regular Perturbation Method for Ordinary Differential Equation]] | 5.6.2 [[Regular Perturbation of Scaled Model for Projectile Launched Vertically From The Earth]] | # 6. Interaction | Definitions | Theorems | Examples | | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------- | | 6.1 [[Law of Mass Action]] | | | | 6.2.1 [[2 x 2 System of First Order Ordinary Differential Equations]] | 6.5 [[Reduction of Order of Scalar Second Order Ordinary Differential Equations]]. You should also be able to convert 2 x 2 systems to second order equations. | | | 6.3.1 [[Autonomous 2 x 2 System of First Order Ordinary Differential Equations]] & [[Stationary Point of Autonomous 2 x 2 System of First Order Ordinary Differential Equations]] | | | | 6.3.2 [[Direction Field]] & [[Phase Portrait]] | 6.3.3 A stationary point of an autonomous 2 x 2 system is a point $(\bar{x}_{1},\bar{x}_{2})\in \mathbb{R}^{2}$ satisfying $(f_{1}(\bar{x}_{1},\bar{x}_{2}), f_{2}(\bar{x}_{1},\bar{x}_{2}))=(0,0).$ The [[Hartman-Grobman Theorem\|Hartman-Grobman theorem]] asserts that the stationary point is stable iff the [[Linearisation near Stationary Point of Autonomous 2 x 2 System of First Order Ordinary Differential Equations\|linearisation of the system near the stationary point]] is stable about the origin. | | | [[Homogenous Linear 2 x 2 System of First Order Ordinary Differential Equations with Real Coefficients]] | 6.4 [[Solution to Homogenous Linear 2 x 2 System of First Order Ordinary Differential Equations with Real Coefficients]] | |