Summary of MAT323 PERSAMAAN PEMBEZAAN II (DIFFERENTIAL EQUATIONS II)

MAT323 PERSAMAAN PEMBEZAAN II (DIFFERENTIAL EQUATIONS II)

The course consists of three important topics in differential equations. First, basic ideas of modern ordinary differential equations in the form of autonomous systems and phase space are introduced. Next, boundary value problems in the context of Sturm-Liouville eigenvalue problems are discussed. Lastly, techniques to analytically solve second-order partial differential equations are elaborated and applied.

Lecturer: PROFESOR MADYA DR. MOHD HAFIZ BIN MOHD

Skill Level: Beginner

Course Outline for MAT323

Announcements

MAT323 Whatsapp Group

MAT323 2023/2024 Course Outline &amp; Planning

16 October - 22 October

Lecture Notes on Section 1.1-1.2.1: Introduction to a System of ODEs

Group Choice

23 October - 29 October

Lecture Notes on Section 1.2.2: Method of Variation of Parameters

Lecture on Section 1.2.2: Method of Variation of Parameters

Lecture on Section 1.3 - Matrix Exponential Method (Example 1.2)

Lecture on Section 1.3 - Matrix Exponential Method (Examples 1.3-1.4)

Lecture Notes on Section 1.4-1.5: Introduction to Nonlinear Systems of ODE and Local Stability Analysis

30 October - 5 November

Lecture on Section 1.4.1: Recap on Summary of Typical Phase Portraits (aka Section 1.5 in Notes)

Lecture on Section 1.4.2 (i): Stability Analysis of Real and Distinct Eigenvalues (Ex.1 - Ex.2)

Lecture on Section 1.4.2(ii): Stability Analysis of Repeated &amp; Complex Eigenvalues (Ex.3 - Ex.6)

Tutorial 1: Linear ODE Systems and Methods of Solving

Tutorial 1: Hints and Partial Answers

Lecture Notes on Section 1.6-1.6.1: Nonlinear Systems with Application to Biology

6 November - 12 November

Lecture Notes on Section 1.6.2: A Nonlinear ODE for Lotka-Volterra (LV) Prey Predator Model

Lecture on Section 1.6.2 (i): A Nonlinear ODE for Lotka-Volterra (LV) Prey Predator Model

Lecture on Section 1.6.2 (ii): Outcomes of LV Prey-Predator Model &amp; Distinct Ecological Systems

Modelling Lab 1 - Section 1.7 (i): Installation of XPPAUT

Modelling Lab 2 - Section 1.7 (ii): Step-by-step to Solve ODE in XPPAUT (Using Nonlinear ODE System)

Code for Section 1.7 (ii): XPPAUT Code for Prey-Predator Model

13 November - 19 November

Lecture Notes on Section 2.1-2.2: Introduction to PDE and Method of Characteristics

Lecture Notes on Section 2.3: Mathematical Physics Equations &amp; Solving Wave Equation Using d’Alembert’s Method

Lecture on Section 2.1.2 (i): Construction &amp; Geometric Interpretation of MOC

Lecture on Section 2.1.2 (ii): Construction &amp; Geometric Interpretation of MOC

Lecture on Section 2.2 (i): Classification of PDE

Lecture on Section 2.2 (ii): Examples 2.2-2.4 for the Classification of PDE

Lecture on Section 2.3: Mathematical Physics Equations

Tutorial 2: Non-Linear ODE Systems, Stability Analysis and their Applications

Tutorial 2: Hints and Partial Answers

20 November - 26 November

Lecture Notes on Section 2.3.1-2.4A

Lecture on Section 2.3.1 (i): Solving a Wave Equation Using the d’Alembert Method of Solution

Lecture on Section 2.3.1 (ii): Solving a Wave Equation Using the d’Alembert Method of Solution

Lecture on Section 2.4A: Solutions of PDE Using Methods of Solutions for ODE (Method 1 - Ex. 2.6)

Lecture on Section 2.4A: Solutions of PDE Using Methods of Solutions for ODE (Method 1 - Ex. 2.7)

Lecture on Section 2.4A: Solutions of PDE Using Methods of Solutions for ODE (Method 2 - Ex. 2.8)

Lecture Notes on Section 2.5-2.6

Lecture on Section 2.5 (i): Some Special Linear Homogeneous PDEs (Proof of Theorem 2.10 - Eq (2.20))

Lecture on Section 2.5 (ii): Some Special Linear Homogeneous PDEs (Proof of Theorem 2.10 - Eq (2.23))

Lecture on Section 2.6 (i): Solving Linear PDE Using Variation of Parameters (Ex. 2.12)

Lecture on Section 2.6 (ii): Solving Linear PDE Using Variation of Parameters (Ex. 2.13A)

Ex. 2.13B: Tutorial 4 - Question 6

Tutorial 3: Introduction to PDEs and Method of Characteristics

Tutorial 3: Tips / Guidance on Question 2

Partial / Final Answers for Tutorial 3

27 November - 3 December

Lecture Notes on Section 2.7: Bernoulli's Separation of Variables Method

Lecture on Section 2.7 (i): Bernoulli's Separation of Variables Method

Lecture on Section 2.7 (ii): Method of Separation of Variables (Further Examples)

Modelling Lab 2: XPPAUT Code for ODE Model Q10 (Tutorial 2)

Extra Manual for Group Further Practice on Parameter Variation (Bifurcation Analysis) in XPPAUT

4 December - 10 December: Mid Sem Break

MAT323 Group Project: Modelling Biological Systems using Non-Linear ODEs

GROUP PROJECT: TOPIC REGISTRATION

MAT323 Group Project Preparation Guidelines: Presentation &amp; Report

11 December - 17 December

Lecture Notes: Section 3.1-3.2

Derivation of FS Coefficients

Lecture Notes: Section 3.3-3.5

18 December - 24 December

Partial / Final Answers for Tutorial 4

Tutorial 4: Methods of Solving PDE

Lecture Notes: Section 3.6-3.7

Lecture on Section 3.7: Vibrating Strings and the One-Dimensional Wave Equation

Lecture on Section 3.7.1: Problem A - Solution of Wave Equation without Initial Velocity

25 December - 31 December

Lecture on Section 3.7.1: Example 9 - Solution of Wave Equation without Initial Velocity

Lecture on Section 3.7.2: Problem B - Solution of Wave Equation with Initial Velocity

Lecture on Section 3.8 (i): Laplace Equation and Steady State Solutions

Lecture on Section 3.8 (ii): General Formulation of Laplace Equation in 2D Space

Lecture on Section 3.8.1: Laplace Equation (Special Case) and Fourier Series Solutions

Group Assignment 2

MAT323 2023/2024 Course Outline & Planning

Lecture on Section 1.4.2(ii): Stability Analysis of Repeated & Complex Eigenvalues (Ex.3 - Ex.6)

Lecture on Section 1.6.2 (ii): Outcomes of LV Prey-Predator Model & Distinct Ecological Systems

Lecture Notes on Section 2.3: Mathematical Physics Equations & Solving Wave Equation Using d’Alembert’s Method

Lecture on Section 2.1.2 (i): Construction & Geometric Interpretation of MOC

Lecture on Section 2.1.2 (ii): Construction & Geometric Interpretation of MOC

MAT323 Group Project Preparation Guidelines: Presentation & Report

Lecture on Section 4.1 (i): Boundary Value Problems, Sturm‐Liouville (SL) Theory & Equations

Lecture on Section 4.1 (ii): SL Equations & Boundary Conditions

Lecture on Section 4.1 (v): SL Equations with Distinct BC (Ex. 7 & 8)

Tutorial 6: Steady State & Transient Solutions of PDE

Test 2 & Group Project Presentation Slots

Revision Ch 2 Q2 & Q3: Past-Year MAT 323 Questions 2018 & Additional Questions