Integral Equation
- Sciences
- 400 level
- 3 credit units
- 77 pages
- 9 units
This course, Integral Equations, explores the fundamental concepts and techniques for solving integral equations. It covers linear integral equations, Volterra and Fredholm equations, and various methods for finding approximate solutions. Students will learn to convert ordinary differential equations into integral equations, work with Eigenfunctions and Eigenvectors, and apply Laplace and Fourier transforms to solve integral equations. The course aims to equip students with the skills to solve a wide range of integral equations.
About this course
- Difficulty
- Intermediate
- Study hours
- 156 hours
- Maths
- Intermediate
- Content
- Theoretical, problem solving
- Practical work
- No
- MTH311: Real Analysis
- MTH315: Complex Analysis
- MTH321: Differential Equations
- Assignments
- Tutor marked assessments
- Final examination
What you'll read
The real module and unit structure of MTH423, taken from the course material NOUN publishes.
One paragraph, so you can see how it reads
MTH423 · UNIT 2: CONVERSIONS OF ORDINARY DIFFERENTIAL EQUATIONS INTO INTEGRAL EQUATIONS
A shop starts selling some goods. It is found that a proportion t K remains unsold at time t after the shop has purchased the goods. It is required to find the stock at which the shop should purchase the goods so that the stock of the goods in the shop remains constant (all processes are deemed to be continuous).
What you should be able to do
- Explain the basic concepts underlying linear integral equations
- Convert ordinary differential equations into integral equations
- Classify linear integral equations
- Solve Volterra integral equations using Resolvent Kernel
- Solve Fredholm equations with degenerate kernels
- Work with Eigenfunctions and Eigenvectors
- Apply Laplace and Fourier transforms to solve integral equations
What it prepares you for
- Applied Mathematician
- Numerical Analyst
- Mathematical Modeler
- Data Scientist
- Research Scientist
- Engineering
- Physics
- Finance
- Computer Science
- Data Analysis
- MATLAB
- Mathematica
- Python (NumPy, SciPy)
Where it gets hard
The units students slow down on, and what makes each one heavy.
- Module 3: Fredholm Equations
Unit 2: Eigenfunctions and Eigenvectors
The abstract nature of Eigenfunctions and Eigenvectors requires a solid foundation in linear algebra and functional analysis.
- Module 4: Integral Transforms
Unit 1: Calculation of 1st Eigenvalue
Applying the convolution theorem and inverse Laplace transforms requires careful manipulation of complex functions and contour integration techniques.
A suggested way through it
13 weeks, about 72 hours in total. Yours will differ.
- Week 1Module 1: Preliminary Concepts
Unit 1: Linear Integral Equation: Preliminary Concepts · 6 hours
Understand the basic concepts of linear integral equations.. Investigate the equations describing the displacement of a loaded elastic string.. Solve shop stocking problems..
- Week 2Module 1: Preliminary Concepts
Unit 2: Conversion of Ordinary Differential Equations into Integral Equations · 6 hours
Convert ordinary differential equations into integral equations.. Transform Sturm-Lowville problems to integral equations.. Practice transformations and conversions..
- Week 3Module 1: Preliminary Concepts
Unit 3: Classification of Linear Integral Equation · 6 hours
Classify linear integral equations.. Find approximate solutions for integral equations.. Solve related exercises..
- Week 4Module 2: Volterra Integral Equation
Unit 1: S2 Volterra Integral Equation · 6 hours
Recognize Volterra integral equations.. Identify the three types of Volterra integral equations.. Determine the Resolvent kernel of a Volterra equation..
- Week 5Module 2: Volterra Integral Equation
Unit 2: Convolution Type Kernels · 6 hours
Solve convolution type kernels of the Volterra integral using Laplace transform.. Practice inverse transforms and convolution..
- Week 6Module 3: Fredholm Equations
Unit 1: Fredholm Equations with Degenerate Kernels · 6 hours
Solve Fredholm equations with degenerate kernels.. Apply the general method of solution of Fredholm equations.. Work through examples..
- Week 7Module 3: Fredholm Equations
Unit 2: Eigenfunctions and Eigenvectors · 6 hours
Work with Eigenfunctions and Eigenvectors.. Prove that symmetric and continuous kernels possess at least one Eigenvalue.. Solve related problems..
- Week 8Module 3: Fredholm Equations
Unit 3: Representation of a Function by a Series of Orthogonal · 6 hours
Prove that functions can be represented by series of orthogonal functions.. Expand K in a series of Eigenfunctions.. Define positive kernels..
- Week 9Module 4: Integral Transforms
Unit 1: Calculation of 1st Eigenvalue · 6 hours
Apply the convolution theorem.. Calculate the first Eigenvalue of an integral equation.. Use the variational formula..
- Week 10Module 4: Integral Transforms
Unit 2: The Application of the Transform · 6 hours
Recognize integral Laplace transforms as transforms.. Derive the solution of integral equations using inverse Laplace transform.. Apply Laplace transform through worked examples..
- Week 11Module 1: Preliminary Concepts
Unit 1: Linear Integral Equation: Preliminary Concepts · 4 hours
Review Module 1: Linear Integral Equations and Preliminary Concepts. Solve additional problems related to preliminary concepts and classifications.
- Week 12Module 2: Volterra Integral Equation
Unit 1: S2 Volterra Integral Equation · 4 hours
Review Module 2: Volterra Integral Equations. Practice solving Volterra equations using Resolvent Kernel and Laplace Transforms.
- Week 13Module 3: Fredholm Equations
Unit 1: Fredholm Equations with Degenerate Kernels · 4 hours
Review Module 3: Fredholm Equations. Practice solving Fredholm equations with degenerate kernels and Eigenfunction expansions.
Preparing for the exam
- Review all unit summaries and key definitions thoroughly
- Practice solving a variety of integral equations from each module
- Focus on mastering Laplace and Fourier transform techniques
- Create concept maps linking different types of integral equations and solution methods
- Work through past exam papers to familiarize yourself with the exam format
- Prioritize understanding the underlying principles rather than memorizing formulas
- Allocate sufficient time for practice problems and review of challenging units
- Form a study group to discuss concepts and solve problems collaboratively
- Ensure a strong understanding of calculus and linear algebra concepts
- Pay close attention to the assumptions and limitations of each solution method
Questions students ask about this course
What is MTH423 about?
This course, Integral Equations, explores the fundamental concepts and techniques for solving integral equations. It covers linear integral equations, Volterra and Fredholm equations, and various methods for finding approximate solutions. Students will learn to convert ordinary differential equations into integral equations, work with Eigenfunctions and Eigenvectors, and apply Laplace and Fourier transforms to solve integral equations. The course aims to equip students with the skills to solve a wide range of integral equations.
How many units does MTH423 have?
MTH423, Integral Equation, has 9 units across 4 modules, over 77 pages of course material. You can read it one unit at a time.
How many credit units is MTH423?
MTH423 carries 3 credit units, at 400 level in Sciences.
Is MTH423 hard?
MTH423 is rated intermediate level, with intermediate mathematical content. It is mostly theoretical and problem solving work.
How long does MTH423 take to study?
About 156 hours of study, spread across its 9 units.
How is MTH423 assessed?
MTH423 is assessed by assignments, tutor marked assessments and final examination.
What do I need before starting MTH423?
MTH311: Real Analysis MTH315: Complex Analysis MTH321: Differential Equations
What can I do with MTH423?
Applied Mathematician, Numerical Analyst, Mathematical Modeler, Data Scientist and Research Scientist.