Numerical Methods and Series solution of Equations

Numerical Solution of ordinary Differential Equations and Series solution of Bessel's and Legendre's Equation
What you'll learn
You will learn Numerical Solution of ordinary Differential Equations of 1st Order
You will learn Runge Kutta method of 4th order and numerical predictor and corrector methods
You will learn the series solution of Bessel's Differential Equation leading to Bessel Functions
You will learn properties of Bessel Functions and series solution of Legendre Differential Equations
Requirements
You need to have a basic knowledge of higher order differentiation and knowledge of Beta and Gamma Functions.. These are available in my earlier courses in case you need it.
Description
Welcome to this course on Numerical Methods and Series solutions of Differential Equations. This course is primarily intended for you if you are studying Math in College and if you are learning Engineering Math. Tips to help you understand Math better.
You will start with a brief introduction to Taylor Series method and how to use Taylor Series method to solve an equation upto 4 decimal places. As a modification of Taylor series method, you will learn modified Euler's theorem and how to use it to solve equations. As a side note, these formulas involve a lot of calculations at the problem solving stage.
Next, you will be introduced to Runge Kutta method of 4th order and how to use it in solving problems. 2 predictor and corrector methods are taught here, namely Milne' s Predictor method and Adam Bashforth methods. The numericals here involve several iterations and have been explained step by step.
In lesson 3, you will be introduced to Bessel's Differential equation and how to solve it. The solution is rather lengthy and has been explained keeping all steps in mind. This will lead you to the Bessel's function at the end . Note the use of Gamma functions in Bessel's function is shown.
Lesson 4 is on Bessel's function and it's properties. The orthogonality property of the Bessel's function is also proved leading to two cases, one of which leads to Lommel's Integral formula.
In Lesson 5, you will learn about Legendre Differential equation and how to solve it using the power series method. As a conclusion to this , you will learn about Legendre functions and how they are derived from Legendre Differential Equations. How Legendre Polynomials lead to  Rodrigue's formula is also shown.
Lesson 6 is a problem solving session where you will learn to use Rodrigue's formula to solve problems.
The course concludes with an assignment which discusses possible questions that can be asked. Note that each of these questions have been discussed during the course.
An important point to keep in mind is that this course is highly theoretical and involves you to write these proofs to gain mastery.
Motivating you to learn Mathematics!


Who this course is for:
Students in college having a math major or Engineering Math students.

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