Computational Numerical Analysis Used in AI and Data Science

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.

You'll also learn what is nth order derivative of a function and how to evaluate it. An interesting and important topic.

Learn the method of finite differences and the forward and backward difference table and how to use it in problem solving. Learn how to solve problems using the Regula Falsi Method in Numerical Analysis.

Learn how to solve equations faster using the Newton Raphson's Method. You'll learn how to apply Simpson's one third Rule, Simpson's three by eighth rule and Weddle's Rule in Numerical Integration.

   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!