Euler and The Corona Pandemic

How someone who lived in 18th century is helping us fight the pandemic today? An introduction to Euler and Graph Theory, on how it is used to #fight CoVID-19 Pandemic.

#BeAProudMathematician

About a few Geometric Shapes and Brachistochrone Problem

What is a Catenary Curve? In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends. Catenary curves were for a long time misunderstood to be parabolas, and has a history behind this shape worth reading! (Continue reading on…

Illustrating Simple Variation Problem

Calculus of variations is a subject where you study the extrema (maximum or minimum) of a function of functions. We know that, to find the extrema (maximum or minimum) of a “functional” of the form we first identify and substitute in Euler Equation (Or sometimes called Euler-Lagrange Equation) to obtain a Ordinary Differential Equation, solving…

Prerequisites: Properties of Resolvent Kernels

This is a bunch of prerequisites to understand section 3.5 of Ram P. Kanwal. There are three properties listed here: is absolutely and uniformly convergent for all values of s and t in the circle It satisfies integrodifferential equations Some Extra Links Compilation of Series Convergence Tests (Wikipedia) Some important sequences and series (Blog) Generalisation…

A Simple Integral Equation

Question Find x such that Points to Ponder Is the solution unique? Leave your thoughts in the comments below!

Selected Solutions (IEL6 Quiz)

Question 1 Question: Let be the solution of the integral equation . Then A couple of solutions to this problem are discussed in Unit Quiz 1 (Question 2, only difference is that options 3 and 4 are about value of phi at x=1), even using methods mentioned in Unit 2. Question 2 For the linear…

Iterated Kernels and Fredholm Theorems (IEL6)

Method of Successive Approximation Solution Method 2 (Neumann Series Solution, Solution by Iterated Kernels) For Fredholm IE For the equation , Put g(s) = f(s) or any appropriate function of s. Find iteratively. where and solution will be of the form For Volterra IE where and solution will be of the form Example Solve the…

Selected Solutions (IEL5-Quiz)

Question 2 Let ϕ be the solution of the integral equation . Then ϕ(0)=20exp(-1) – 8 ϕ(0)=20e-8 ϕ(0)=22 – 8e ϕ(0)=22 – 8exp(-1) Solution 1 Let . Substituting in the equation, Hence, Concluding, ϕ(0)=20exp(-1) – 8. Solution 2 For a Fredholm integral equation of 2nd Kind, using , where Here, . Hence Hence the solution…

Separable Kernels and Approximation Methods (IEL5)

Recollect: General Linear Integral Equation where h(s) and f(s) are known functions g(s) is the unknown function K(s,t) is called the kernel If f(s) = 0 and h(s) = 1, is called the eigenvalue. Classification Based on limits Fredholm: Limits are constant Volterra: Limits are functions of s Singular: Either one, or both the limits…