## Storytelling to Introduce Applications of Laplace Transforms

“…and they couldn’t send the satellite up! Those big solar panels! Won’t fit in the rocket! If, in any way, they can pack it in the rocket – like we lazy souls pack our dresses when we go for a trip – and pull it out in the Outerspace and iron it back to the…

## Integral Equations and Calculus of Variations: Lecture Notes

Lecture Notes for IECV Coursework at MCC (From archives)

## 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 Prerequisites Property 1 Direct Comparison Test (Wikipedia): If and then is also absolutely convergent. (Make note…

## 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…

## Classification of Kernels (IEL4)

Kernels are classified based on the following criteria: Separable/Degenerate: Symmetric or Hermitian: , where * denotes the conjugate Convolution Type: Note that a kernel may be of two types at the same time. For example, is of both separable and of convolution type. It is not symmetric as . You will find more examples in…

## Classification of Integral Equations (IEL3)

Recollect Recollect discussion about integral equations of the form where h(s), f(s) and K(s,t) are known real/complex functions, g(s) is the unknown function K(s,t) is called the kernel is a non-zero real or complex parameter. Limits, a is constant and b may be fixed or a variable. Classification of Integral Equations Integral equations are classified…