## 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 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…

## 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…

## Linear Integral Equations (IEL2)

Recollect An operator is called linear if . Integral Operator We can write an integral equations in terms of a linear operator, as in following examples. Let . Then we can write as . Similarly, can be rewritten in terms of a operator as well. Linear Integral Operator An integral operator is linear if it…