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…

Integral Equations: Introduction (L1)

We are familiar with differential equations: an equation involving one (or more) unknown function and its derivatives. Similarly, an integral equation is an equaiton in which an unknown function appears under one or more integral signs. For example, for , the equations where the function g(s) is the unknown function while all the other functions…