Classification of Integral Equations (IEL3)

Recollect

Recollect discussion about integral equations of the form

h(s)g(s)=f(s)+\lambda \int_a^b K(s,t)g(t)dt

where

  1. h(s), f(s) and K(s,t) are known real/complex functions,
  2. g(s) is the unknown function
  3. K(s,t) is called the kernel
  4. \lambda is a non-zero real or complex parameter.
  5. Limits, a is constant and b may be fixed or a variable.

Classification of Integral Equations

Integral equations are classified based on two criteria:

  1. Classification 1: Whether limits of integral (a and b) are fixed, variable or \pm\infty or based on inifnities of Kernel.
  2. Classification 2: Presence/Absence of known functions h(s) and f(s).
Classification 1: Based on region of integration/limits and infinities of Kernel

An linear integral equations can be classified into 3 types based on this criteria:

  1. Fredholm Integral Equations: An equation is called Fredholm integral equation if the limits of integration, a and b are fixed. These type of integral equations were first studied by Erik Ivar Fredholm, hence was named after him. For example, g(s)=s+ \lambda \int_1^2 st^2g(t)dt has fixed limits 1 and 2 hence is a Fredholm Integral Equation.
  2. Volterra Integral Equations: An equation is called Volterra Integral Equations if b is a variable. For example, g(s)=s+\lambda\int_1^s (s+t)g(t)dt is a Volterra Integral Equation.
  3. Singular Integral Equations: An integral equation is called singular integral equation if (a) upper or lower limits or both are \pm\infty (b) Kernel becomes infinite at one or more points within the range of integration. For example, g(s)=f(s)+\lambda\int_{-\infty}^{\infty}(exp-|s-t|)g(t)dt and f(s)=\int_0^s\left[\frac{1}{(s-t)^\alpha}\right]g(t)dt are singular integral equations.
Classification 2: Based on presence/absence of known functions

Again, there are three special types:

  1. First Kind: h(s) = 0. Then integral equation is of the form f(s)+\int_a^b K(s,t)g(t)dt = 0
  2. Second Kind: h(s) = 1. Then integral equation is of the form g(s) = f(s)+\int_a^b K(s,t)g(t)dt
  3. Homogenous Equation of Second Kind: h(s) = 1 and f(s) = 0. Then integral equation is of the form g(s) = \int_a^b K(s,t)g(t)dt
  4. Carleman’s Thrid Kind: A general linear integral equation given at the beginning of the document is called Carleman’s Third Kind.

h(s)g(s)=f(s)+\lambda \int_a^b K(s,t)g(t)dt

An integral equation that is Fredholm and of First kind, is called Fredholm IE of First Kind. Similarly, an IE that is Volterra and of second kind is called Volterra Equation of Second Kind and so on.

Extra Reading

  1. Wikipedia: Fredholm Integral Equations (link)
  2. Wikipedia: Erik Ivar Fredholm (link)
  3. Wikipedia: Volterra Integral Equations (link)
  4. Wikipedia: Vito Volterra (link) (Check out the Biography Section – yet another revolutionary mathematician!))
  5. Gaurav Tiwari’s Notes: Same content, explained differently.

NPTEL: Parallel Lecture

This lecture plays from 10th minute. Do watch the video from beginning, where professor explains how Integral Equations occur in mechanics.

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