Numerical Solution of Integral Equations

Integral Equations

by numerical-methods.com


In their simplest form, integral equations are equations in one variable (say t) that involve an integral over a domain of another variable (s) of the product of a kernel function K(s,t) and another (unknown) function (f(s)). The purpose of the numerical solution is to determine the unknown function f. If the limit(s) on the integration domain are fixed then it is said to be a Fredholm Equation. If the limit(s) on the integration domain are not fixed then it is said to be a Volterra Equation.

 

Integral Equations

Integral equations are solved by replacing the integral by a numerical integration or quadrature formula. The integral equation is then reduced to a linear equation with the values of f at the quadrature points being unknown at the outset. The solution of the linear equation(s) gives the approximate values of f at the quadrature points.

Fredholm Equations

Fredholm equations occur in two forms: Fredholm equations of the first kind and Fredholm equations of the second kind. There is a close analogy between Fredholm equations and linear systems of equations; the functions can be viewed as vectors, the integration over the kernel function as a matrix. The solution generally involves replacing the integral equation by a linear system and then solving it. Clearly the accuracy of this method depends partially on the accuracy of the numerical integration method; the more quadrature points the better.

Fredholm Equations of the First Kind have the form

g(t) =


b

a 

K(s,t) f(s) ds

where only the function f is unknown at the outset. Usually an approximation to f(t) for a t b is required.

On applying an integral equation method, such an equation is replaced by a linear system of equations

K f = g

where K is a matrix and f and g are vectors.

Unfortunately, such equations have poor numerical properties if the kernel function is smooth at s = t. The matrix is often severely ill-conditioned; solution is hopeless.

In the special cases where K(s,t) is non-smooth at s = t, for example a singular function (e.g. K(s,t) = [1/( s-t)]), then the condition of the matrix is better and solution is possible.

Fredholm Equations of the Second Kind have the form

f(t) = l


b

a 

K(s,t) f(s) ds + g(t)

where only the function f is unknown at the outset. Usually an approximation to f(t) for a t b is required.

On applying an integral equation method, such an equation is replaced by a linear system of equations

f = lK f + g

where K is a matrix and f and g are vectors.

Solution of Fredholm Integral Equations by Collocation :: Excel Spreadsheet solution of example problem

Fredholm integral equations formulate a range of physical problems. Elliptic partial diffential equations like the Laplace or Helmholtz equations can be reformulated as Fredholm integral equations and are solved as part of the boundary element method.

Volterra Equations

A Volterra equation has the form

f(t) =


t

a 

K(t,s) f(s) ds + g(t) ,

usually we need to find f(t) in a domain [a,b].

Most methods for solving volterra equations divide the range into discrete time windows with time points t0 = a, t1,..., tn = b. Starting with f(t0) = g(t0), by using some integration method, f(t1), f(t2),... can be found in turn.

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