Integral Equations Inhomogeneous Fredholm integral equations of the second kind,
involve solving for , where is the integral kernel, is a parameter, and is the inhomogeneous term [1, 2]. As a specific example, consider Love's equation
which arises in electrostatics [3, 4]. An excellent approximation to the solution can be found in the form of a
The coefficients for , are determined by solving numerically the simultaneous linear algebraic equations that result by evaluating the integral equation (2) at the Here is a direct implementation for . The Chebyshev points are calculated using arbitrary-precision arithmetic.
Instead of computing the points using Table, we have used the Listable attribute of Cos. We introduce the undetermined coefficients .
We evaluate the left-hand side of equation (2) at the Chebyshev points.
Note that we have replaced the sum in the Chebyshev-series (3) by a dot product and have used the Listable attribute of ChebyshevT to compute the value at
To compute the right-hand side of equation (2) we could use a Chebyshev quadrature formula [5, 6]. Instead we use NIntegrate for simplicity, again using arbitrary-precision arithmetic.
Solving the resulting simultaneous linear algebraic equations we obtain the coefficients for .
The solution is obtained from the Chebyshev-series (3).
To see how well the Chebyshev-series solution satisfies Love's equation (2), we substitute it back into the equation and plot the difference of the left- and right-hand sides. The Chebyshev points are also displayed and it is apparent that the
As an aside, we can get a bound for the
Copyright © 2002 Wolfram Media, Inc. All rights reserved. |