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]. Mathematica does not include built-in routines or packages for integral equations. However, it is straightforward to implement quite general methods for solving such equations.
As a specific example, consider Love's equation
The coefficients for , are determined by solving numerically the simultaneous linear algebraic equations that result by evaluating the integral equation (2) at the Chebyshev points for , using the Chebyshev-series (3) [5, 6].
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 all the Chebyshev points at once.
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 minimum error in the Chebyshev-series solution occurs at these points.
As an aside, we can get a bound for the maximum error directly from the PlotRange of this plot. We could use this error-bound in a recursive procedure so as to find a solution for a given maximum error.
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