Numerical Solution of Fujita’s Equation

The Fujita equation is a special Fredholm integral equation of the first kind with compact integral operator and as such is a typical example of a severely ill-posed problem. It is proved that a class of Fredholm integral equation of the first kind (containing the Fujita equation) has injective integral operators and thus solutions coinciding with the corresponding minimum norm least squares solution. An application of an algorithm for a good choice of the regularization parameter gives satisfactory results for an example of Fujita’s equation. Moreover, the use of Sobolev norms for a partial regularization of the problem may improve these results considerably. The same example shows that collocation methods based on equidistant points are numerically instable and inaccurate.