Enclosure of the Solutions to Interval Fredholm Equations of the First and Second Kind Bart F. Zalewski and Robert L. Mullen Department of Civil Engineering Case Western Reserve University 10900 Euclid Ave. Cleveland, Ohio 44106, USA [email protected] Abstract Integral equations often result from a weak form formulation when computing solutions to physical systems. There exist rare cases where exact solutions to integral equations can be computed, however, most of the solutions to integral equations are obtained approximately by numerical methods. Using conventional methods, the degree of accuracy of the numerical solution cannot be verified since in general the exact solution is unknown. The method presented in this paper obtains guaranteed enclosure of the solutions to Fredholm Equation of the First and Second Kind using the developed Interval Kernel Splitting Technique. The technique was used successfully by the authors [5] to obtain guaranteed bounds on the discretization error to boundary element method. The method is general and does not require any special property of the Fredholm Equations other than that the solutions are unique and finite over their domains. The uniqueness requirement allows for solvability and the finite solution requirement allows for obtaining meaningful enclosure and is not necessary for the guarantee of the enclosure. In this paper the technique is extended to obtain guaranteed enclosure to Fredholm Equation of the Second Kind with the same restrictions on the solutions. Standard numerical approximations of the solutions are used to obtain interval linear system of equations and a transformation of the resulting system to the standard interval linear system of equations is presented. The system is parameterized with respect to the location at which the approximate solution is calculated in order to achieve finite interval bounds. Parameterized Krawczyk iteration [3,4] is reviewed for clarity and further parameterization is utilized to improve the quality of the enclosure. Examples are presented for equations of both types to illustrate the method and its behavior with increasing number of domain subdivision.

1

References: [1] Dobner, H.; Kernel-Splitting Technique for Enclosing the Solution of Fredholm Equations of the First Kind, Reliable Computing 8 (2002), pp. 469479. [2] Moore, R. E.; Interval Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J, 1966. [3] Neumaier, A.; Interval Methods for Systems of Equations, Cambridge University Press, 1990. [4] Rump, S. M.; Self-validating Methods, Linear Algebra and its Applications 324 (2001), pp. 3-13. [5] Zalewski, B. F. and Mullen, R. L.; Discretizaton Error in Boundary Element Analysis using Interval Methods, SAE 2007 Transactions Journal of Passenger Cars: Mechanical Systems V116-6, paper No. 2007-01-1482. Keywords: Interval Fredholm Equation, Interval Enclosure

2