F-TRANSFORM FOR NUMERICAL SOLUTION OF TWO-POINT BOUNDARY VALUE PROBLEM

Document Type: Research Paper

Authors

1 University of Ostrava, Centre of Excellence IT4Innovations, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 701 03 Ostrava 1, Czech Republic

2 University of Ostrava, Centre of Excellence IT4Innovations, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 701 03 Ostrava 1, Czech Republic

Abstract

We propose a fuzzy-based approach aiming at finding numerical solutions to some classical problems. We use the technique of F-transform to solve a second-order ordinary differential equation with boundary conditions. We reduce the problem to a system of linear equations and make experiments that demonstrate applicability of the proposed method. We estimate the order of accuracy of the proposed method. We show that the F-transform-based approach does not only extend the set of its applications, but has a certain advantage in the solution of ill-posed problems.

Keywords


[1] V. I. Arnold and R. Cook, Ordinary Diff erential Equations, Springer Textbook, Springer,
Berlin Heidelberg, 1992. ISBN 9783540548133.
[2] U. Ascher, Collocation for two-point boundary value problems revisited, SIAM J. Numer.
Anal. 23 (1986), 596{609.
[3] K. E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., John Wiley, Iowa, USA,
1988.
[4] N. S. Bakhvalov, N. P. Zhidkov and G. M. Kobelkov, Numerical Methods, Nauka, Moskva,
(1987), 600 p.
[5] W. Chen and Y. Schen, Approximate solution for a class of second-order ordinary diff erenci-
tal equations by the fuzzy transform, Intelligent and Fuzzy Systems. 27 (2014), 73{82.
[6] H. Keller, Numerical Methods for Two-Point Boundary Value Problems, Ginn (Blaisdell),
Boston, 1968.
[7] A. Khastan, I. Perfi lieva and Z. Alijani, A new fuzzy approximation method to Cauchy prob-
lems by fuzzy transform, Fuzzy Sets and Systems, 288 (2016), 75{95.

[8] A. Khastan, Z. Alijani and I. Perfi lieva, Fuzzy transform to approximate solution of two-
point boundary value problems, Mathematical Methods in the Applied Sciences. (2016),
http://dx.doi.org/10.1002/mma.3832.
[9] I. Perfi lieva, Fuzzy transforms: Theory and applications, Fuzzy Sets and Systems, 157 (2006),
993{1023.
[10] I. Perfi lieva, Fuzzy approach to solution of diff erential equations with imprecise data: appli-
cation to reef growth problem, in: R.V. Demicco, G.J. Klir (Eds.), Fuzzy Logic in Geology,
Academic Press, Amsterdam, (2003), 275-300.
[11] I. Perfi lieva, M. Holcapek and V. Kreinovich, A new reconstruction from the F-transform
components, Fuzzy Sets and Systems, 288 (2016), 3{25.
[12] A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Springer, New York, 2007.
[13] P. Stevuliakova, I. Perfi lieva and R. Valasek, R., A New Approach to Fuzzy Boundary Value
Problem, in: World Scientifi c Proceedings Series on Computer Engineering and Information
Science, 10 (2016), 276{281.