Numerical solution for interval initial value problems based on interactive arithmetic

Document Type : Research Paper

Authors

1 Department of Applied Mathematics, Institute of Mathematics, Statistics and Scientific Computing, University of Campinas, Campinas, Brazil

2 Ilum School of Science, Brazilian Center for Research in Energy and Materials, Campinas, Brazil

10.22111/ijfs.2022.7206

Abstract

This work studies Interval Initial Value Problems (IIVPs), where the derivative is given by the generalized Hukuhara derivative ($gH$-derivative) and the initial condition is given by an interval. The focus of the paper is to provide the numerical approximations for the solutions associated with the $gH$-derivative of IIVPs. This article considers the Euler numerical method, where the classical arithmetic operation is adapted for intervals. The arithmetic considered here is obtained using sup-$J$ extension principle, where $J$ is a particular family of joint possibility distributions. This family gives raise to different types of interactivity and this work shows what kind of interactivity is necessary in the numerical method, in order to approximate the solution via $gH$-derivative. To illustrate the results, the paper focuses in the decay Malthusian model.

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