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Numerical Solution for Fuzzy Partial Differential Equations with Interactive Fuzzy Boundary Conditions

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Explainable AI and Other Applications of Fuzzy Techniques (NAFIPS 2021)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 258))

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Abstract

This paper presents a numerical method to solve fuzzy partial differential equations, where the initial and boundary conditions are given by interactive fuzzy numbers. Interactivity is a relationship between fuzzy numbers that is associated with the notion of joint possibility distribution. From this approach, different solutions are presented to the fuzzy partial differential equation. In addition to the method, an algorithm is proposed to determine which of these solutions is most suitable for a given problem. The numerical solution is given by the finite difference method, adapted for the arithmetic operations of interactive fuzzy numbers. The method is applied to the heat equation, in order to illustrate the results.

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Author information

Authors and Affiliations

  1. National Center for Research in Energy and Materials, São Paulo State University, São Paulo, Brazil

    Vinícius F. Wasques

Authors
  1. Vinícius F. Wasques
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Editors and Affiliations

  1. Purdue University, West Lafayette, IN, USA

    Julia Rayz

  2. Purdue University, West Lafayette, IN, USA

    Victor Raskin

  3. University of Alberta, Edmonton, AB, Canada

    Scott Dick

  4. University of Texas at El Paso, El Paso, TX, USA

    Vladik Kreinovich

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Wasques, V.F. (2022). Numerical Solution for Fuzzy Partial Differential Equations with Interactive Fuzzy Boundary Conditions. In: Rayz, J., Raskin, V., Dick, S., Kreinovich, V. (eds) Explainable AI and Other Applications of Fuzzy Techniques. NAFIPS 2021. Lecture Notes in Networks and Systems, vol 258. Springer, Cham. https://doi.org/10.1007/978-3-030-82099-2_44

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