Available online at www.sciencedirect.com Fuzzy Sets and Systems 219 (2013) 68 – 80 www.elsevier.com/locate/fss On fuzzy solutions for partial differential equations Ana Maria Bertone a , Rosana Motta Jafelice a ∗ , Laécio Carvalho de Barros b , Rodney Carlos Bassanezi c a FAMAT, Federal Universityof Uberlândia, 38408-100 Uberlândia MG, Brazil b IMECC, State University of Campinas, 13083-859 Campinas SP, Brazil c CMCC, Federal University of ABC, 09210-170 São Paulo SP, Brazil Received 20 January 2012; received in revised form 5 December 2012; accepted 6 December 2012 Available online 28 December 2012 Abstract In this study we investigate heat, wave and Poisson equations as classical models of partial differential equations (PDEs) with uncertain parameters, considering the parameters as fuzzy numbers. The fuzzy solution is built from fuzzification of the deterministic solution. The continuity of the Zadeh extension is used to obtain qualitative properties on regular -cuts of the fuzzy solution. We prove the stability with respect to the initial boundary data, and show that as time goes to zero, the diameter of the fuzzy solution converges to zero and, as a consequence, to the cylindrical surface determined by the curve of the degree of membership. Numerical simulations are used to obtain a graphical representation of the fuzzy solution and a defuzzification of this solution is obtained using the center of gravity method. We theoretically show that the surface obtained by defuzzification with the plane determined by fixing time is indeed the solution of the same initial boundary problem for this time-point for the heat and Poisson equations and, in a particular case, for the wave equation. The deterministic solution and the defuzzified surface intercept are numerically compared using the Euclidean distance. © 2013 Elsevier B.V. All rights reserved. Keywords: Fuzzy numbers; Partial differential equations; Analysis 1. Introduction Physical models often have some uncertainty in their parameters and estimates are usually based on statistical methods and experimental data. Since Zadeh [1] introduced the concept of fuzzy sets, there has been a great deal of research in this area, including studies of fuzzy partial differential equations (PDEs). Some studies considered application of PDEs with fuzzy parameters obtained through fuzzy rule-based systems [2,3]. Oberguggenberger described weak and fuzzy solutions for PDEs [4] and Chen et al. presented a new inference method with applications to PDEs [5]. More recently Leite and Bassanezi used Zadeh’s extension principle to determine a fuzzy solution for a PDE with initial fuzzy conditions [6]. ∗ Corresponding author. E-mail addresses: anamaria@famat.ufu.br (A. Maria Bertone), rmotta@ufu.br (R. Motta Jafelice), laeciocb@ime.unicamp.br (L. Carvalho de Barros), rodney@ime.unicamp.br (R. Carlos Bassanezi). 0165-0114/$-see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2012.12.002