Non-probabilistic uncertainty analysis of forest fire model by solving fuzzy hyperbolic reaction–diffusion equation Smita Tapaswini, S. Chakraverty n Department of Mathematics, National Institute of Technology Rourkela, Odisha 769008, India article info Article history: Received 21 October 2013 Received in revised form 22 February 2014 Accepted 5 April 2014 Available online 24 April 2014 Keywords: Fuzzy number Trapezoidal fuzzy number Gaussian fuzzy number Fuzzy hyperbolic reaction–diffusion equations abstract This paper investigates the uncertain rate of burning trees by solving fuzzy hyperbolic reaction–diffusion equation with different uncertain initial conditions. Uncertainties present in the initial conditions are modelled through trapezoidal and Gaussian convex normalised fuzzy sets. Obtained solutions are depicted in term of figures and tables to show the efficiency and reliability of the present analysis. Comparisons have been made with the existing results in special cases. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Forest fire is the most common hazard in forests and so it is an important and critical issue in our environment. Every year, large area of forest is burnt in various countries throughout the world. Forest fires are mainly due to natural causes such as lightning, high atmospheric temperature or by human negligence, viz. cigarette, electric spark or any other source of ignition. This destruction represents a huge amount of direct cost. Simulation of forest fire propagation can serve several obvious purposes. Forest fire is a complex large scale natural phenomenon that takes into account both the chemicophysical aspect of the combustion of the forest stratum that produces heat and the local meteorological forecast. Prediction of forest fire behaviour is an important element in the management of forests as well as in assessing ecological effects. Conceptually, fire prediction models fall into two categories such as deterministic and uncertain. Most fire prediction models are deterministic, incorporating physical mechanisms for fire spread. An early influential model of this type is the “Rothermel model” [1]. FARSITE [2] is the most popular of the deterministic and mechanistic models of forest fire growth and spread. Another important deterministic model is Prometheus [3]. Also an example of deterministic lattice is given by Berjak and Hearne [4]. As regards, various researchers are trying to develop efficient dynamical model for fire spread or fire propagation in forests. Prevedel [5] discussed the mapping and monitoring of the forest fire area. Assessment of vegetation change has been investigated by Jakubauska et al. [6] whereas restoration of burned areas is studied by Greer [7]. Perez et al. [8] analysed the effect of wind and slope when scaling the forest fire rate of spread by laboratory experiments. Viegas and Simeoni [9] described the eruptive behaviour of forest fires to minimise accidents. Grishin [10] explained the static and dynamic models of forest fire hazard. Static models provide the prediction for all fire hazard seasons, and dynamic one, models for each day of the fire hazard season. In terms of prediction reliability, dynamic models are more prefer- able. A new feature regarding the forest fire safety was developed by Zhong et al. [11] to analyse the statistical data on forest fires in China. Based on their experiment, Walker and Stocks [12] mea- sured true flame temperatures in the forest fire field. Boychuk et al. [13] considered stochastic forest fire growth model to determine the behaviour of the rate of spread of forest fire. Reaction–diffusion equations are also being used by different researchers [14–18] for the modelling of forest fire propagation. Also Mendez and Llebot [19] used hyperbolic reaction–diffusion equations for a forest fire model. Some experimental works have also been done by various researchers for forest fire modelling. In this regard Dupuy [20] conducted laboratory fire experiments in order to investigate the effect of slope on fire behaviour for different levels of fuel load. Mendes-Lopes et al. [21] carried out an extensive set of experiments in order to collect data to validate fire propagation. Simeoni [22] explains the experimental study of wildland fires and develops fire- Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/firesaf Fire Safety Journal http://dx.doi.org/10.1016/j.firesaf.2014.04.002 0379-7112/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. Tel.: þ91 661 2462713; fax: þ91 661 2462713 2701. E-mail addresses: smitatapaswini@gmail.com (S. Tapaswini), sne_chak@yahoo.com (S. Chakraverty). Fire Safety Journal 66 (2014) 8–14