On fuzzy solutions for partial differential equations

The main goal of this work is to study the classical models of Partial Differential Equations (PDE) as the heat, wave and Poisson with uncertain parameters, considering the parameter as a fuzzy number. The fuzzy solution is built from the fuzzification of the deterministic solution. The continuity of the Zadeh extension is used to obtain qualitative properties upon the regular α − cuts of the fuzzy solution. The stability with respect to the initial-boundary data is proven, showing that, when 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 maximal degree of membership. Numerical simulations are used to obtain a graphical representation of the fuzzy solution and the average of this fuzzy solution, obtained using the center of gravity method. It is theoretically shown that the surface obtained by defuzzification, intersection with the plane t = t is indeed the solution of the same initial-boundary problem for the time t, for the heat and Poisson equations and, in a particular case, for the wave equation. The deterministic solution at the time that the defuzzificated surface is intercepted, are numerically compared to each other via the Euclidean distance.