A Structure-Preserving Modified Exponential Method for the Fisher–Kolmogorov Equation

In this work, we propose an exponential-type discretization of the well-known Fisher’s equation from population dynamics. Only non-negative, bounded and monotone solutions are physically relevant in this note, and the discretization that we provide is able to preserve these properties. The method is a modified explicit exponential technique which has the advantage of requiring a small amount of computational resources and computer time. It is worthwhile to notice that our technique has the advantage over other exponentiallike methodologies that it yields no singularities. In addition, the preservation of the properties of non-negativity, boundedness and monotonicity are distinctive features of our method. As consequences of the analytical properties of the technique, the method is capable of preserving the spatial and the temporal monotonicity of solutions. Qualitative and quantitative numerical simulations assess the convergence properties of the finite-difference scheme proposed in this manuscript

Note on a Picard-like Method for Caputo Fuzzy Fractional Differential Equations

A Picard-like approach which has been used to solve a class of Volterra integro-differential equations, is extended in this manuscript to solve fuzzy fractional differential equations. Such technique uses quadrature rules and Picard’s iterations in the fuzzy context. In spite of this, it is conceived to become a non-recursive scheme, in terms of operational matrices, in the linear regime. Some properties of the method are thoroughly discussed, and some numerical examples are provided in order to illustrate the effectiveness of the approach