Assessment of the performance of the hyperbolic-NILT method to solve fractional differential equations

The performance of the hyperbolic–numerical inverse Laplace transform (hyperbolic-NILT) method is evaluated when it is used to solve time-fractional ordinary and partial differential equations. With this purpose, the formalistic fractionalization approach of Gompertz and diffusion equations are used as model problems, i.e., in the Gompertz and diffusion equations the integer-order time derivative is replaced by the Caputo or Atangana–Baleanu fractional derivative or the Caputo–Fabrizio non-integer order operator. The accuracy, stability and convergence of the numerical solutions are analyzed by comparing the numerical and exact solutions. From our analysis, we obtain an independent formula of the fractional order, which together with the initial condition is used to optimize the parameter of the hyperbolic-NILT method. This expression can be implemented in linear fractional differential equations with non-homogeneous initial condition. Finally, we show that the value of the parameter is transferred throughout the time domain with the certainty that the accuracy of the inverted solution remains between certain orders of magnitude. In fact, everything indicates that this conclusion fits well with model problems that are similar (fractional linear differential equations) to those studied here and for which we highly recommended the hyperbolic-NILT method to solve them.