This paper presents the algorithms for solving the inverse problems on models with the fractional derivative. The presented algorithm is based on the Real Ant Colony Optimization algorithm. In this paper, the examples of the algorithm application for the inverse heat conduction problem on the model with the fractional derivative of the Caputo type is also presented. Based on those examples, the authors are comparing the proposed algorithm with the iteration method presented in the paper: Zhang, Z. An undetermined coefficient problem for a fractional diffusion equation. Inverse Probl. 2016, 32.

This paper presents the application of the swarm intelligence algorithm for solving the inverse problem concerning the parameter identification. The paper examines the two-dimensional Riesz space fractional diffusion equation. Based on the values of the function (for the fixed points of the domain) which is the solution of the described differential equation, the order of the Riesz derivative and the diffusion coefficient are identified. The paper includes numerical examples illustrating the algorithm’s accuracy.

The paper presents an application of the homotopy analysis method for solving the one-phase fractional inverse Stefan design problem. The problem was to determine the temperature distribution in the domain and functions describing the temperature and the heat flux on one of the considered area boundaries. It was demonstrated that if the series constructed for the method is convergent then its sum is a solution of the considered equation. The sufficient condition of this convergence was also presented as well as the error of the approximate solution estimation. The paper also includes the example presenting the application of the described method. The obtained results show the usefulness of the proposed method. The method is stable for the input data disturbances and converges quickly. The big advantage of this method is the fact that it does not require discretization of the area and the solution is a continuous function.

The method of evaluating the integrals through use of the matrix inversion, presented here, was introduced by J.W. Rogers and then generalized by Matlak, Słota and Wituła. This method is still developed and one of its other possible applications is presented in this paper. This application concerns a new way of evaluating the integral sec 2n+1 x dx on the basis of the discussed method. Additionally, many other applications of the obtained original recursive formula for this type of integral are given here. Some of them are used to generate the interesting identities for inverses of the central binomial coefficients and the trigonometric limits. The historical view is also presented as well as the connections between the received and previously known identities.

The aim of this paper is the adaptation of the alternating phase truncation (APT) method for solving the two-phase time-fractional Stefan problem. The aim was to determine the approximate temperature distribution in the domain with the moving boundary between the solid and the liquid phase. The adaptation of the APT method is a kind of method that allows us to consider the enthalpy distribution instead of the temperature distribution in the domain. The method consists of reducing the whole considered domain to liquid phase by adding sufficient heat at each point of the solid and then, after solving the heat equation transformed to the enthalpy form in the obtained region, subtracting the heat that has been added. Next the whole domain is reduced to the solid phase by subtracting the sufficient heat from each point of the liquid. The heat equation is solved in the obtained region and, after that, the heat that had been subtracted is added at the proper points. The steps of the APT method were adapted to solve the equations with the fractional derivatives. The paper includes numerical examples illustrating the application of the described method.