This workshop represents the final moment of the research project Special functions and PDEs: Algorithms and Software Development, funded by the National Group for Scientific Computing (GNCS) of INdAM. The project involved the study of some fundamental equations in mathematical physics, which represent a significant field of research in both mathematics and physics.
The approach is based on some specific features that emerge in the study of certain classes of equations, relative to the investigation methodology; in particular, while series solution techniques (Frobenius method) allow deriving solutions for many ordinary differential equations, but only under certain conditions, on the other hand, there is a need to construct solutions with numerical techniques that allow the development of predictive tools, such as equilibrium codes, that include a broader range of parameters.
Many of the results obtained, both analytically and numerically have led to the definition of new classes of special functions with useful relationships in various fields of mathematics and physics and beyond. Given that for some families of ordinary second-order equations, only one numerically satisfactory solution has been deduced between the two required fundamental solutions, and that the solutions are local (in the sense that they have narrow regions of convergence and fail for large parameters), one of the goals of the project is to expand the coverage region of the solutions and achieve their numerical stability.
From a numerical perspective, we studied the accuracy of the algorithm for the solution functions implemented in Wolfram Mathematica and test the algorithm's performance for both new parameter combinations and real-world applications. An important aspect studied concerns the problem of connection formulas for different solutions of the equations under study (such as Heun-type equations) generated at different singular points. The computational efficiency of the algorithm implemented in Mathematica has been examined and compared with that of alternative methods, to allow for the assessment of both the robustness and the overall reliability of the implemented algorithm. In particular, computational techniques based on Radial Basis Functions (RBFs) have been developed and calibrated for the numerical solution of these problems. The proposed models, the numerical discretization used, and the related solution procedures has been studied; the software resulting from the implementation of these techniques has been compared with NDSolve.