Uncertain M-fractional differential problems: existence, uniqueness, and approximations using Hilbert reproducing technique provisioner with the case application: series resistor-inductor circuit

In this article, the principle of characterization is proposed as a new tool for solving uncertain M-fractional differential problems under firmly generalized differentiability. The study demonstrates the solvability of such issues by presenting theoretical implications on the existence and uniqueness of two uncertain M-solutions. Additionally, the study provides quantitative solutions in a novel uncertain framework using two Hilbert spaces that are combined through the kernel-based Gram-Schmidt orthogonalization technique. The proposed uncertain problems and algorithms are examined, with a focus on analyzing the solution collection, assessing convergence, and evaluating errors. The debatable Hilbert approach can solve numerous M-fractional differential problems under uncertainty, and the numerical results demonstrate the accuracy and effectiveness of the algorithm. Based on the figures, tables, and quantitative analysis, our work significantly enhances mathematical tools for solving complex M-fractional differential problems under uncertainty. By utilizing the numerical pseudocode; this advancement has the potential to make an impact on various scientific and engineering fields. The final section presents numerical notes, along with recommendations for future research directions. Additionally, an evaluation of the study's findings is provided based on the conducted analysis.