In this study, the fractional sine-Gordon model in the time-dependent variable domain using Caputo non-integer order basis derivative is presented. The main purpose is to utilize the adaptation of reproducing kernel Hilbert algorithm to construct pointwise numerical solution to variant forms of fractional sine-Gordon model in fullness of overdetermination Dirichlet boundary condition. Allocates theoretical requirements are employed to interpret pointwise numerical solutions to such fractional models on the space of Sobolev. In addendum, the convergence of the pointwise numerical algorithm and error estimates are promoted by global convergence treatises. This handling pointwise numerical solution depending on the orthogonalization Schmidt process that can be straightway carried out to generate Fourier expansion within a fast convergence rate. The soundness and powerfulness of the discussed algorithm are expounded by testing the solvability of a couple of time-fractional sine-Gordon models. Some schematic plots and tabulated results outcomes signalize that the algorithm procedure is accurate and convenient in the field of fractional sense. Ultimately, future remarks and concluding are acted with the most focused used references.

In this research study, fuzzy fractional differential equations in presence of the Atangana-Baleanu-Caputo differential operators are analytically and numerically treated using extended reproducing Kernel Hilbert space technique. With the utilization of a fuzzy strongly generalized differentiability form, a new fuzzy characterization theorem beside two fuzzy fractional solutions is constructed and computed. To besetment the attitude of fuzzy fractional numerical solutions; analysis of convergence and conduct of error beyond the reproducing kernel theory are explored and debated. In this tendency, three computational algorithms and modern trends in terms of analytic and numerical solutions are propagated. Meanwhile, the dynamical characteristics and mechanical features of these fuzzy fractional solutions are demonstrated and studied during two applications via three-dimensional graphs and tabulated numerical values. In the end, highlights and future suggested research work are eluded.

Mathematical modeling of uncertain FIDEs is an extremely significant topic in electric circuits, signal processing, electromagnetics, and anomalous diffusion systems. Based on the RKA, a touching numerical approach is considering in this study to solve groups of FFIDEs with ABC fractional distributed order derivatives. The solution-based approach ‎lies in generating infinite orthogonal basis from kernel functions, where uncertain condition is fulfilled. Thereafter, an orthonormal basis is erected to figurate fuzzy ABC solutions with series shape in idioms of η-cut extrapolation in Hilbert space A(D) and B(D). In this orientation, fuzzy ABC fractional integral, fuzzy ABC fractional derivative, and fuzzy ABC FIDE are utilized and comprised. The competency and accuracy of the suggested approach are indicating by employing several experiments. From theoretical viewpoints, the acquired results signalize that the utilize approach has several merits in feasibility and opportunity for treating with many fractional ABC distributed order models. In the end, highlights and future suggested research work are eluded.