Semi-Analytical View Of Time-Fractional Pdes With Proportional Delays Pertaining To Index And Mittag-Leffler Memory Interacting With Hybrid Transforms

This paper focuses on the theoretical and computational investigation of the innovative nonlinear time-fractional PDEs incorporating the Caputo and Atangana–Baleanu fractional derivatives in the Caputo context using the q̂-homotopy analysis transform method (HATM). The expected strategy employs a combination of q̂-HATM and the Jafari transform with the assistance of Caputo and Atangana–Baleanu fractional derivative operators to obtain the methodology permits of PDEs with proportional delay. The fractional operators are employed in this research to demonstrate how crucial they are in generalizing frames involving singular and nonsingular kernels. The proposed series of solutions are closely in agreement with an exact solution. Several important challenges can be addressed to illustrate the validity of the proposed method. The outcomes of the proposed framework are displayed and assessed using numerical and graphical outputs. Furthermore, the results of our suggested strategy were compared to earlier outcomes. The proposed method requires less computation and has significantly better performance. Finally, the analysis shows that the enhanced technique is both reliable and meticulous when evaluating the impact of nonlinearities in science and technology.