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The present work introduces a novel approach, the Adomian Decomposition Formable Transform Method (ADFTM), and its application to solve the fractional order Sharma-Tasso-Olver problem. The method’s distinctive outcomes are highlighted through a comparative analysis with established non-local Caputo fractional derivatives and the non-singular Atangana–Baleanu (ABC) fractional derivatives. To provide a comprehensive understanding, the proposed ADFTM’s approximate solution is compared with the homotopy perturbation method (HPM) and residual power series method (RPSM). Further, numerical and graphical results demonstrate the reliability and accuracy of the ADFTM approach. The novel outcomes presented in this work emphasize its capability to address complex engineering problems effectively. By demonstrating its efficacy in solving the fractional order problems, the new ADFTM proves to be a valuable tool in solving scientific problems.

The present paper deals with the study of a generalized Mittag-Leffler function operator. This paper is based on the generalized Mittag-Leffler function introduced and studied by Saxena and Nishimoto (J. Fract. Calc. 37:43-52, 2010). Laplace and Mellin transforms of this new operator are investigated. The results are useful where the Mittag-Leffler function occurs naturally. The boundedness and composition properties of this operator are established. The importance of the derived results further lies in the fact that the results of the generalized Mittag-Leffler function defined by Prabhakar (Yokohama Math. J. 19:7-15, 1971), Shukla and Prajapati (J. Math. Anal. Appl. 336:797-811, 2007), and the multiindex Mittag-Leffler function due to Kiryakova (Fract. Calc. Appl. Anal. 2:445-462, 1999; J. Comput. Appl. Math. 118:214-259, 2000; J. Fract. Calc. 40:29-41, 2011) readily follow as a special case of our findings. Further the results obtained are of general nature and include the results given earlier by Prajapati et al. (J. Inequal. Appl. 2013:33, 2013) and Srivastava and Tomovski (Appl. Math. Comput. 211:198-210, 2009). Some special cases of the established results are also given as corollaries.

The fractional ordered mathematical model offers more insights compared to integer order models. In this work, we analyzed fractional order rat bite fever model. We employ the Adams–Bashforth–Moulton method in conjunction with fractional-order derivatives in the Caputo sense to study the model. The work demonstrates how fractional derivative models offer an increased degree of flexibility to investigate memory effects and illness dynamics for a particular data set. Further, an analysis of the aforementioned model including its existence, uniqueness, and stability is considered. The distinct parameter estimation for every value of the fractional order highlights the importance of this work.

In this paper, we propose a solution of fractional logistic equation by using properties of Mittag-Leffler function.