Abstract
This manuscript assesses a semi-analytical method in connection with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). Moreover, we use the aforesaid strategy to address the time-fractional Fornberg–Whitham equation (FWE) under gH-differentiability by employing different initial conditions (IC). Several algebraic aspects of the fuzzy Caputo fractional derivative (CFD) and fuzzy Atangana–Baleanu (AB) fractional derivative operator in the Caputo sense, with respect to the Elzaki transform, are presented to validate their utilities. Apart from that, a general algorithm for fuzzy Caputo and AB fractional derivatives in the Caputo sense is proposed. Some illustrative cases are demonstrated to understand the algorithmic approach of FWE. Taking into consideration the uncertainty parameter ζ∈[0,1] and various fractional orders, the convergence and error analysis are reported by graphical representations of FWE that have close harmony with the closed form solutions. It is worth mentioning that the projected approach to fuzziness is to verify the supremacy and reliability of configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.
Highlights
We demonstrate the series solutions with the aid of Elzaki Adomian decomposition method (EADM) concerning different initial conditions by employing fuzzy Caputo and ABC fractional derivative operators, respectively
The fuzzy solutions of Fornberg–Whitham equation (FWE) proposed for such flows are characterized by EADM
We illustrated two test examples of the evolutionary method to gain deeper insight into the exact–approximate solutions to validate the projected technique to attain a parametric solution for each case of the fuzzy (Caputo and ABC) fractional derivative operator
Summary
The ADM is a semi-analytical approach to solving linear-nonlinear FDEs by advantageously creating a functional series solution, initially presented by Adomian [48] Later, this approach was used with numerous transformations (such as the Sumudu, Aboodh, Laplace, and Mohand transforms), as shown in [49,50,51,52,53,54,55,56,57,58]. Our intention is to establish an efficacious algorithm for generating estimated solutions of fuzzy fractional FWE, the general FWE arising in wave breaking subject to uncertainty in IC by EADM that models the dynamics of the system being analyzed.
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