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In this paper, we examine the time-dependent two-dimensional cable equation of fractional order in terms of the Caputo fractional derivative. This cable equation plays a vital role in diverse areas of electrophysiology and modeling neuronal dynamics. This paper conveys a precise semi-analytical method called the q-homotopy analysis transform method to solve the fractional cable equation. The proposed method is based on the conjunction of the q-homotopy analysis method and Laplace transform. We explained the uniqueness of the solution produced by the suggested method with the help of Banach’s fixed-point theory. The results obtained through the considered method are in the form of a series solution, and they converge rapidly. The obtained outcomes were in good agreement with the exact solution and are discussed through the 3D plots and graphs that express the physical representation of the considered equation. It shows that the proposed technique used here is reliable, well-organized and effective in analyzing the considered non-homogeneous fractional differential equations arising in various branches of science and engineering. [ABSTRACT FROM AUTHOR]

This work proposes a semi-analytical new hybrid approach, so-called differential -transform method (D TM), to evaluate the behavior of n-space dimensional fractional-nonlinear hyperbolic-like wave equations, where time-fractional derivative is considered in Caputo format. The D TM is the hybrid method in which projected differential transform is implemented after imposing the recently introduced integral transform, i.e., so-called transform [W. Zhao and S. Maitama, J. Appl. Anal. Comput. 10, 1223 (2020)]. The efficiency and applicability of the proposed D TM had been tested by considering three different test examples of the Caputo time-fractional nonlinear hyperbolic-like wave equations in terms of absolute error norms, and the different order D TM solutions are compared with exact solution behaviors and the existing results, for the large time level τ ∈ [ 0 , 1 0 ]. In addition, the convergence analysis of D TM is studied theoretically and verified it numerically as well as graphically, which confirms that the numerical experiments via D TM for distinct fractional orders support the theoretical findings excellently, and the presented D TM results converge to their exact solution behavior, very fast. The evaluated series approximations are expressed in the compact form of Mittag-Leffler functions. [ABSTRACT FROM AUTHOR]

The pivotal aim of the present study is to find the solution for a nonlinear system describing the aggregation process of cellular slime mold by using q -homotopy analysis transform method. The coupled system is considered within the frame of the Caputo fractional operator. We examine three different cases with distinct values of sensitivity function χ ρ = 1 , ρ and ρ 2 to exemplify the efficiency and applicability of the considered scheme. We capture the nature of the obtained results with respect to the fractional order, with distinct initial conditions, and illustrate them using 2D and 3D plots for particular values of the parameters. The considered scheme offers parameters, which help to adjust the convergence region, and we plotted the ℏ-curves to dissipate the effect in the present framework. Moreover, some simulating and important behavior of the considered model using attained results shows the prominence of the hired operator while analyzing the coupled equations and confirms the competence of the projected scheme. [ABSTRACT FROM AUTHOR]

The present study is concerned with studying the dynamical behavior of two space-dimensional nonlinear time-fractional models governing the unsteady-flow of polytropic-gas (in brief, pGas) that occurred in cosmology and astronomy. For this purpose, two efficient hybrid methods so-called optimal homotopy analysis J -transform method ( O HA J TM) and J -variational iteration transform method (J -VITM) have been adopted. The OHA J TM is the hybrid method, where optimal-homotopy analysis method ( O HAM) is utilized after implementing the properties of J -transform (J T), and in J -VITM is the J -transform-based variational iteration method. Banach's fixed point approach is adopted to analyze the convergence of these methods. It is demonstrated that J -VITM is T -stable, and the evaluated dynamics of pGas are described in terms of Mittag–Leffler functions. The proposed evaluation confirms that the implemented methods perform better for the referred model equation of pGas. In addition, for a given iteration, the proposed behavior via O HA J TM performs better in producing more accurate behavior in comparison to J -VITM and the methods introduced recently. [ABSTRACT FROM AUTHOR]

The main goal of this study is to find solutions for the generalized coupled Ramani equation with the fractional order using the fractional natural decomposition method (FNDM). Four distinct cases are chosen to illustrate and validate the effectiveness of the considered method. The simulations in terms of numeric have been illustrated to confirm the reliability and proficiency of the projected scheme. Moreover, the behavior of the obtained results is captured for distinct fractional order. The comparison study is illustrated to verify the accuracy of the projected procedure. The achieved results exemplify that the projected solution procedure offers a simple algorithm and is also very efficient to analyze the nature of the coupled differential equations with arbitrary order situated in associated areas of Science and Engineering. [ABSTRACT FROM AUTHOR]

The main purpose of the present investigation is to find the solution of fractional coupled equations describing the romantic relationships using q -homotopy analysis transform method (q -HATM). The considered scheme is a unification of q -homotopy analysis technique with Laplace transform (LT). More preciously, we scrutinized the behavior of the obtained solution for the considered model with fractional-order, in order to elucidate the effectiveness of the proposed algorithm. Further, for the different fractional-order and parameters offered by the considered method, the physical natures have been apprehended. The obtained consequences evidence that the proposed method is very effective and highly methodical to study and examine the nature and its corresponding consequences of the system of fractional order differential equations describing the real word problems. [ABSTRACT FROM AUTHOR]