A Comparative Analysis of Fractional Space-Time Advection-Dispersion Equation via Semi-Analytical Methods

Abstract

The approximate solutions of the time fractional advection-dispersion equation are presented in this article. The nonlocal nature of solute movement and the nonuniformity of fluid flow velocity in the advection-dispersion process lead to the formation of a heterogeneous system, which can be modeled using a fractional advection-dispersion equation, which generalizes the classical advection-dispersion equation and replaces the time derivative with the fractional Caputo derivative. Researchers use a variety of numerical techniques to study such fractional models, but the nonlocality of the derivative having fractional order leads to high computation complexity and complex calculations, so the task is to find an efficient technique that requires less computation and provides greater accuracy when numerically solving such models. A innovative techniques, homotopy perturbation method and new iteration method, are used in connection with the Elzaki transform to solve the “fractional advection-dispersion equation” which provides the solution in the convergent series form. When the homotopy perturbation method is used with the Elzaki transform, fast convergent series solutions can be obtained with less computation. By solving some cases of time-fractional advection-dispersion equation with varied initial conditions with the help of new iterative transform method and homotopy perturbation transform method demonstrates the usefulness of the proposed methods.

A Comparative Study of the Fractional-Order Belousov–Zhabotinsky System

“…By doing so, it is possible to improve the accuracy of analysis in related fields through the use of FDEs. Numerous approaches have been devised to address this issue; One of the methods employed is the Adomian decomposition method (DM) [36], the reduced differential transform method [37], variational iteration method [38], the Elzaki DM [39,40], the iterative transformation technique [41], the Natural DT [42], the homotopy perturbation technique (HPT) [43], and so on [44][45][46][47][48].…”

An Efficient Numerical Scheme for Solving Multiorder Tempered Fractional Differential Equations via Operational Matrix

“…where f ðxÞ is the unknown solution, T D ðα r ,β r Þ x is tempered fractional derivatives, q r ∈ ℝ, r = 1, ⋯, l are constants, and α r , β r ≥ 0 are real derivative orders which β denotes the tempered coefficient, while hðxÞ is the unhomogeneous terms. To date, various numerical or analytical methods were derived to find the solution for different fractional calculus problems, such as [11][12][13]. On top of that, the operational matrix method via different types of the polynomial is one of the common numerical schemes which had been widely used in solving various types of fractional calculus problems, such as the poly-Bernoulli operational matrix for solving fractional delay differential equation [14], poly-Genocchi operational matrix for solving fractional differential equation [15], Jacobi wavelet operational matrix of fractional integration for solving fractional integro-differential equation [16], and Fibonacci wavelet operational matrix of integration for solving of nonlinear Stratonovich Volterra integral equations [17].…”

Fractional Leindler’s Inequalities via Conformable Calculus

“…During the last few years, by using conformable fractional calculus, authors proved some integral inequalities, such as Hardy's inequality [11], Hermite-Hadamard's inequality [12][13][14][15][16][17], Opial's inequality [18,19], Steffensen's inequality [20], and Chebyshev's inequality [21]. Additionally, over several decads, many generalizations, extensions and refinements of other types of integral inequalities have been studied we refer the reader to the papers [22][23][24][25][26].…”