An Accurate Approximate-Analytical Technique for Solving Time-Fractional Partial Differential Equations

Abstract

The demand of many scientific areas for the usage of fractional partial differential equations (FPDEs) to explain their real-world systems has been broadly identified. The solutions may portray dynamical behaviors of various particles such as chemicals and cells. The desire of obtaining approximate solutions to treat these equations aims to overcome the mathematical complexity of modeling the relevant phenomena in nature. This research proposes a promising approximate-analytical scheme that is an accurate technique for solving a variety of noninteger partial differential equations (PDEs). The proposed strategy is based on approximating the derivative of fractional-order and reducing the problem to the corresponding partial differential equation (PDE). Afterwards, the approximating PDE is solved by using a separation-variables technique. The method can be simply applied to nonhomogeneous problems and is proficient to diminish the span of computational cost as well as achieving an approximate-analytical solution that is in excellent concurrence with the exact solution of the original problem. In addition and to demonstrate the efficiency of the method, it compares with two finite difference methods including a nonstandard finite difference (NSFD) method and standard finite difference (SFD) technique, which are popular in the literature for solving engineering problems.

Fractional order ecological system for complexities of interacting species with harvesting threshold in imprecise environment

“…To start with, the Caputo-type fractional operator on the left-hand side of each system (3)-(6) is approximated by using fuzzy fractional Laplace transform [31,34,35]. Let L denote FFLT, then FFLT of Caputo-type fractional derivative of order 0 < σ ≤ 1 of the functions x(t) andỹ(t) is expanded as follows:…”

“…for (i)-fgH-differentiable and (ii)-fgH-differentiable ofỹ(t), respectively. Following the method of linearization [34,35], we get the linearized form of p σ as follows:…”

“…with the same initial conditions as for system (2). (30,35,40), and initial conditionsx 0 = (0.3, 0.5, 0.9) andỹ 0 = (0.4, 0.5, 0.7) are shown in Table 1. It interprets the constant population densities of prey and predator with respect to fractional variation in time.…”

“…Furthermore, Tables 2-5 ratify stability and originality of the numerical solutions obtained by utilizing the well-known methods, namely FFRK, GL [15], and ABM [38]. (30,35,40). As the parameters are considered to be triangular fuzzy numbers, we also depict all the possible values of these parameters graphically in Figs.…”

“…Thus, this fractional rate of change further remarkably discusses these minor changes along with the fuzziness occurring within the body, instead of directly calculating a whole change. Moreover, to cope with the fractional operator, we implement the parametric expansion of fuzzy fractional Laplace transform [31,34,35]. The expansion of fractional Laplace transform greatly converts the fractional order derivative into the integer order, which can be further scrutinized easily using any appropriate numerical-analytical methods.…”