Abstract
The current work highlights the following issues: a brief survey of the development in the theory of fractional differential equations has been raised. A very recent technique based on the generalized Taylor series called—residual power series (RPS)—is introduced in detailed manner. The time-fractional foam drainage equation is considered as a target model to test the validity of the RPS method. Analysis of the obtained approximate solution of the fractional foam model reveals that RPS is an alternative method to be added for the fractional theory and computations and considered to be a significant method for exploring nonlinear fractional models.
Highlights
The last four decades witness fundamental works and developments on the fractional derivative and fractional differential equations
Alquran Department of Mathematics and Statistics, Sultan Qaboos University, P.O.Box(36), PC 123, Al-Khod, Muscat, Oman form an introduction to the theory of fractional differential equations and provide a systematic understanding of the fractional calculus such as the existence and the uniqueness of solutions
: we present in details the derivation of the residual power series solution to the generalized fractional DSW system
Summary
The last four decades witness fundamental works and developments on the fractional derivative and fractional differential equations. 3, we derive a residual power series solution to the time-fractional foam drainage equation. We present in details the derivation of the residual power series solution to the generalized fractional DSW system. The aim of this section is to construct power series solution to the time-fractional Foam drainage model by substituting its power series (PS) expansion among its truncated residual function [29, 30]. A recursion formula for the computation of the coefficients is derived, while the coefficients in the fractional PS expansion can be computed recursively by recurrent fractional differentiation of the truncated residual function. To clarify the RPS technique, we substitute the k-th truncated series of uðx; tÞ into Eq (3.6), find the fractional derivative formula DðtkÀ1Þa of both Resu;kðx; tÞ; k 1⁄4 1; 2; 3; . By the above recurrence relations, we are ready to present some graphical results regarding the time-fractional Foam drainage model
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