Abstract
In this paper, we introduced a new generalization method to solve fractional convection–diffusion equations based on the well-known variational iteration method (VIM) improved by an auxiliary parameter. The suggested method was highly effective in controlling the convergence region of the approximate solution. By solving some fractional convection–diffusion equations with a propounded method and comparing it with standard VIM, it was concluded that complete reliability, efficiency, and accuracy of this method are guaranteed. Additionally, we studied and investigated the convergence of the proposed method, namely the VIM with an auxiliary parameter. We also offered the optimal choice of the auxiliary parameter in the proposed method. It was noticed that the approach could be applied to other models of physics.
Highlights
Over the past few years, fractional calculus has emerged in physical phenomena.Fractional derivatives prepared an effective tool for the elucidation of memory and patrimonial confidants of disparate materials and processes [1,2]
Fractional differential equations (FDEs) have attracted the attention of many researchers owing to their varied applications in science and engineering such as acoustics, control, viscoelasticity, edge detection, and signal processing [3,4,5]
The fractional convection–diffusion Equation (1) is considered, where the unknown function u is vanishing for t < 0, i.e., it is a causal function of time [49]
Summary
Over the past few years, fractional calculus has emerged in physical phenomena. Fractional derivatives prepared an effective tool for the elucidation of memory and patrimonial confidants of disparate materials and processes [1,2]. There are prepared solution methods for differential equations of optional real order, and applications of the demonstrated methods in several fields which give a systematic presentation of the applications, methods, and ideas on fractional calculus. These works have played an significant role in the expansion of the theory of fractional order [2,9,10,11]. Where N (u) that has been selected as a potential energy, is a nonlinear operator, c is a constant parameter and a constant α describes the fractional derivative This type of equations are obtained from the usual convection–diffusion equation; the difference is that the first-order time derivative term has become a fractional derivative of order α > 0. Abbasbandy et al [51] proposed fractional-order Legendre functions to solve the time-fractional convection diffusion equation
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