New Perspective on the Conventional Solutions of the Nonlinear Time-Fractional Partial Differential Equations

Hijaz Ahmad,Predrag S Stanimirović,Yu-Ming Chu,Tufail A Khan,Ali Akgül,Mostafa M A Khater

doi:10.1155/2020/8829017

Abstract

The role of integer and noninteger order partial differential equations (PDE) is essential in applied sciences and engineering. Exact solutions of these equations are sometimes difficult to find. Therefore, it takes time to develop some numerical techniques to find accurate numerical solutions of these types of differential equations. This work aims to present a novel approach termed as fractional iteration algorithm-I for finding the numerical solution of nonlinear noninteger order partial differential equations. The proposed approach is developed and tested on nonlinear fractional-order Fornberg–Whitham equation and employed without using any transformation, Adomian polynomials, small perturbation, discretization, or linearization. The fractional derivatives are taken in the Caputo sense. To assess the efficiency and precision of the suggested method, the tabulated numerical results are compared with the standard variational iteration method and the exact solution as well. In addition, numerical results for different cases of the fractional-order α are presented graphically, which show the effectiveness of the proposed procedure and revealed that the proposed scheme is very effective, suitable for fractional PDEs, and may be viewed as a generalization of the existing methods for solving integer and noninteger order differential equations.

Highlights

Introduction e nonlinear partial differential equations (PDE) have become a hot topic in the field of nonlinear science, which has been used to describe the problems in many fields, such as quantum mechanics, image processing, ecology and economic system, and epidemiology
PDEs are broadly emerging in different physical applications like dispersing and propagation of waves, magnetic resonance imaging, computational fluid dynamics, magnetohydrodynamic move through pipes, phenomena of supersonic and turbulence flow, acoustic transmission, and traffic
For some complex problems in these fields, the fractional PDE is more accurate than integer-order partial differential equation

Summary

Fractional Iteration Algorithm-I

The main idea of the fractional iteration algorithm-I is illustrated by considering a nonlinear differential equation of the generic form. For an appropriate given initial condition U0(ψ), series Uk+1(ψ), which approximates the solution of equation (13), can be obtained as ψ. The nonlinear terms have to be considered as restricted variations for obtaining the value of Lagrange multiplier, and a correction functional can be constructed after determining the identified value of corresponding nonlinear terms. It is worth mentioning that the proposed algorithm may be considered as a nice refinement in existing analytical and numerical methods, where the discretization, transformations, and linearization are not required, and the numerical solution of the given differential equations can be obtained in series form as. Equation (20) has two obvious advantages; one is the limited step, which is needed for better accuracy, while the other is an auxiliary parameter (ρ), which ensures the convergence, and a more accurate solution can be gained after a higher iteration process. UN(ψ, t) ≔ w0(ψ, t) + 􏽐Nk 1 wk(ψ, t, ρ) the maximum error norm can be assessed as

Implementation of the Algorithm and Examples

Conclusion