Solving nonlinear systems of fractional-order partial differential equations using an optimization technique based on generalized polynomials

2020 ◽  
Vol 39 (4) ◽  
Author(s):  
H. Hassani ◽  
J. A. Tenreiro Machado ◽  
E. Naraghirad ◽  
B. Sadeghi
Keyword(s):  
Partial Differential Equations ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Fractional Order ◽  
Optimization Technique ◽  
Partial Differential ◽  
Generalized Polynomials

scholarly journals Efficient Approaches for Solving Systems of Nonlinear Time-Fractional Partial Differential Equations

2022 ◽  
Vol 6 (1) ◽  
pp. 32
Author(s):  
Hegagi Mohamed Ali ◽  
Hijaz Ahmad ◽  
Sameh Askar ◽  
Ismail Gad Ameen
Keyword(s):  
Partial Differential Equations ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Fractional Order ◽  
Adomian Decomposition Method ◽  
Partial Differential ◽  
Fractional Pdes ◽  
Adomian Decomposition ◽  
Vorticity Transport ◽  
The Given

In this work, we present a modified generalized Mittag–Leffler function method (MGMLFM) and Laplace Adomian decomposition method (LADM) to get an analytic-approximate solution for nonlinear systems of partial differential equations (PDEs) of fractional-order in the Caputo derivative. We apply the MGMLFM and LADM on systems of nonlinear time-fractional PDEs. Precisely, we consider some important fractional-order nonlinear systems, namely Broer–Kaup (BK) and Burgers, which have found major significance because they arise in many physical applications such as shock wave, wave processes, vorticity transport, dispersal in porous media, and hydrodynamic turbulence. The analysis of these methods is implemented on the BK, Burgers systems and solutions have been offered in a simple formula. We show our results in figures and tables to demonstrate the efficiency and reliability of the used methods. Furthermore, our outcome converges rapidly to the given exact solutions.

Invariant solutions of fractional-order spatio-temporal partial differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nkosingiphile Mnguni ◽  
Sameerah Jamal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Growth Models ◽  
Spatial Patterning ◽  
Partial Differential ◽  
Symmetry Approach ◽  
Time Component ◽  
Spatio Temporal ◽  
Lie Symmetry Approach

Abstract This paper considers two categories of fractional-order population growth models, where a time component is defined by Riemann–Liouville derivatives. These models are studied under the Lie symmetry approach, and we reduce the fractional partial differential equations to nonlinear ordinary differential equations. Subsequently, solutions of the latter are determined numerically or with the aid of Laplace transforms. Graphical representations for integral and trigonometric solutions are presented. A key feature of these models is the connection between spatial patterning of organisms versus competitive coexistence.

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