Solving time fractional Keller–Segel type diffusion equations with symmetry analysis, power series method, invariant subspace method and q-homotopy analysis method

2021 ◽  
Author(s):  
Xiaoyu Cheng ◽  
Lizhen Wang ◽  
Jie Hou
Keyword(s):  
Power Series ◽  
Invariant Subspace ◽  
Homotopy Analysis Method ◽  
Symmetry Analysis ◽  
Diffusion Equations ◽  
Subspace Method ◽  
Series Method ◽  
Analysis Method ◽  
Power Series Method ◽  
Homotopy Analysis

scholarly journals Homotopy analysis method applied to second-order frequency mixing in nonlinear optical dielectric media

2019 ◽  
Vol 28 (04) ◽  
pp. 1950033 ◽  
Author(s):  
Nathan J. Dawson ◽  
Moussa Kounta
Keyword(s):  
Power Series ◽  
Homotopy Analysis Method ◽  
Nonlinear Optical ◽  
Classical Problem ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Wave Mixing ◽  
Analysis Method ◽  
Generic Polynomial ◽  
Homotopy Analysis

The classical problem of three-wave mixing in a nonlinear optical medium is investigated using the homotopy analysis method (HAM). We show that the power series basis builds a generic polynomial expression that can be used to study three-wave mixing for arbitrary input parameters. The phase-mismatched and perfectly phase matched cases are investigated. Parameters that result in generalized sum- and difference-frequency generation are studied using HAM with a power series basis and compared to an explicit finite-difference approximation. The convergence region is extended by increasing the auxiliary parameter.

Approximate symmetry analysis of nonlinear Rayleigh-wave equation

2018 ◽  
Vol 15 (04) ◽  
pp. 1850055
Author(s):  
Saeede Rashidi ◽  
S. Reza Hejazi ◽  
Elham Dastranj
Keyword(s):  
Wave Equation ◽  
Power Series ◽  
Small Parameter ◽  
Conservation Laws ◽  
Rayleigh Wave ◽  
Symmetry Analysis ◽  
Approximate Symmetry ◽  
Series Method ◽  
Power Series Method ◽  
New Exact Solutions

In this paper, the Lie approximate symmetry analysis is applied to investigate the new exact solutions of the Rayleigh-wave equation. The power series method is employed to solve some of the obtained reduced ordinary differential equations with a small parameter. We yield the new analytical solutions with small parameter which is effectively obtained by the proposed method. The concept of nonlinear self-adjointness is used to construct the conservation laws for Rayleigh-wave equation. It is shown that this equation is approximately nonlinearly self-adjoint and therefore desired conservation laws can be found using appropriate formal Lagrangians.

scholarly journals Analytical Solution for Multi-Energy Groups of Neutron Diffusion Equations by a Residual Power Series Method

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 633 ◽  
Author(s):  
Mohammed Shqair ◽  
Ahmad El-Ajou ◽  
Mazen Nairat
Keyword(s):  
Power Series ◽  
Diffusion Theory ◽  
Analytical Description ◽  
Diffusion Equations ◽  
Neutron Diffusion ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Residual Power

In this paper, a multi-energy groups of a neutron diffusion equations system is analytically solved by a residual power series method. The solution is generalized to consider three different geometries: slab, cylinder and sphere. Diffusion of two and four energy groups of neutrons is specifically analyzed through numerical calculation at certain boundary conditions. This study revels sufficient analytical description for radial flux distribution of multi-energy groups of neutron diffusion theory as well as determination of each nuclear reactor dimension in criticality case. The generated results are compatible with other different methods data. The generated results are practically efficient for neutron reactors dimension.

scholarly journals Symmetry Analysis and Exact Solutions to the Space-Dependent Coefficient PDEs in Finance

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Hanze Liu
Keyword(s):  
Power Series ◽  
Exact Solutions ◽  
Variable Coefficients ◽  
Similarity Solutions ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Series Method ◽  
Lie Symmetry Analysis ◽  
Power Series Method ◽  
Generalized Power Series

The variable-coefficients partial differential equations (vc-PDEs) in finance are investigated by Lie symmetry analysis and the generalized power series method. All of the geometric vector fields of the equations are obtained; the symmetry reductions and exact solutions to the equations are presented, including the exponentiated solutions and the similarity solutions. Furthermore, the exact analytic solutions are provided by the transformation technique and generalized power series method, which has shown that the combination of Lie symmetry analysis and the generalized power series method is a feasible approach to dealing with exact solutions to the variable-coefficients PDEs.

scholarly journals Application of Residual Power Series Method to Fractional Coupled Physical Equations Arising in Fluids Flow

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Anas Arafa ◽  
Ghada Elmahdy
Keyword(s):  
Power Series ◽  
Power Efficiency ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Homotopy Analysis ◽  
Residual Power Series ◽  
Residual Power

The approximate analytical solution of the fractional Cahn-Hilliard and Gardner equations has been acquired successfully via residual power series method (RPSM). The approximate solutions obtained by RPSM are compared with the exact solutions as well as the solutions obtained by homotopy perturbation method (HPM) and q-homotopy analysis method (q-HAM). Numerical results are known through different graphs and tables. The fractional derivatives are described in the Caputo sense. The results light the power, efficiency, simplicity, and reliability of the proposed method.

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