scholarly journals Direct algebraic method for solving fractional Fokas equation

2021 ◽  
pp. 111-111
Author(s):  
Yi Tian ◽  
Jun Liu
Keyword(s):  
Exact Solution ◽  
Solution Process ◽  
Algebraic Method ◽  
Characteristic Set ◽  
Fractional Complex Transform

Fractional Fokas equation is studied, its exact solution is obtained by the direct algebraic method. The solution process is elucidated step by step, and the fractional complex transform and the characteristic set algorithm are emphasized.

An Exact Solution to the Local Fractional Richards' Equation for Unsaturated Soils and Porous Fabrics

2013 ◽  
Vol 843 ◽  
pp. 97-101
Author(s):  
Zheng Biao Li ◽  
Yin Shan Yun ◽  
Hong Ying Luo
Keyword(s):  
Exact Solution ◽  
Porous Media ◽  
Water Content ◽  
Unsaturated Soils ◽  
Fractal Dimensions ◽  
Volumetric Water Content ◽  
Richards Equation ◽  
Fractional Complex Transform ◽  
Porous Soil

A local fractional Richards equation is derived by considering the soil as fractal porous media, and an exact solution is obtained by a generalized Boltzmann transform and the fractional complex transform. The new theory predicts that the volumetric water content depends on the ratio (distance)2a /(time), where a is the value of fractal dimensions of the porous soil, and its value is recommended for various soils.

scholarly journals Construction of Benchmark Problems for Solution of Ordinary Differential Equations

1994 ◽  
Vol 1 (5) ◽  
pp. 403-414 ◽  
Author(s):  
Sangchul Lee ◽  
John L. Junkins
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Numerical Solution ◽  
Solution Process ◽  
Optimal Choice ◽  
Benchmark Problems ◽  
Tuning Parameters ◽  
Forcing Function ◽  
The Relationship

An inverse method is introduced to construct benchmark problems for the numerical solution of initial value problems. Benchmark problems constructed in this fashion have a known exact solution, even though analytical solutions are generally not obtainable. The process leading to the exact solution makes use of an initially available approximate numerical solution. A smooth interpolation of the approximate solution is forced to exactly satisfy the differential equation by analytically deriving a small forcing function to absorb all of the errors in the interpolated approximate solution. Using this special case exact solution, it is possible to directly investigate the relationship between global errors of a candidate numerical solution process and the associated tuning parameters for a given code and a given problem. Under the assumption that the original differential equation is well-posed with respect to the small perturbations, we thereby obtain valuable information about the optimal choice of the tuning parameters and the achievable accuracy of the numerical solution. Five illustrative examples are presented.

scholarly journals A New Reconstruction of Variational Iteration Method and Its Application to Nonlinear Volterra Integrodifferential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
K. Maleknejad ◽  
M. Tamamgar
Keyword(s):  
Exact Solution ◽  
Iteration Method ◽  
Decomposition Method ◽  
Solution Process ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Integrodifferential Equations ◽  
Variational Iteration ◽  
Adomian Decomposition

We reconstruct the variational iteration method that we call, parametric iteration method (PIM). The purposed method was applied for solving nonlinear Volterra integrodifferential equations (NVIDEs). The solution process is illustrated by some examples. Comparisons are made between PIM and Adomian decomposition method (ADM). Also exact solution of the 3rd example is obtained. The results show the simplicity and efficiency of PIM. Also, the convergence of this method is studied in this work.

scholarly journals The exact solution of the nonlinear Schrödinger equation by the exp-function method

2021 ◽  
pp. 88-88
Author(s):  
Qiaoling Chen ◽  
Zhiqiang Sun
Keyword(s):  
Exact Solution ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Function Method ◽  
Solution Process ◽  
Nonlinear Schrodinger Equation ◽  
Dinger Equation ◽  
Nonlinear Schrödinger ◽  
Free Parameters

This paper elucidates the main advantages of the exp-function method in finding the exact solution of the nonlinear Schr?dinger equation. The solution process is extremely simple and accessible, and the obtained solution contains some free parameters.

scholarly journals On fractal space-time and fractional calculus

2016 ◽  
Vol 20 (3) ◽  
pp. 773-777 ◽  
Author(s):  
Yue Hu ◽  
Ji-Huan He
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Physical Meaning ◽  
Solution Process ◽  
Space Time ◽  
Continuum Models ◽  
Nano Structure ◽  
Fractional Complex Transform ◽  
Fractal Space ◽  
Whitham Equation

This paper gives an explanation of fractional calculus in fractal space-time. On observable scales, continuum models can be used, however, when the scale tends to a smaller threshold, a fractional model has to be adopted to describe phenomena in micro/nano structure. A time-fractional Fornberg-Whitham equation is used as an example to elucidate the physical meaning of the fractional order, and its solution process is given by the fractional complex transform.

scholarly journals Numerical method for fractional Zakharov-Kuznetsov equations with He’s fractional derivative

2019 ◽  
Vol 23 (4) ◽  
pp. 2163-2170 ◽  
Author(s):  
Kang-Le Wang ◽  
Shao-Wen Yao
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Numerical Method ◽  
Homotopy Perturbation Method ◽  
Solution Process ◽  
Fractional Complex Transform ◽  
Homotopy Perturbation

In this paper, a fractional Zakharov-Kuznetsov equation with He's fractional derivative is studied by the fractional complex transform and He's homotopy perturbation method. The solution process is elucidated step by step to show its simplicity and effectiveness of the proposed method.

Exact optical solutions of the (2+1) dimensions Kundu–Mukherjee–Naskar model via the new extended direct algebraic method

2020 ◽  
Vol 34 (22) ◽  
pp. 2050225 ◽  
Author(s):  
Hatıra Günerhan ◽  
Farid Samsami Khodadad ◽  
Hadi Rezazadeh ◽  
Mostafa M. A. Khater
Keyword(s):  
Exact Solution ◽  
Algebraic Method ◽  
Powerful Method

In this work, the researchers for computing an exact solution of the (2[Formula: see text]+[Formula: see text]1) dimensions Kundu–Mukherjee–Naskar (KMN) equation used a newly developed technique, namely, the new extended direct algebraic method. This powerful method gives an exact solution to the KMN equation considered in this paper. The results are new and attest to the efficiency of the proposed method.

scholarly journals The Adomian decomposition method and the fractional complex transform for fractional Bratu-type equation

2017 ◽  
Vol 21 (4) ◽  
pp. 1713-1717 ◽  
Author(s):  
Huan-Huan Wang ◽  
Yue Hu ◽  
Kang-Le Wang
Keyword(s):  
Fractional Derivative ◽  
Decomposition Method ◽  
Solution Process ◽  
Adomian Decomposition Method ◽  
Type Equation ◽  
Practical Applications ◽  
Fractional Complex Transform ◽  
Analytical Results ◽  
Adomian Decomposition

In this paper, the Adomian decomposition method and the fractional complex transform are adopted to solve a fractional Bratu-type equations based on He?s fractional derivative. The solution process is elucidated and analytical results can be directly used in practical applications.

An algebraic method for solving Hartree-Fock-Roothaan equations

1990 ◽  
Vol 87 ◽  
pp. 2017-2025 ◽  
Author(s):  
Lac Malbouisson ◽  
JDM Vianna
Keyword(s):  
Algebraic Method ◽  
Hartree Fock
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