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17 pages, 643 KiB

Open AccessArticle

On the Modified Numerical Methods for Partial Differential Equations Involving Fractional Derivatives

by Fahad Alsidrani, Adem Kılıçman and Norazak Senu

Axioms 2023, 12(9), 901; https://doi.org/10.3390/axioms12090901 - 21 Sep 2023

Cited by 1 | Viewed by 2346

This paper provides both analytical and numerical solutions of (PDEs) involving time-fractional derivatives. We implemented three powerful techniques, including the modified variational iteration technique, the modified Adomian decomposition technique, and the modified homotopy analysis technique, to obtain an approximate solution for the bounded [...] Read more.

ν

. The Laplace transformation is used in the time-fractional derivative operator to enhance the proposed numerical methods’ performance and accuracy and find an approximate solution to time-fractional Fornberg–Whitham equations. To confirm the accuracy of the proposed methods, we evaluate homogeneous time-fractional Fornberg–Whitham equations in terms of non-integer order and variable coefficients. The obtained results of the modified methods are shown through tables and graphs. Full article

(This article belongs to the Special Issue Advances in Generalized Hypergeometric Functions, Integral Transforms and Number Theory)

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18 pages, 2823 KiB

Open AccessArticle

Numerical Solutions of the Nonlinear Dispersive Shallow Water Wave Equations Based on the Space–Time Coupled Generalized Finite Difference Scheme

by Po-Wei Li, Shenghan Hu and Mengyao Zhang

Appl. Sci. 2023, 13(14), 8504; https://doi.org/10.3390/app13148504 - 23 Jul 2023

Cited by 5 | Viewed by 1778

Abstract

This study applies the space–time generalized finite difference scheme to solve nonlinear dispersive shallow water waves described by the modified Camassa–Holm equation, the modified Degasperis–Procesi equation, the Fornberg–Whitham equation, and its modified form. The proposed meshless numerical scheme combines the space–time generalized finite difference method, the two-step Newton’s method, and the time-marching method. The space–time approach treats the temporal derivative as a spatial derivative. This enables the discretization of all partial derivatives using a spatial discretization method and efficiently handles mixed derivatives with the proposed mesh-less numerical scheme. The space–time generalized finite difference method is derived from Taylor series expansion and the moving least-squares method. The numerical discretization process only involves functional data and weighting coefficients on the central and neighboring nodes. This results in a sparse matrix system of nonlinear algebraic equations that can be efficiently solved using the two-step Newton’s method. Additionally, the time-marching method is employed to advance the space–time domain along the time axis. Several numerical examples are presented to validate the effectiveness of the proposed space–time generalized finite difference scheme. Full article

(This article belongs to the Special Issue Computer Methods in Mechanical, Civil and Biomedical Engineering)

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15 pages, 783 KiB

Open AccessArticle

The Comparative Study for Solving Fractional-Order Fornberg–Whitham Equation via ρ-Laplace Transform

by Pongsakorn Sunthrayuth, Ahmed M. Zidan, Shao-Wen Yao, Rasool Shah and Mustafa Inc

Symmetry 2021, 13(5), 784; https://doi.org/10.3390/sym13050784 - 1 May 2021

Cited by 46 | Viewed by 3470

Abstract

In this article, we also introduced two well-known computational techniques for solving the time-fractional Fornberg–Whitham equations. The methods suggested are the modified form of the variational iteration and Adomian decomposition techniques by

ρ

-Laplace. Furthermore, an illustrative scheme is introduced to verify the [...] Read more.

ρ

-Laplace. Furthermore, an illustrative scheme is introduced to verify the accuracy of the available methods. The graphical representation of the exact and derived results is presented to show the suggested approaches reliability. The comparative solution analysis via graphs also represented the higher reliability and accuracy of the current techniques. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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14 pages, 427 KiB

Open AccessArticle

The Variational Iteration Transform Method for Solving the Time-Fractional Fornberg–Whitham Equation and Comparison with Decomposition Transform Method

by Nehad Ali Shah, Ioannis Dassios, Essam R. El-Zahar, Jae Dong Chung and Somaye Taherifar

Mathematics 2021, 9(2), 141; https://doi.org/10.3390/math9020141 - 11 Jan 2021

Cited by 15 | Viewed by 2872

Abstract

In this article, modified techniques, namely the variational iteration transform and Shehu decomposition method, are implemented to achieve an approximate analytical solution for the time-fractional Fornberg–Whitham equation. A comparison is made between the results of the variational iteration transform method and the Shehu decomposition method. The solution procedure reveals that the variational iteration transform method and Shehu decomposition method is effective, reliable and straightforward. The variational iteration transform methods solve non-linear problems without using Adomian’s polynomials and He’s polynomials, which is a clear advantage over the decomposition technique. The solutions achieved are compared with the corresponding exact result to show the efficiency and accuracy of the existing methods in solving a wide variety of linear and non-linear problems arising in various science areas. Full article

(This article belongs to the Special Issue Dynamical Systems in Engineering)

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16 pages, 1040 KiB

Open AccessArticle

Periodic Wave Solutions and Their Asymptotic Property for a Modified Fornberg–Whitham Equation

by Yiren Chen

Symmetry 2020, 12(9), 1517; https://doi.org/10.3390/sym12091517 - 15 Sep 2020

Cited by 1 | Viewed by 2032

Abstract

Recently, periodic traveling waves, which include periodically symmetric traveling waves of nonlinear equations, have received great attention. This article uses some bifurcations of the traveling wave system to investigate the explicit periodic wave solutions with parameter

α

and their asymptotic property for the [...] Read more.

α

and their asymptotic property for the modified Fornberg–Whitham equation. Furthermore, when

α

tends to given parametric values, the elliptic periodic wave solutions become the other three types of nonlinear wave solutions, which include the trigonometric periodic blow-up solution, the hyperbolic smooth solitary wave solution, and the hyperbolic blow-up solution. Full article

(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)

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