MDPI - Publisher of Open Access Journals

15 pages, 2709 KiB

Open AccessArticle

A New Iterative Method for Investigating Modified Camassa–Holm and Modified Degasperis–Procesi Equations within Caputo Operator

by Humaira Yasmin, Yousuf Alkhezi and Khaled Alhamad

Symmetry 2023, 15(12), 2172; https://doi.org/10.3390/sym15122172 - 7 Dec 2023

Cited by 2 | Viewed by 1624

Abstract

In this paper, we employ the new iterative method to investigate two prominent nonlinear partial differential equations, namely the modified Camassa–Holm (mCH) equation and the modified Degasperis–Procesi (mDP) equation, both within the framework of the Caputo operator. The mCH and mDP equations are fundamental in studying wave propagation and soliton dynamics, exhibiting complex behavior and intriguing mathematical structures. The new iterative method (NIM), a powerful numerical technique, is utilized to obtain analytical and numerical solutions for these equations, offering insights into their dynamic properties and behavior. Through systematic analysis and computation, we unveil the unique features of the mCH and the mDP equations, shedding light on their applicability in various scientific and engineering domains. This research contributes to the ongoing exploration of nonlinear wave equations and their solutions, emphasizing the versatility of the new iterative method in tackling complex mathematical problems. Numerical results and comparative analyses are presented to validate the effectiveness of the new iterative method in solving these equations, highlighting its potential for broader applications in mathematical modeling and analysis. Full article

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications, 3rd Edition)

► Show Figures

Figure 1

15 pages, 299 KiB

Open AccessArticle

Convergence Analysis of the Strang Splitting Method for the Degasperis-Procesi Equation

by Runjie Zhang and Jinwei Fang

Axioms 2023, 12(10), 946; https://doi.org/10.3390/axioms12100946 - 4 Oct 2023

Viewed by 1421

Abstract

This article is concerned with the convergence properties of the Strang splitting method for the Degasperis-Procesi equation, which models shallow water dynamics. The challenges of analyzing splitting methods for this equation lie in the fact that the involved suboperators are both nonlinear. In [...] Read more.

L^{2}

for the proposed method directly, we first show that the Strang splitting method has first order convergence in

H^{2}

. In the analysis, the Lie derivative bounds for the local errors are crucial. The obtained first order convergence result provides the

H^{2}

boundedness of the approximate solutions, thereby enabling us to subsequently establish the second order convergence in

L^{2}

for the Strang splitting method. Full article

(This article belongs to the Topic Numerical Methods for Partial Differential Equations)

13 pages, 442 KiB

Open AccessArticle

Peakons and Persistence Properties of Solution for the Interacting System of Popowicz

by Yaohong Li and Chunyan Qin

Mathematics 2023, 11(16), 3529; https://doi.org/10.3390/math11163529 - 15 Aug 2023

Viewed by 1050

Abstract

This paper focuses on a two-component interacting system introduced by Popowicz, which has the coupling form of the Camassa–Holm and Degasperis–Procesi equations. Using distribution theory, single peakon solutions and several double peakon solutions of the system are described in an explicit expression. Moreover, dynamic behaviors of several types of double peakon solutions are illustrated through figures. In addition, the persistence properties of the solutions to the Popowicz system in weighted

L^{p}

spaces is considered via a large class of moderate weights. Full article

(This article belongs to the Section E4: Mathematical Physics)

► Show Figures

Figure 1

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)

► Show Figures

Figure 1

17 pages, 693 KiB

Open AccessArticle

Finite-Difference Hermite WENO Scheme for Degasperis-Procesi Equation

by Liang Li, Yapu Feng, Yanmeng Wang, Liuyong Pang and Jun Zhu

Processes 2023, 11(5), 1536; https://doi.org/10.3390/pr11051536 - 17 May 2023

Viewed by 1550

Abstract

We present a fifth-order finite-difference Hermite weighted essentially non-oscillatory (HWENO) method for solving the Degasperis–Procesi (DP) equation in this paper. First, the DP equation can be rewritten as a system of equations consisting of hyperbolic equations and elliptic equations by introducing an auxiliary variable, since the equations contain nonlinear higher order derivative terms. Then, the auxiliary variable equations are solved using the Hermite interpolation, while the HWENO scheme is performed for the hyperbolic equations. Compared with the popular WENO-type scheme, the most important feature of the HWENO scheme mentioned in this paper is the compactness of its spatial reconstruction stencil, which can achieve the fifth-order accuracy of the expected design with only three points, while the WENO method requires five points. Finally, we demonstrate the effectiveness of the HWENO method in various aspects by conducting some benchmark numerical tests. Full article

(This article belongs to the Special Issue Numerical Simulation of Nonlinear Dynamical Systems)

► Show Figures

Figure 1

13 pages, 3433 KiB

Open AccessArticle

On New Solutions of Time-Fractional Wave Equations Arising in Shallow Water Wave Propagation

by Rajarama Mohan Jena, Snehashish Chakraverty and Dumitru Baleanu

Mathematics 2019, 7(8), 722; https://doi.org/10.3390/math7080722 - 8 Aug 2019

Cited by 32 | Viewed by 3670

Abstract

The primary objective of this manuscript is to obtain the approximate analytical solution of Camassa–Holm (CH), modified Camassa–Holm (mCH), and Degasperis–Procesi (DP) equations with time-fractional derivatives labeled in the Caputo sense with the help of an iterative approach called fractional reduced differential transform method (FRDTM). The main benefits of using this technique are that linearization is not required for this method and therefore it reduces complex numerical computations significantly compared to the other existing methods such as the perturbation technique, differential transform method (DTM), and Adomian decomposition method (ADM). Small size computations over other techniques are the main advantages of the proposed method. Obtained results are compared with the solutions carried out by other technique which demonstrates that the proposed method is easy to implement and takes small size computation compared to other numerical techniques while dealing with complex physical problems of fractional order arising in science and engineering. Full article

(This article belongs to the Special Issue Modelling Problems Arising in Science and Engineering with Fractional Differential Operators: Beyond the Power-Law Limit)

► Show Figures

Figure 1

Search Results (6)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (6)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI