Mathematics

Research

26 pages, 1259 KiB

Open AccessArticle

A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam Polynomials

by Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori and Ahmed Gamal Atta

Mathematics 2025, 13(2), 300; https://doi.org/10.3390/math13020300 - 17 Jan 2025

Viewed by 417

This paper proposes a numerical algorithm for the nonlinear fifth-order Korteweg–de Vries equations. This class of equations is known for its significance in modeling various complex wave phenomena in physics and engineering. The approximate solutions are expressed in terms of certain shifted Horadam [...] Read more.

This paper proposes a numerical algorithm for the nonlinear fifth-order Korteweg–de Vries equations. This class of equations is known for its significance in modeling various complex wave phenomena in physics and engineering. The approximate solutions are expressed in terms of certain shifted Horadam polynomials. A theoretical background for these polynomials is first introduced. The derivatives of these polynomials and their operational metrics of derivatives are established to tackle the problem using the typical collocation method to transform the nonlinear fifth-order Korteweg–de Vries equation governed by its underlying conditions into a system of nonlinear algebraic equations, thereby obtaining the approximate solutions. This paper also includes a rigorous convergence analysis of the proposed shifted Horadam expansion. To validate the proposed method, we present several numerical examples illustrating its accuracy and effectiveness. Full article

(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)

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25 pages, 3296 KiB

Open AccessArticle

A Novel Approximation Method for Solving Ordinary Differential Equations Using the Representation of Ball Curves

by Abdul Hadi Bhatti, Sharmila Karim, Ala Amourah, Ali Fareed Jameel, Feras Yousef and Nidal Anakira

Mathematics 2025, 13(2), 250; https://doi.org/10.3390/math13020250 - 13 Jan 2025

Viewed by 504

Abstract

Numerical methods are frequently developed to investigate concepts for approximately solving ordinary differential equations (ODEs). To achieve minimal error and higher accuracy in approximate solutions, researchers have focused on developing algorithms using various numerical techniques. This study proposes the application of Ball curves, [...] Read more.

Numerical methods are frequently developed to investigate concepts for approximately solving ordinary differential equations (ODEs). To achieve minimal error and higher accuracy in approximate solutions, researchers have focused on developing algorithms using various numerical techniques. This study proposes the application of Ball curves, specifically the Said–Ball curve, for estimating solutions to higher-order ODEs. To obtain the best control points of the Said–Ball curve, the least squares method is used. These control points are calculated by minimizing the residual error through the sum of the squares of the residual functions. To demonstrate the proposed method, several boundary value problems are presented, and their performance is compared with existing methods in terms of error accuracy. The numerical results indicate that the proposed method improves error accuracy compared to existing studies, including those employing Bézier curves and the steepest descent method. Full article

(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)

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17 pages, 4679 KiB

Open AccessArticle

Exploration of Soliton Solutions to the Special Korteweg–De Vries Equation with a Stability Analysis and Modulation Instability

by Abdulrahman Alomair, Abdulaziz S. Al Naim and Ahmet Bekir

Mathematics 2025, 13(1), 54; https://doi.org/10.3390/math13010054 - 27 Dec 2024

Viewed by 553

Abstract

This work is concerned with Hirota bilinear,

{exp}_{a}

function, and Sardar sub-equation methods to find the breather-wave, 1-Soliton, 2-Soliton, three-wave, and new periodic-wave results and some exact solitons of the special (1 + 1)-dimensional Korteweg–de Vries (KdV) equation. The model of concern [...] Read more.

This work is concerned with Hirota bilinear,

{exp}_{a}

function, and Sardar sub-equation methods to find the breather-wave, 1-Soliton, 2-Soliton, three-wave, and new periodic-wave results and some exact solitons of the special (1 + 1)-dimensional Korteweg–de Vries (KdV) equation. The model of concern is a partial differential equation that is used as a mathematical model of waves on shallow water surfaces. The results are attained as well as verified by Mathematica and Maple softwares. Some of the obtained solutions are represented in three-dimensional (3-D) and contour plots through the Mathematica tool. A stability analysis is performed to verify that the results are precise as well as accurate. Modulation instability is also performed for the steady-state solutions to the governing equation. The solutions are useful for the development of corresponding equations. This work shows that the methods used are simple and fruitful for investigating the results for other nonlinear partial differential models. Full article

(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)

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15 pages, 2705 KiB

Open AccessArticle

Numerical Solution of Emden–Fowler Heat-Type Equations Using Backward Difference Scheme and Haar Wavelet Collocation Method

by Mohammed N. Alshehri, Ashish Kumar, Pranay Goswami, Saad Althobaiti and Abdulrahman F. Aljohani

Mathematics 2024, 12(23), 3692; https://doi.org/10.3390/math12233692 - 25 Nov 2024

Viewed by 605

Abstract

In this study, we introduce an algorithm that utilizes the Haar wavelet collocation method to solve the time-dependent Emden–Fowler equation. This proposed method effectively addresses both linear and nonlinear partial differential equations. It is a numerical technique where the differential equation is discretized [...] Read more.

In this study, we introduce an algorithm that utilizes the Haar wavelet collocation method to solve the time-dependent Emden–Fowler equation. This proposed method effectively addresses both linear and nonlinear partial differential equations. It is a numerical technique where the differential equation is discretized using Haar basis functions. A difference scheme is also applied to approximate the time derivative. By leveraging Haar functions and the difference scheme, we form a system of equations, which is then solved for Haar coefficients using MATLAB software. The effectiveness of this technique is demonstrated through various examples. Numerical simulations are performed, and the results are presented in graphical and tabular formats. We also provide a convergence analysis and an error analysis for this method. Furthermore, approximate solutions are compared with those obtained from other methods to highlight the accuracy, efficiency, and computational convenience of this technique. Full article

(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)

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21 pages, 11735 KiB

Open AccessArticle

Dynamical Behaviors and Abundant New Soliton Solutions of Two Nonlinear PDEs via an Efficient Expansion Method in Industrial Engineering

by Ibrahim Alraddadi, M. Akher Chowdhury, M. S. Abbas, K. El-Rashidy, J. R. M. Borhan, M. Mamun Miah and Mohammad Kanan

Mathematics 2024, 12(13), 2053; https://doi.org/10.3390/math12132053 - 30 Jun 2024

Viewed by 1266

Abstract

In this study, we discuss the dynamical behaviors and extract new interesting wave soliton solutions of the two significant well-known nonlinear partial differential equations (NPDEs), namely, the Korteweg–de Vries equation (KdVE) and the Jaulent–Miodek hierarchy equation (JMHE). This investigation has applications in pattern [...] Read more.

In this study, we discuss the dynamical behaviors and extract new interesting wave soliton solutions of the two significant well-known nonlinear partial differential equations (NPDEs), namely, the Korteweg–de Vries equation (KdVE) and the Jaulent–Miodek hierarchy equation (JMHE). This investigation has applications in pattern recognition, fluid dynamics, neural networks, mechanical systems, ecological systems, control theory, economic systems, bifurcation analysis, and chaotic phenomena. In addition, bifurcation analysis and the chaotic behavior of the KdVE and JMHE are the main issues of the present research. As a result, in this study, we obtain very effective advanced exact traveling wave solutions with the aid of the proposed mathematical method, and the solutions involve rational functions, hyperbolic functions, and trigonometric functions that play a vital role in illustrating and developing the models involving the KdVE and the JMHE. These new exact wave solutions lead to utilizing real problems and give an advanced explanation of our mentioned mathematical models that we did not yet have. Some of the attained solutions of the two equations are graphically displayed with 3D, 2D, and contour panels of different shapes, like periodic, singular periodic, kink, anti-kink, bell, anti-bell, soliton, and singular soliton wave solutions. The solutions obtained in this study of our considered equations can lead to the acceptance of our proposed method, effectively utilized to investigate the solutions for the mathematical models of various important complex problems in natural science and engineering. Full article

(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)

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22 pages, 9411 KiB

Open AccessArticle

Chaotic Phenomena, Sensitivity Analysis, Bifurcation Analysis, and New Abundant Solitary Wave Structures of The Two Nonlinear Dynamical Models in Industrial Optimization

by M. Mamun Miah, Faisal Alsharif, Md. Ashik Iqbal, J. R. M. Borhan and Mohammad Kanan

Mathematics 2024, 12(13), 1959; https://doi.org/10.3390/math12131959 - 24 Jun 2024

Viewed by 1248

Abstract

In this research, we discussed the different chaotic phenomena, sensitivity analysis, and bifurcation analysis of the planer dynamical system by considering the Galilean transformation to the Lonngren wave equation (LWE) and the (2 + 1)-dimensional stochastic Nizhnik–Novikov–Veselov System (SNNVS). These two important equations [...] Read more.

In this research, we discussed the different chaotic phenomena, sensitivity analysis, and bifurcation analysis of the planer dynamical system by considering the Galilean transformation to the Lonngren wave equation (LWE) and the (2 + 1)-dimensional stochastic Nizhnik–Novikov–Veselov System (SNNVS). These two important equations have huge applications in the fields of modern physics, especially in the electric signal in data communication for LWE and the mechanical signal in a tunnel diode for SNNVS. A different chaotic nature with an additional perturbed term was also highlighted. Concerning the theory of the planer dynamical system, the bifurcation analysis incorporating phase portraits of the dynamical systems of the declared equations was performed. Additionally, a sensitivity analysis was used to monitor the sensitivity of the mentioned equations. Also, we extracted new, abundant solitary wave structures with the graphical phenomena of the mentioned nonlinear mathematical models. By conducting an expansion method on the abovementioned equations, we generated three types of soliton structures, which are rational function, trigonometric function, and hyperbolic function. By simulating the 3D, contour, and 2D graphs of these obtained solitons, we scrutinized the behavior of the waves affecting the nonlinear terms. The figures show that the solitary waves obtained from LWE are efficient in analyzing electromagnetic wave signals in the cable lines, and the solitary waves from SNNVS are essential in any stochastic system like a sound wave. Moreover, by taking some values of the parameters, we found some interesting soliton shapes, such as compaction soliton, singular periodic solution, bell-shaped soliton, anti-kink-shaped soliton, one-sided kink-shaped soliton, and some flat kink-shaped solitons, etc. This article will have a great impact on nonlinear science due to the new solitary wave structures with different complex phenomena, sensitivity analysis, and bifurcation analysis. Full article

(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)

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13 pages, 408 KiB

Open AccessArticle

Painlevé Analysis of the Traveling Wave Reduction of the Third-Order Derivative Nonlinear Schrödinger Equation

by Nikolay A. Kudryashov and Sofia F. Lavrova

Mathematics 2024, 12(11), 1632; https://doi.org/10.3390/math12111632 - 23 May 2024

Viewed by 1160

Abstract

The second partial differential equation from the Kaup–Newell hierarchy is considered. This equation can be employed to model pulse propagation in optical fiber, wave propagation in plasma, or high waves in the deep ocean. The integrability of the explored equation in traveling wave [...] Read more.

The second partial differential equation from the Kaup–Newell hierarchy is considered. This equation can be employed to model pulse propagation in optical fiber, wave propagation in plasma, or high waves in the deep ocean. The integrability of the explored equation in traveling wave variables is investigated using the Painlevé test. Periodic and solitary wave solutions of the studied equation are presented. The investigated equation belongs to the class of generalized nonlinear Schrödinger equations and may be used for the description of optical solitons in a nonlinear medium. Full article

(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)

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19 pages, 7906 KiB

Open AccessArticle

Abundant New Optical Soliton Solutions to the Biswas–Milovic Equation with Sensitivity Analysis for Optimization

by Md Nur Hossain, Faisal Alsharif, M. Mamun Miah and Mohammad Kanan

Mathematics 2024, 12(10), 1585; https://doi.org/10.3390/math12101585 - 19 May 2024

Cited by 4 | Viewed by 1365

Abstract

This study extensively explores the Biswas–Milovic equation (BME) with Kerr and power law nonlinearity to extract the unique characteristics of optical soliton solutions. These optical soliton solutions have different applications in the field of precision in optical switching, applications in waveguide design, exploration [...] Read more.

This study extensively explores the Biswas–Milovic equation (BME) with Kerr and power law nonlinearity to extract the unique characteristics of optical soliton solutions. These optical soliton solutions have different applications in the field of precision in optical switching, applications in waveguide design, exploration of nonlinear optical effects, imaging precision, reduced intensity fluctuations, suitability for optical signal processing in optical physics, etc. Through the powerful (

G^{'} / G

,

1 / G

)-expansion analytical method, a variety of soliton solutions are expressed in three distinct forms: trigonometric, hyperbolic, and rational expressions. Rigorous validation using Mathematica software ensures precision, while dynamic visual representations vividly portray various soliton patterns such as kink, anti-kink, singular soliton, hyperbolic, dark soliton, and periodic bright soliton solutions. Indeed, a sensitivity analysis was conducted to assess how changes in parameters affect the exact solutions, aiding in the understanding of system behavior and informing decision-making, especially in accurately designing or analyzing real-world optical phenomena. This investigation reveals the significant influence of parameters

λ

,

τ

,

c

,

B

, and

Κ

on the precise solutions in Kerr and power law nonlinearities within the BME. Notably, parameter

λ

exhibits consistently high sensitivity across all scenarios, while parameters

τ

and

c

demonstrate pronounced sensitivity in scenario III. The outcomes derived from this method are distinctive and carry significant implications for the dynamics of optical fibers and wave phenomena across various optical systems. Full article

(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)

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10 pages, 1597 KiB

Open AccessArticle

Analyzing Soliton Solutions of the Extended (3 + 1)-Dimensional Sakovich Equation

by Rubayyi T. Alqahtani and Melike Kaplan

Mathematics 2024, 12(5), 720; https://doi.org/10.3390/math12050720 - 29 Feb 2024

Cited by 2 | Viewed by 1032

Abstract

This work focuses on the utilization of the generalized exponential rational function method (GERFM) to analyze wave propagation of the extended (3 + 1)-dimensional Sakovich equation. The demonstrated effectiveness and robustness of the employed method underscore its relevance to a wider spectrum of [...] Read more.

This work focuses on the utilization of the generalized exponential rational function method (GERFM) to analyze wave propagation of the extended (3 + 1)-dimensional Sakovich equation. The demonstrated effectiveness and robustness of the employed method underscore its relevance to a wider spectrum of nonlinear partial differential equations (NPDEs) in physical phenomena. An examination of the physical characteristics of the generated solutions has been conducted through two- and three-dimensional graphical representations. Full article

(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)

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13 pages, 555 KiB

Open AccessArticle

Simulation of a Combined (2+1)-Dimensional Potential Kadomtsev–Petviashvili Equation via Two Different Methods

by Muath Awadalla, Arzu Akbulut and Jihan Alahmadi

Mathematics 2024, 12(3), 427; https://doi.org/10.3390/math12030427 - 29 Jan 2024

Viewed by 921

Abstract

This paper presents an investigation into original analytical solutions of the (2+1)-dimensional combined potential Kadomtsev–Petviashvili and B-type Kadomtsev–Petviashvili equations. For this purpose, the generalized Kudryashov technique (GKT) and exponential rational function technique (ERFT) have been applied to deal with the equation. These two [...] Read more.

This paper presents an investigation into original analytical solutions of the (2+1)-dimensional combined potential Kadomtsev–Petviashvili and B-type Kadomtsev–Petviashvili equations. For this purpose, the generalized Kudryashov technique (GKT) and exponential rational function technique (ERFT) have been applied to deal with the equation. These two methods have been applied to the model for the first time, and the the generalized Kudryashov method has an important place in the literature. The characteristics of solitons are unveiled through the use of three-dimensional, two-dimensional, contour, and density plots. Furthermore, we conducted a stability analysis on the acquired results. The results obtained in the article were seen to be different compared to other results in the literature and have not been published anywhere before. Full article

(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)

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12 pages, 901 KiB

Open AccessArticle

On the Dynamics of the Complex Hirota-Dynamical Model

by Arzu Akbulut, Melike Kaplan, Rubayyi T. Alqahtani and W. Eltayeb Ahmed

Mathematics 2023, 11(23), 4851; https://doi.org/10.3390/math11234851 - 2 Dec 2023

Cited by 2 | Viewed by 1082

Abstract

The complex Hirota-dynamical Model (HDM) finds multifarious applications in fields such as plasma physics, fusion energy exploration, astrophysical investigations, and space studies. This study utilizes several soliton-type solutions to HDM via the modified simple equation and generalized and modified Kudryashov approaches. Modulation instability [...] Read more.

The complex Hirota-dynamical Model (HDM) finds multifarious applications in fields such as plasma physics, fusion energy exploration, astrophysical investigations, and space studies. This study utilizes several soliton-type solutions to HDM via the modified simple equation and generalized and modified Kudryashov approaches. Modulation instability (MI) analysis is investigated. We also offer visual representations for the HDM. Full article

(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)

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11 pages, 1073 KiB

Open AccessArticle

New (3+1)-Dimensional Kadomtsev–Petviashvili–Sawada– Kotera–Ramani Equation: Multiple-Soliton and Lump Solutions

by Abdul-Majid Wazwaz, Ma’mon Abu Hammad, Ali O. Al-Ghamdi, Mansoor H. Alshehri and Samir A. El-Tantawy

Mathematics 2023, 11(15), 3395; https://doi.org/10.3390/math11153395 - 3 Aug 2023

Cited by 6 | Viewed by 1519

Abstract

In this investigation, a novel (3+1)-dimensional Lax integrable Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation is constructed and analyzed analytically. The Painlevé integrability for the mentioned model is examined. The bilinear form is applied for investigating multiple-soliton solutions. Moreover, we employ the positive quadratic function method to create [...] Read more.

In this investigation, a novel (3+1)-dimensional Lax integrable Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation is constructed and analyzed analytically. The Painlevé integrability for the mentioned model is examined. The bilinear form is applied for investigating multiple-soliton solutions. Moreover, we employ the positive quadratic function method to create a class of lump solutions using distinct parameters values. The current study serves as a guide to explain many nonlinear phenomena that arise in numerous scientific domains, such as fluid mechanics; physics of plasmas, oceans, and seas; and so on. Full article

(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)

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