MDPI - Publisher of Open Access Journals

21 pages, 363 KB

Open AccessArticle

A Fast High-Order Compact Difference Scheme for Time-Fractional KS Equation with the Generalized Burgers’ Type Nonlinearity

by Huifa Jiang and Da Xu

Fractal Fract. 2025, 9(4), 218; https://doi.org/10.3390/fractalfract9040218 - 30 Mar 2025

Viewed by 406

Abstract

This work integrates the fast Alikhanov method with a compact scheme to solve the time-fractional Kuramoto–Sivashinsky (KS) equation with the generalized Burgers’ type nonlinearity. Initially, the Alikhanov algorithm, designed to handle the Caputo fractional derivative on non-uniform time grids, effectively avoids the initial [...] Read more.

This work integrates the fast Alikhanov method with a compact scheme to solve the time-fractional Kuramoto–Sivashinsky (KS) equation with the generalized Burgers’ type nonlinearity. Initially, the Alikhanov algorithm, designed to handle the Caputo fractional derivative on non-uniform time grids, effectively avoids the initial singularity. Additionally, the combination of the Alikhanov method with the sum-of-exponentials (SOE) technique significantly reduces both computational cost and memory requirements. By discretizing the spatial direction using a compact finite difference method, a fully discrete scheme is developed, achieving fourth-order convergence in the spatial domain. Stability and convergence are analyzed through energy methods. Several numerical examples are provided to validate the theoretical framework, demonstrating that the proposed algorithm is accurate, stable, and efficient. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)

► Show Figures

Figure 1

17 pages, 3878 KB

Open AccessArticle

Qualitative Analysis and Novel Exact Soliton Solutions to the Compound Korteweg–De Vries–Burgers Equation

by Abdulrahman Alomair, Abdulaziz Saud Al Naim and Mustafa Bayram

Fractal Fract. 2024, 8(12), 752; https://doi.org/10.3390/fractalfract8120752 - 21 Dec 2024

Viewed by 971

Abstract

This paper deals with the exact wave results of the (1+1)-dimensional nonlinear compound Korteweg–De Vries and Burgers (KdVB) equation with a truncated M-fractional derivative. This model represents the generalization of Korteweg–De Vries-modified Korteweg–De Vries and Burgers equations. We obtained periodic, combo singular, dark–bright, [...] Read more.

This paper deals with the exact wave results of the (1+1)-dimensional nonlinear compound Korteweg–De Vries and Burgers (KdVB) equation with a truncated M-fractional derivative. This model represents the generalization of Korteweg–De Vries-modified Korteweg–De Vries and Burgers equations. We obtained periodic, combo singular, dark–bright, and other wave results with the use of the extended sinh-Gordon equation expansion (EShGEE) and modified

(G^{'} / G^{2})

-expansion techniques. The use of the effective fractional derivative makes our results much better than the existing results. The obtained solutions are useful as well as applicable in various fields, including mathematical physics, plasma physics, ocean engineering, optics, etc. The obtained solutions are demonstrated by 2D, 3D, and contour plots. The achieved results will be fruitful for future research on this equation. Stability analysis is used to check that the results are precise as well as exact. Modulation instability (MI) analysis is performed to find stable steady-state solutions of the abovementioned model. In the end, it is concluded that the methods used are easy and reliable. Full article

(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)

► Show Figures

Figure 1

22 pages, 2331 KB

Open AccessArticle

A Novel Hybrid Computational Technique to Study Conformable Burgers’ Equation

by Abdul-Majeed Ayebire, Atul Pasrija, Mukhdeep Singh Manshahia and Shelly Arora

Math. Comput. Appl. 2024, 29(6), 114; https://doi.org/10.3390/mca29060114 - 5 Dec 2024

Viewed by 1047

Abstract

A fully discrete computational technique involving the implicit finite difference technique and cubic Hermite splines is proposed to solve the non-linear conformable damped Burgers’ equation with variable coefficients numerically. The proposed scheme is capable of solving the equation having singularity at [...] Read more.

A fully discrete computational technique involving the implicit finite difference technique and cubic Hermite splines is proposed to solve the non-linear conformable damped Burgers’ equation with variable coefficients numerically. The proposed scheme is capable of solving the equation having singularity at

t = 0

. The space direction is discretized using cubic Hermite splines, whereas the time direction is discretized using an implicit finite difference scheme. The convergence, stability and error estimates of the proposed scheme are discussed in detail to prove the efficiency of the technique. The convergence of the proposed scheme is found to be of order

h^{2}

in space and order

{(Δ t)}^{α}

in the time direction. The efficiency of the proposed scheme is verified by calculating error norms in the Eucledian and supremum sense. The proposed technique is applied on conformable damped Burgers’ equation with different initial and boundary conditions and the results are presented as tables and graphs. Comparison with results already in the literature also validates the application of the proposed technique. Full article

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

► Show Figures

Figure 1

25 pages, 1009 KB

Open AccessArticle

Solution for Time-Fractional Coupled Burgers Equations by Generalized-Laplace Transform Methods

by Hassan Eltayeb and Said Mesloub

Fractal Fract. 2024, 8(12), 692; https://doi.org/10.3390/fractalfract8120692 - 25 Nov 2024

Viewed by 850

Abstract

In this work, nonlinear time-fractional coupled Burgers equations are solved utilizing a computational method, which is called the double and triple generalized-Laplace transform and decomposition method. We discuss the proof of triple generalized-Laplace transform for a Caputo fractional derivative. We have given four [...] Read more.

In this work, nonlinear time-fractional coupled Burgers equations are solved utilizing a computational method, which is called the double and triple generalized-Laplace transform and decomposition method. We discuss the proof of triple generalized-Laplace transform for a Caputo fractional derivative. We have given four examples to show the precision and adequacy of the suggested approach. The results show that this method is easy and accurate when compared to the A domain decomposition method (ADM), homotopy perturbation method (HPM), and generalized differential transform method (GDTM). Finally, we have sketched the graphics for all these examples. Full article

(This article belongs to the Special Issue Fractional Elliptic and Parabolic Equations: Analysis and Related Topics)

► Show Figures

Figure 1

22 pages, 6282 KB

Open AccessArticle

Quadrature Solution for Fractional Benjamin–Bona–Mahony–Burger Equations

by Waleed Mohammed Abdelfattah, Ola Ragb, Mokhtar Mohamed, Mohamed Salah and Abdelfattah Mustafa

Fractal Fract. 2024, 8(12), 685; https://doi.org/10.3390/fractalfract8120685 - 22 Nov 2024

Viewed by 695

Abstract

In this work, we present various novelty methods by employing the fractional differential quadrature technique to solve the time and space fractional nonlinear Benjamin–Bona–Mahony equation and the Benjamin–Bona–Mahony–Burger equation. The novelty of these methods is based on the generalized Caputo sense, classical differential [...] Read more.

In this work, we present various novelty methods by employing the fractional differential quadrature technique to solve the time and space fractional nonlinear Benjamin–Bona–Mahony equation and the Benjamin–Bona–Mahony–Burger equation. The novelty of these methods is based on the generalized Caputo sense, classical differential quadrature method, and discrete singular convolution methods based on two different kernels. Also, the solution strategy is to apply perturbation analysis or an iterative method to reduce the problem to a series of linear initial boundary value problems. Consequently, we apply these suggested techniques to reduce the nonlinear fractional PDEs into ordinary differential equations. Hence, to validate the suggested techniques, a solution to this problem was obtained by designing a MATLAB code for each method. Also, we compare this solution with the exact ones. Furthermore, more figures and tables have been investigated to illustrate the high accuracy and rapid convergence of these novel techniques. From the obtained solutions, it was found that the suggested techniques are easily applicable and effective, which can help in the study of the other higher-D nonlinear fractional PDEs emerging in mathematical physics. Full article

(This article belongs to the Section Numerical and Computational Methods)

► Show Figures

Figure 1

19 pages, 1775 KB

Open AccessArticle

The Four-Dimensional Natural Transform Adomian Decomposition Method and (3+1)-Dimensional Fractional Coupled Burgers’ Equation

by Huda Alsaud and Hassan Eltayeb

Fractal Fract. 2024, 8(4), 227; https://doi.org/10.3390/fractalfract8040227 - 15 Apr 2024

Cited by 2 | Viewed by 1561

Abstract

This research article introduces the four-dimensional natural transform Adomian decomposition method (FNADM) for solving the (3+1)-dimensional time-singular fractional coupled Burgers’ equation, along with its associated initial conditions. The FNADM approach represents a fusion of four-dimensional natural transform techniques and Adomian decomposition methodologies. In [...] Read more.

This research article introduces the four-dimensional natural transform Adomian decomposition method (FNADM) for solving the (3+1)-dimensional time-singular fractional coupled Burgers’ equation, along with its associated initial conditions. The FNADM approach represents a fusion of four-dimensional natural transform techniques and Adomian decomposition methodologies. In order to observe the influence of time-Caputo fractional derivatives on the outcomes of the aforementioned models, two examples are illustrated along with their three-dimensional figures. The effectiveness and reliability of this approach are validated through the analysis of these examples related to the (3+1)-dimensional time-singular fractional coupled Burgers’ equations. This study underscores the method’s applicability and effectiveness in addressing the complex mathematical models encountered in various scientific and engineering domains. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 2nd Edition)

► Show Figures

Figure 1

18 pages, 18693 KB

Open AccessArticle

A Study of The Stochastic Burgers’ Equation Using The Dynamical Orthogonal Method

by Mohamed El-Beltagy, Ragab Mahdi and Adeeb Noor

Axioms 2023, 12(2), 152; https://doi.org/10.3390/axioms12020152 - 1 Feb 2023

Cited by 3 | Viewed by 1866

Abstract

In the current work, the stochastic Burgers’ equation is studied using the Dynamically Orthogonal (DO) method. The DO presents a low-dimensional representation for the stochastic fields. Unlike many other methods, it has a time-dependent property on both the spatial basis and stochastic coefficients, [...] Read more.

In the current work, the stochastic Burgers’ equation is studied using the Dynamically Orthogonal (DO) method. The DO presents a low-dimensional representation for the stochastic fields. Unlike many other methods, it has a time-dependent property on both the spatial basis and stochastic coefficients, with flexible representation especially in the strongly transient and nonstationary problems. We consider a computational study and application of the DO method and compare it with the Polynomial Chaos (PC) method. For comparison, both the stochastic viscous and inviscid Burgers’ equations are considered. A hybrid approach, combining the DO and PC is proposed in case of deterministic initial conditions to overcome the singularities that occur in the DO method. The results are verified with the stochastic collocation method. Overall, we observe that the DO method has a higher rate of convergence as the number of modes increases. The DO method is found to be more efficient than PC for the same level of accuracy, especially for the case of high-dimensional parametric spaces. The inviscid Burgers’ equation is analyzed to study the shock wave formation when using the DO after suitable handling of the convective term. The results show that the sinusoidal wave shape is distorted and sharpened as the time evolves till the shock wave occurs. Full article

(This article belongs to the Special Issue Advances in Numerical Analysis and Scientific Computing)

► Show Figures

Figure 1

20 pages, 3999 KB

Open AccessArticle

The Sensitive Visualization and Generalized Fractional Solitons’ Construction for Regularized Long-Wave Governing Model

by Riaz Ur Rahman, Waqas Ali Faridi, Magda Abd El-Rahman, Aigul Taishiyeva, Ratbay Myrzakulov and Emad Ahmad Az-Zo’bi

Fractal Fract. 2023, 7(2), 136; https://doi.org/10.3390/fractalfract7020136 - 1 Feb 2023

Cited by 34 | Viewed by 2506

Abstract

The solution of partial differential equations has generally been one of the most-vital mathematical tools for describing physical phenomena in the different scientific disciplines. The previous studies performed with the classical derivative on this model cannot express the propagating behavior at heavy infinite [...] Read more.

The solution of partial differential equations has generally been one of the most-vital mathematical tools for describing physical phenomena in the different scientific disciplines. The previous studies performed with the classical derivative on this model cannot express the propagating behavior at heavy infinite tails. In order to address this problem, this study addressed the fractional regularized long-wave Burgers problem by using two different fractional operators, Beta and M-truncated, which are capable of predicting the behavior where the classical derivative is unable to show dynamical characteristics. This fractional equation is first transformed into an ordinary differential equation using the fractional traveling wave transformation. A new auxiliary equation approach was employed in order to discover new soliton solutions. As a result, bright, periodic, singular, mixed periodic, rational, combined dark–bright, and dark soliton solutions were found based on the constraint relation imposed on the auxiliary equation parameters. The graphical visualization of the obtained results is displayed by taking the suitable parametric values and predicting that the fractional order parameter is responsible for controlling the behavior of propagating solitary waves and also providing the comparison between fractional operators and the classical derivative. We are confident about the vital applications of this study in many scientific fields. Full article

(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)

► Show Figures

Figure 1

10 pages, 2639 KB

Open AccessArticle

Painlevé Test and Exact Solutions for (1 + 1)-Dimensional Generalized Broer–Kaup Equations

by Sheng Zhang and Bo Xu

Mathematics 2022, 10(3), 486; https://doi.org/10.3390/math10030486 - 2 Feb 2022

Cited by 4 | Viewed by 2275

Abstract

In this paper, the Painlevé integrable property of the (1 + 1)-dimensional generalized Broer–Kaup (gBK) equations is first proven. Then, the Bäcklund transformations for the gBK equations are derived by using the Painlevé truncation. Based on a special case of the derived Bäcklund [...] Read more.

In this paper, the Painlevé integrable property of the (1 + 1)-dimensional generalized Broer–Kaup (gBK) equations is first proven. Then, the Bäcklund transformations for the gBK equations are derived by using the Painlevé truncation. Based on a special case of the derived Bäcklund transformations, the gBK equations are linearized into the heat conduction equation. Inspired by the derived Bäcklund transformations, the gBK equations are reduced into the Burgers equation. Starting from the linear heat conduction equation, two forms of N-soliton solutions and rational solutions with a singularity condition of the gBK equations are constructed. In addition, the rational solutions with two singularity conditions of the gBK equation are obtained by considering the non-uniqueness and generality of a resonance function embedded into the Painlevé test. In order to understand the nonlinear dynamic evolution dominated by the gBK equations, some of the obtained exact solutions, including one-soliton solutions, two-soliton solutions, three-soliton solutions, and two pairs of rational solutions, are shown by three-dimensional images. This paper shows that when the Painlevé test deals with the coupled nonlinear equations, the highest negative power of the coupled variables should be comprehensively considered in the leading term analysis rather than the formal balance between the highest-order derivative term and the highest-order nonlinear term. Full article

(This article belongs to the Special Issue Partial Differential Equations with Applications: Analytical Methods)

► Show Figures

Figure 1

24 pages, 1542 KB

Open AccessArticle

Analysis of the Time Fractional-Order Coupled Burgers Equations with Non-Singular Kernel Operators

by Noufe H. Aljahdaly, Ravi P. Agarwal, Rasool Shah and Thongchai Botmart

Mathematics 2021, 9(18), 2326; https://doi.org/10.3390/math9182326 - 19 Sep 2021

Cited by 45 | Viewed by 3119

Abstract

In this article, we have investigated the fractional-order Burgers equation via Natural decomposition method with nonsingular kernel derivatives. The two types of fractional derivatives are used in the article of Caputo–Fabrizio and Atangana–Baleanu derivative. We employed Natural transform on fractional-order Burgers equation followed [...] Read more.

In this article, we have investigated the fractional-order Burgers equation via Natural decomposition method with nonsingular kernel derivatives. The two types of fractional derivatives are used in the article of Caputo–Fabrizio and Atangana–Baleanu derivative. We employed Natural transform on fractional-order Burgers equation followed by inverse Natural transform, to achieve the result of the equations. To validate the method, we have considered a two examples and compared with the exact results. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

► Show Figures

Figure 1

12 pages, 241 KB

Open AccessArticle

Singularity Formation in the Inviscid Burgers Equation

by Giuseppe Maria Coclite and Lorenzo di Ruvo

Symmetry 2021, 13(5), 848; https://doi.org/10.3390/sym13050848 - 11 May 2021

Viewed by 1716

Abstract

We provide a lower bound for the blow up time of the

H^{2}

norm of the entropy solutions of the inviscid Burgers equation in terms of the

H^{2}

norm of the initial datum. This shows an interesting symmetry of the Burgers [...] Read more.

We provide a lower bound for the blow up time of the

H^{2}

norm of the entropy solutions of the inviscid Burgers equation in terms of the

H^{2}

norm of the initial datum. This shows an interesting symmetry of the Burgers equation: the invariance of the space

H^{2}

under the action of such nonlinear equation. The argument is based on a priori estimates of energy and stability type for the (viscous) Burgers equation. Full article

(This article belongs to the Section Mathematics)

21 pages, 341 KB

Open AccessArticle

A Note on Double Conformable Laplace Transform Method and Singular One Dimensional Conformable Pseudohyperbolic Equations

by Hassan Eltayeb, Said Mesloub, Yahya T. Abdalla and Adem Kılıçman

Mathematics 2019, 7(10), 949; https://doi.org/10.3390/math7100949 - 12 Oct 2019

Cited by 7 | Viewed by 2399

Abstract

The purpose of this article is to obtain the exact and approximate numerical solutions of linear and nonlinear singular conformable pseudohyperbolic equations and conformable coupled pseudohyperbolic equations through the conformable double Laplace decomposition method. Further, the numerical examples were provided in order to [...] Read more.

The purpose of this article is to obtain the exact and approximate numerical solutions of linear and nonlinear singular conformable pseudohyperbolic equations and conformable coupled pseudohyperbolic equations through the conformable double Laplace decomposition method. Further, the numerical examples were provided in order to demonstrate the efficiency, high accuracy, and the simplicity of present method. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)

► Show Figures

Figure 1

12 pages, 1821 KB

Open AccessArticle

Analytical Wave Solutions for Foam and KdV-Burgers Equations Using Extended Homogeneous Balance Method

by U.M. Abdelsalam and M. G. M. Ghazal

Mathematics 2019, 7(8), 729; https://doi.org/10.3390/math7080729 - 9 Aug 2019

Cited by 9 | Viewed by 3446

Abstract

In this paper, extended homogeneous balance method is presented with the aid of computer algebraic system Mathematica for deriving new exact traveling wave solutions for the foam drainage equation and the Kowerteg-de Vries–Burgers equation which have many applications in industrial applications and plasma [...] Read more.

In this paper, extended homogeneous balance method is presented with the aid of computer algebraic system Mathematica for deriving new exact traveling wave solutions for the foam drainage equation and the Kowerteg-de Vries–Burgers equation which have many applications in industrial applications and plasma physics. The method is effective to construct a series of analytical solutions including many types like periodical, rational, singular, shock, and soliton wave solutions for a wide class of nonlinear evolution equations in mathematical physics and engineering sciences. Full article

► Show Figures

Figure 1

13 pages, 264 KB

Open AccessArticle

On Conformable Double Laplace Transform and One Dimensional Fractional Coupled Burgers’ Equation

by Hassan Eltayeb, Imed Bachar and Adem Kılıçman

Symmetry 2019, 11(3), 417; https://doi.org/10.3390/sym11030417 - 21 Mar 2019

Cited by 25 | Viewed by 3256

Abstract

In the present work we introduced a new method and name it the conformable double Laplace decomposition method to solve one dimensional regular and singular conformable functional Burger’s equation. We studied the existence condition for the conformable double Laplace transform. In order to [...] Read more.

In the present work we introduced a new method and name it the conformable double Laplace decomposition method to solve one dimensional regular and singular conformable functional Burger’s equation. We studied the existence condition for the conformable double Laplace transform. In order to obtain the exact solution for nonlinear fractional problems, then we modified the double Laplace transform and combined it with the Adomian decomposition method. Later, we applied the new method to solve regular and singular conformable fractional coupled Burgers’ equations. Further, in order to illustrate the effectiveness of present method, we provide some examples. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory, Methods and Applications)

Search Results (14)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (14)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI