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24 pages, 4704 KiB

Open AccessArticle

An Unconditionally Stable Numerical Scheme for 3D Coupled Burgers’ Equations

by Gonca Çelikten

Symmetry 2025, 17(3), 452; https://doi.org/10.3390/sym17030452 - 18 Mar 2025

Viewed by 349

In this study, we sought numerical solutions for three-dimensional coupled Burgers’ equations. Burgers’ equations are fundamental partial differential equations in fluid mechanics. They integrate the characteristics of both the first-order wave equation and the heat conduction equation, serving as crucial tools for modeling [...] Read more.

In this study, we sought numerical solutions for three-dimensional coupled Burgers’ equations. Burgers’ equations are fundamental partial differential equations in fluid mechanics. They integrate the characteristics of both the first-order wave equation and the heat conduction equation, serving as crucial tools for modeling the interaction between convection and diffusion. First, the fractional step method was applied to decompose the equations into one-dimensional forms. Then, implicit finite difference approximations were used to solve the resulting one-dimensional equations. To assess the accuracy of the proposed approach, we tested it on two benchmark problems and compared the results with existing methods in the literature. Additionally, the symmetry of the solution graphs was analyzed to gain deeper insight into the results. Stability analysis using the von Neumann method confirmed that the proposed approach is unconditionally stable. The results obtained in this study strongly support the effectiveness and reliability of the proposed method in solving three-dimensional coupled Burgers’ equations. Full article

(This article belongs to the Special Issue Symmetry in Fluid Mechanics: New Challenges in Fluid–Structure Interaction)

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21 pages, 1488 KiB

Open AccessArticle

Exploring Fractional Damped Burgers’ Equation: A Comparative Analysis of Analytical Methods

by Azzh Saad Alshehry and Rasool Shah

Fractal Fract. 2025, 9(2), 107; https://doi.org/10.3390/fractalfract9020107 - 10 Feb 2025

Viewed by 866

Abstract

This investigation focuses on the study of the fractional damped Burgers’ equation by using the natural residual power series method coupled with the new iteration transform method in the context of the Caputo operator. The equation of Burgers under the damped context is [...] Read more.

This investigation focuses on the study of the fractional damped Burgers’ equation by using the natural residual power series method coupled with the new iteration transform method in the context of the Caputo operator. The equation of Burgers under the damped context is useful when studying one-dimensional nonlinear waves involving damping effect, and is used in fluid dynamics, among other applications. Two new mathematical methods that can be used to obtain an approximate solution to this complex non-linear problem are the natural residual power series method and the new iteration transform method. Therefore, it can be deduced that the Caputo operator aids in modeling of the fractional derivatives, as it provides a better description of the physical realities. Thus, the objective of the present work is to advance the knowledge accumulated on the behavior of solutions to the damped Burgers’ equation, as well as to check the applicability of the proposed approaches to other nonlinear fractional partial differential equations. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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19 pages, 1775 KiB

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 1519

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)

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15 pages, 298 KiB

Open AccessArticle

The Solution of Coupled Burgers’ Equation by G-Laplace Transform

by Reem K. Alhefthi and Hassan Eltayeb

Symmetry 2023, 15(9), 1764; https://doi.org/10.3390/sym15091764 - 15 Sep 2023

Cited by 3 | Viewed by 1316

Abstract

The coupled Burgers’ equation is a fundamental partial differential equation with applications in various scientific fields. Finding accurate solutions to this equation is crucial for understanding physical phenomena and mathematical models. While different methods have been explored, this work highlights the importance of [...] Read more.

The coupled Burgers’ equation is a fundamental partial differential equation with applications in various scientific fields. Finding accurate solutions to this equation is crucial for understanding physical phenomena and mathematical models. While different methods have been explored, this work highlights the importance of the G-Laplace transform. The G-transform is effective in solving a wide range of non-constant coefficient differential equations, setting it apart from the Laplace, Sumudu, and Elzaki transforms. Consequently, it stands as a powerful tool for addressing differential equations characterized by variable coefficients. By applying this transformative approach, the study provides reliable and exact solutions for both homogeneous and non-homogeneous coupled Burgers’ equations. This innovative technique offers a valuable tool for gaining deeper insights into this equation’s behavior and significance in diverse disciplines. Full article

(This article belongs to the Special Issue Various Approaches for Generalized Integral Transforms)

16 pages, 348 KiB

Open AccessArticle

An Efficient Non-Standard Numerical Scheme Coupled with a Compact Finite Difference Method to Solve the One-Dimensional Burgers’ Equation

by Komalpreet Kaur and Gurjinder Singh

Axioms 2023, 12(6), 593; https://doi.org/10.3390/axioms12060593 - 15 Jun 2023

Cited by 4 | Viewed by 1819

Abstract

This article proposes a family of non-standard methods coupled with compact finite differences to numerically integrate the non-linear Burgers’ equation. Firstly, a family of non-standard methods is derived to deal with a system of ordinary differential equations (ODEs) arising from the semi-discretization of [...] Read more.

This article proposes a family of non-standard methods coupled with compact finite differences to numerically integrate the non-linear Burgers’ equation. Firstly, a family of non-standard methods is derived to deal with a system of ordinary differential equations (ODEs) arising from the semi-discretization of initial-boundary value partial differential equations (PDEs). Further, a method of this family is considered as a special case and coupled with a fourth-order compact finite difference resulting in a combined numerical scheme to solve initial-boundary value PDEs. The combined scheme has first-order accuracy in time and fourth-order accuracy in space. Some basic characteristics of the scheme are analysed and a section concerning the numerical experiments is presented demonstrating the good performance of the combined numerical scheme. Full article

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16 pages, 921 KiB

Open AccessArticle

A Comparative Study of the Fractional Coupled Burgers and Hirota–Satsuma KdV Equations via Analytical Techniques

by Humaira Yasmin and Naveed Iqbal

Symmetry 2022, 14(7), 1364; https://doi.org/10.3390/sym14071364 - 2 Jul 2022

Cited by 17 | Viewed by 2273

Abstract

This paper applies modified analytical methods to the fractional-order analysis of one and two-dimensional nonlinear systems of coupled Burgers and Hirota–Satsuma KdV equations. The Atangana–Baleanu fractional derivative operator and the Elzaki transform will be used to solve the proposed problems. The results of [...] Read more.

This paper applies modified analytical methods to the fractional-order analysis of one and two-dimensional nonlinear systems of coupled Burgers and Hirota–Satsuma KdV equations. The Atangana–Baleanu fractional derivative operator and the Elzaki transform will be used to solve the proposed problems. The results of utilizing the proposed techniques are compared to the exact solution. The technique’s convergence is successfully presented and mathematically proven. To demonstrate the efficacy of the suggested techniques, we compared actual and analytic solutions using figures, which are in strong agreement with one another. Furthermore, the solutions achieved by applying the current techniques at different fractional orders are compared to the integer order. The proposed methods are appealing, simple, and accurate, indicating that they are appropriate for solving partial differential equations or systems of partial differential equations. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry Applied in Nonlinear Analysis)

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10 pages, 2639 KiB

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 2240

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)

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7 pages, 233 KiB

Open AccessArticle

Generalized Symmetries and mCK Method Analysis of the (2+1)-Dimensional Coupled Burgers Equations

by Gangwei Wang, Yixing Liu, Shuxin Han, Hua Wang and Xing Su

Symmetry 2019, 11(12), 1473; https://doi.org/10.3390/sym11121473 - 3 Dec 2019

Cited by 8 | Viewed by 2094

Abstract

In this paper, generalized symmetries and mCK method are employed to analyze the (2+1)-dimensional coupled Burgers equations. Firstly, based on the generalized symmetries method, the corresponding symmetries of the (2+1)-dimensional coupled Burgers equations are derived. And then, using the mCK method, symmetry transformation [...] Read more.

In this paper, generalized symmetries and mCK method are employed to analyze the (2+1)-dimensional coupled Burgers equations. Firstly, based on the generalized symmetries method, the corresponding symmetries of the (2+1)-dimensional coupled Burgers equations are derived. And then, using the mCK method, symmetry transformation group theorem is presented. From symmetry transformation group theorem, a great many of new solutions can be derived. Lastly, Lie algebra for given symmetry group are considered. Full article

(This article belongs to the Special Issue Advances in Nonlinear, Discrete, Continuous and Hamiltonian Systems)

21 pages, 341 KiB

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 2378

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)

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13 pages, 264 KiB

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 3235

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)

18 pages, 12970 KiB

Open AccessArticle

Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method

by Imtiaz Ahmad, Muhammad Ahsan, Iltaf Hussain, Poom Kumam and Wiyada Kumam

Symmetry 2019, 11(3), 394; https://doi.org/10.3390/sym11030394 - 18 Mar 2019

Cited by 32 | Viewed by 4036

Abstract

In this paper, a local meshless differential quadrature collocation method based on radial basis functions is proposed for the numerical simulation of one-dimensional Klein–Gordon, two-dimensional coupled Burgers’, and regularized long wave equations. Both local and global meshless collocation procedures are used for spatial [...] Read more.

In this paper, a local meshless differential quadrature collocation method based on radial basis functions is proposed for the numerical simulation of one-dimensional Klein–Gordon, two-dimensional coupled Burgers’, and regularized long wave equations. Both local and global meshless collocation procedures are used for spatial discretization, which convert the mentioned partial differential equations into a system of ordinary differential equations. The obtained system has been solved by the forward Euler difference formula. An upwind technique is utilized in the case of the convection-dominated coupled Burgers’ model equation. Having no need for the mesh in the problem domain and being less sensitive to the variation of the shape parameter as compared to global meshless methods are the salient features of the local meshless method. Both rectangular and non-rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy, and the ease of implementation of the proposed method are shown via test problems. Full article

(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)

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21 pages, 2118 KiB

Open AccessArticle

Variable Shape Parameter Strategy in Local Radial Basis Functions Collocation Method for Solving the 2D Nonlinear Coupled Burgers’ Equations

by Hananeh Nojavan, Saeid Abbasbandy and Tofigh Allahviranloo

Mathematics 2017, 5(3), 38; https://doi.org/10.3390/math5030038 - 21 Jul 2017

Cited by 3 | Viewed by 4543

Abstract

This study aimed at investigating a local radial basis function collocation method (LRBFCM) in the reproducing kernel Hilbert space. This method was, in fact, a meshless one which applied the local sub-clusters of domain nodes for the approximation of the arbitrary field. For [...] Read more.

This study aimed at investigating a local radial basis function collocation method (LRBFCM) in the reproducing kernel Hilbert space. This method was, in fact, a meshless one which applied the local sub-clusters of domain nodes for the approximation of the arbitrary field. For time-dependent partial differential equations (PDEs), it would be changed to a system of ordinary differential equations (ODEs). Here, we intended to decrease the error through utilizing variable shape parameter (VSP) strategies. This method was an appropriate way to solve the two-dimensional nonlinear coupled Burgers’ equations comprised of Dirichlet and mixed boundary conditions. Numerical examples indicated that the variable shape parameter strategies were more efficient than constant ones for various values of the Reynolds number. Full article

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