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Search Results (5)

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Keywords = damped Burgers’ equation

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21 pages, 1488 KiB  
Article
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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22 pages, 2331 KiB  
Article
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 1015
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 h2 in space and order (Δt)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)
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8 pages, 860 KiB  
Article
Dynamics and Exact Traveling Wave Solutions of the Sharma–Tasso–Olver–Burgers Equation
by Yan Zhou and Jinsen Zhuang
Symmetry 2022, 14(7), 1468; https://doi.org/10.3390/sym14071468 - 18 Jul 2022
Cited by 6 | Viewed by 1927
Abstract
In this paper, to study the Sharma–Tasso–Olver–Burgers equation, we focus on the geometric properties and the exact traveling wave solutions. The corresponding traveling system is a cubic oscillator with damping, and it has time-dependent and time-independent first integral. For all bounded orbits of [...] Read more.
In this paper, to study the Sharma–Tasso–Olver–Burgers equation, we focus on the geometric properties and the exact traveling wave solutions. The corresponding traveling system is a cubic oscillator with damping, and it has time-dependent and time-independent first integral. For all bounded orbits of the traveling system, we give the exact explicit kink wave solutions. Full article
(This article belongs to the Special Issue Recent Advances in Symmetries Methods and Other Approaches to Nonlinear Differential Equations)
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12 pages, 2889 KiB  
Article
On the Multistage Differential Transformation Method for Analyzing Damping Duffing Oscillator and Its Applications to Plasma Physics
by Noufe H. Aljahdaly and S. A. El-Tantawy
Mathematics 2021, 9(4), 432; https://doi.org/10.3390/math9040432 - 22 Feb 2021
Cited by 54 | Viewed by 2898
Abstract
The multistage differential transformation method (MSDTM) is used to find an approximate solution to the forced damping Duffing equation (FDDE). In this paper, we prove that the MSDTM can predict the solution in the long domain as compared to differential transformation method (DTM) [...] Read more.
The multistage differential transformation method (MSDTM) is used to find an approximate solution to the forced damping Duffing equation (FDDE). In this paper, we prove that the MSDTM can predict the solution in the long domain as compared to differential transformation method (DTM) and more accurately than the modified differential transformation method (MDTM). In addition, the maximum residual errors for DTM and its modification methods (MSDTM and MDTM) are estimated. As a real application to the obtained solution, we investigate the oscillations in a complex unmagnetized plasma. To do that, the fluid govern equations of plasma species is reduced to the modified Korteweg–de Vries–Burgers (mKdVB) equation. After that, by using a suitable transformation, the mKdVB equation is transformed into the forced damping Duffing equation. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
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11 pages, 5234 KiB  
Article
Optimal Perturbation Iteration Method for Solving Fractional Model of Damped Burgers’ Equation
by Sinan Deniz, Ali Konuralp and Mnauel De la Sen
Symmetry 2020, 12(6), 958; https://doi.org/10.3390/sym12060958 - 4 Jun 2020
Cited by 19 | Viewed by 2794
Abstract
The newly constructed optimal perturbation iteration procedure with Laplace transform is applied to obtain the new approximate semi-analytical solutions of the fractional type of damped Burgers’ equation. The classical damped Burgers’ equation is remodeled to fractional differential form via the Atangana–Baleanu fractional derivatives [...] Read more.
The newly constructed optimal perturbation iteration procedure with Laplace transform is applied to obtain the new approximate semi-analytical solutions of the fractional type of damped Burgers’ equation. The classical damped Burgers’ equation is remodeled to fractional differential form via the Atangana–Baleanu fractional derivatives described with the help of the Mittag–Leffler function. To display the efficiency of the proposed optimal perturbation iteration technique, an extended example is deeply analyzed. Full article
(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)
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