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Keywords = fractional-in-space Burgers equation

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22 pages, 6282 KiB  
Article
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 685
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)
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20 pages, 1815 KiB  
Article
Coupled Fixed Point and Hybrid Generalized Integral Transform Approach to Analyze Fractal Fractional Nonlinear Coupled Burgers Equation
by Souhail Mohammed Bouzgarrou, Sami Znaidia, Adeeb Noor, Shabir Ahmad and Sayed M. Eldin
Fractal Fract. 2023, 7(7), 551; https://doi.org/10.3390/fractalfract7070551 - 16 Jul 2023
Cited by 5 | Viewed by 1539
Abstract
In this manuscript, the nonlinear Burgers equations are studied via a fractal fractional (FF) Caputo operator. The results of coupled fixed point theorems in cone metric space are used to discuss the uniqueness of solution to the proposed coupled equations. The solution of [...] Read more.
In this manuscript, the nonlinear Burgers equations are studied via a fractal fractional (FF) Caputo operator. The results of coupled fixed point theorems in cone metric space are used to discuss the uniqueness of solution to the proposed coupled equations. The solution of the proposed equation is computed via Natural transform associated with the Adomian decomposition method (NADM). The acquired results are graphically presented for some values of fractional order and fractal dimensions. The accuracy and consistency of the applied method is verified through error analysis. Full article
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14 pages, 468 KiB  
Article
On Exact Solutions of Some Space–Time Fractional Differential Equations with M-truncated Derivative
by Ayten Özkan, Erdoĝan Mehmet Özkan and Ozgur Yildirim
Fractal Fract. 2023, 7(3), 255; https://doi.org/10.3390/fractalfract7030255 - 10 Mar 2023
Cited by 14 | Viewed by 2095
Abstract
In this study, the extended G′/G method is used to investigate the space–time fractional Burger-like equation and the space–time-coupled Boussinesq equation with M-truncated derivative, which have an important place in fluid dynamics. This method is efficient and produces soliton solutions. [...] Read more.
In this study, the extended G′/G method is used to investigate the space–time fractional Burger-like equation and the space–time-coupled Boussinesq equation with M-truncated derivative, which have an important place in fluid dynamics. This method is efficient and produces soliton solutions. A symbolic computation program called Maple was used to implement the method in a dependable and effective way. There are also a few graphs provided for the solutions. Using the suggested method to solve these equations, we have provided many new exact solutions that are distinct from those previously found. By offering insightful explanations of many nonlinear systems, the study’s findings add to the body of literature. The results revealed that the suggested method is a valuable mathematical tool and that using a symbolic computation program makes these tasks simpler, more dependable, and quicker. It is worth noting that it may be used for a wide range of nonlinear evolution problems in mathematical physics. The study’s findings may have an influence on how different physical problems are interpreted. Full article
(This article belongs to the Special Issue Advances in Nonlinear Differential Equations with Applications)
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12 pages, 291 KiB  
Article
Blow-Up of Solutions to Fractional-in-Space Burgers-Type Equations
by Munirah Alotaibi, Mohamed Jleli and Bessem Samet
Fractal Fract. 2021, 5(4), 249; https://doi.org/10.3390/fractalfract5040249 - 1 Dec 2021
Cited by 1 | Viewed by 2118
Abstract
We consider fractional-in-space analogues of Burgers equation and Korteweg-de Vries-Burgers equation on bounded domains. Namely, we establish sufficient conditions for finite-time blow-up of solutions to the mentioned equations. The obtained conditions depend on the initial value and the boundary conditions. Some examples are [...] Read more.
We consider fractional-in-space analogues of Burgers equation and Korteweg-de Vries-Burgers equation on bounded domains. Namely, we establish sufficient conditions for finite-time blow-up of solutions to the mentioned equations. The obtained conditions depend on the initial value and the boundary conditions. Some examples are provided to illustrate our obtained results. In the proofs of our main results, we make use of the test function method and some integral inequalities. Full article
(This article belongs to the Special Issue Probabilistic Solutions and Stochastic Representation of Fractional Differential Equations)
20 pages, 478 KiB  
Article
Numerical Solution of Fractional Order Burgers’ Equation with Dirichlet and Neumann Boundary Conditions by Reproducing Kernel Method
by Onur Saldır, Mehmet Giyas Sakar and Fevzi Erdogan
Fractal Fract. 2020, 4(2), 27; https://doi.org/10.3390/fractalfract4020027 - 19 Jun 2020
Cited by 8 | Viewed by 3057
Abstract
In this research, obtaining of approximate solution for fractional-order Burgers’ equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms. Then an iterative approach is constructed by using kernel functions. [...] Read more.
In this research, obtaining of approximate solution for fractional-order Burgers’ equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms. Then an iterative approach is constructed by using kernel functions. The convergence of this approach and its error estimates are given. The numerical algorithm of the method is presented. Furthermore, numerical outcomes are shown with tables and graphics for some examples. These outcomes demonstrate that the proposed method is convenient and effective. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order)
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