SOLVING BURGER’S AND COUPLED BURGER’S EQUATIONS WITH CAPUTO-FABRIZIO FRACTIONAL OPERATOR

Hijaz Ahmad, Hassan Kamil Jassim

DOI Number
https://doi.org/10.22190/FUMI210327017A
First page
241
Last page
252

Abstract


In this paper, we apply Daftardar-Jafari method (DJM) to obtain approximate solutions of the nonlinear Burgers (NBE) and coupled nonlinear Burger’s equations (CNBEs) with Caputo-Fabrizio fractional operator (CFFO). The efficiency of the considered method is illustrated by some examples. Graphical results are utilized and discussed quantitatively to illustrate the solution. The results reveal that the suggested algorithm is very effective and simple and can be applied for other problems in sciences and engineering.


Keywords

nonlinear equations, fractional operator, approximate solutions.

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References


I. Podlubny: Fractional Differential Equations. Mathematics in Science and Engineering, Academic Press, San Diego, California, USA, 1999.

J. H. He: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178 (1999) 257-262.

V. Gill, K. Modi and Y. Singh: Analytic solutions of fractional differential equation associated with RLC electrical circuit. Journal of Statistics and Management Systems 21(4) (2018) 575-582.

Y. Yu, P. Perdikaris and G. E. Karniadakis: Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms. Journal of Computational Physics 323 (2016) 219-242.

R. Hilfer: Applications of Fractional Calculus in Physics. Orlando (1999).

H. Sheng, Y. Chen and T. Qiu: Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications. Springer, Berlin (2011).

I. Ahmad, H. Ahmad, P. Thounthong, Y.-M. Chu and C. Cesarano: Solution of multi-term time-fractional PDE models arising in mathematical biology and physics by local meshless method. Symmetry 12.7 (2020), 1195.

H. Ahmad, T. A. Khan, I. Ahmad, P. S. Stanimirovic and Yu-Ming Chu: A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations. Results in Physics 19 (2020), 103462.

H. Ahmad, T. A. Khan, P. S. Stanimirovic, Y.-M. Chu and I. Ahmad: Modied variational iteration algorithm-II: convergence and applications to diffusion models. Complexity 2020 (2020), 1-14.

A. Imtiaz, A. Hijaz, I. Mustafa, Y. Shao-Wen and A. Bandar: Application of local meshless method for the solution of two term time fractional-order multidimensional PDE arising in heat and mass transfer. Thermal Science 24 Suppl. 1 (2020), 95-105.

M. N. Khan, I. Ahmad, A. Akgul, H. Ahmad and P. Thounthong : Numerical solution of time-fractional coupled Kortewegde Vries and Klein-Gordon equations by local meshless method. Pramana 95.1 (2021), 1-13.

J.-F. Li, I. Ahmad, H. Ahmad, D. Shah, Y.-M. Chu, P. Thounthong and M. Ayaz: Numerical solution of two-term time-fractional PDE models arising in mathematical physics using local meshless method. Open Physics 18.1 (2020), 1063-1072.

M. Shakeel, I. Hussain, H. Ahmad, I. Ahmad, P. Thounthong and Y.-F. Zhang: Meshless technique for the solution of time-fractional partial differential equations having real-world applications. Journal of Function Spaces 2020 (2020), 8898309.

H. Srivastava, Hijaz Ahmad, I. Ahmad, P. Thounthong and M. Khan: Numerical simulation of three-dimensional fractional-order convection-diffusion PDEs by a local meshless method. Thermal Science 25(1) (2021), 347-358.

M. Inc, M. N. Khan, I. Ahmad, S.-W. Yao, H. Ahmad and P. Thounthong: Analysing time-fractional exotic options via efficient local meshless method. Results in Physics 19 (2020), 103385.

I. Ahmad, M. N. Khan, M. Inc, H. Ahmad and K.S. Nisar: Numerical simulation of simulate an anomalous solute transport model via local meshless method. Alexandria Engineering Journal 59.4 (2020), 2827-2838.

Ahmad, Imtiaz and Sakhi Zaman: Local meshless differential quadrature collocation method for time-fractional PDEs. Discrete & Continuous Dynamical Systems-S 13.10 (2020): 2641.

H. Jafari, M. Ghorbani and S. Ghasempour: A note on exact solutions for nonlinear integral equations by a modied homotopy perturbation method. New Trends Math. Sci. 1(2) (2013), 22-26.

H. K. Jassim and D. Baleanu: A novel approach for Korteweg-de Vries equation of fractional order. Journal of Applied Computational Mechanics 5(2) (2019), 192-198.

S. P. Yan, H. Jafari and H. K. Jassim: Local Fractional Adomian Decomposition and Function Decomposition Methods for Solving Laplace Equation within Local Fractional Operators. Advances in Mathematical Physics 2014 (2014), 1-7, ID 161580.

H. K. Jassim and W. A. Shahab: Fractional variational iteration method to solve one dimensional second order hyperbolic telegraph equations. Journal of Physics: Conference Series 1032 (2018), 1-9.

H. Jafari, H. K. Jassim and J. Vahidi: Reduced Differential Transform and Variational Iteration Methods for 3D Diffusion Model in Fractal Heat Transfer within Local Fractional Operators. Thermal Science 22 (2018) S301-S307.

H. Jafari, H. K. Jassim, S. P. Moshokoa, V. M. Ariyan and F. Tchier: Reduced differential transform method for partial differential equations within local fractional derivative operators. Advances in Mechanical Engineering 8 (2016), 1-6.

D. Baleanu and H. K. Jassim: A Modication Fractional Homotopy Perturbation Method for Solving Helmholtz and Coupled Helmholtz Equations on Cantor Sets. Fractaland Fractional 3(30) (2019), 1-8.

JF Gomez-Aguilar, H Yepez-Martinez, J Torres-Jimenez, T Cordova-Fraga, RF Escobar-Jimenez and VH Olivares-Peregrino: Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel. Advances in Difference Equations 2017, 68 (2017).

D. Baleanu, H. K. Jassim and M. Al Qurashi: Solving Helmholtz Equation with Local Fractional Derivative Operators, Fractal and Fractional 3(43) (2019), 1-13.

H. K. Jassim, M. G. Mohammed and S. A. Khafif: The Approximate solutions of time-fractional Burger's and coupled time-fractional Burger's equations. International Journal of Advances in Applied Mathematics and Mechanics 6(4) (2019), 64-70.

D. Baleanu and H. K. Jassim: Approximate Solutions of the Damped Wave Equation and Dissipative Wave Equation in Fractal Strings. Fractal and Fractional 3(26) (2019), 1-12.

H. K. Jassim: Analytical Approximate Solutions for Local Fractional Wave Equations. Mathematical Methods in the Applied Sciences 43(2) (2020) 939-947.

S. Q. Wang, Y. J. Yang and H. K. Jassim: Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative. Abstract and Applied Analysis 2014 (2014), 1-7.

H. K. Jassim, C. Unlu, S. P. Moshokoa and C. M. Khalique: Local Fractional Laplace Variational Iteration Method for Solving Diffusion and Wave Equations on Cantor Sets within Local Fractional Operators. Mathematical Problems in Engineering 2015 (2015), 1-9.

Z. P. Fan, H. K. Jassim, R. K. Rainna and X. J. Yang: Adomian Decomposition Method for Three-Dimensional Diffusion Model in Fractal Heat Transfer Involving Local Fractional Derivatives. Thermal Science 19 (2015), S137-S141.

S. Xu, X. Ling, Y. Zhao and H. K. Jassim: A Novel Schedule for Solving the Two-Dimensional Diffusion in Fractal Heat Transfer. Thermal Science 19 (2015), S99-S103.

H. K. Jassim: New Approaches for Solving Fokker Planck Equation on Cantor Sets within Local Fractional Operators. Journal of Mathematics 2015 (2015), 1-8.

H. K. Jassim: The Approximate Solutions of Three-Dimensional Diffusion and Wave Equations within Local Fractional Derivative Operator. Abstract and Applied Analysis 2016 (2016), 1-5.

M. Caputo and M. Fabrizio: A new Denition of Fractional Derivative without Singular Kernel. Progress in Fractional Differentiation and Applications 1(2) (2015), 73-85.

M. Al-Refai and K. Pal: New Aspects of Caputo-Fabrizio Fractional Derivative. Progress in Fractional Differentiation and Applications 5(2) (2019), 157-166.




DOI: https://doi.org/10.22190/FUMI210327017A

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