Cubic B-spline collocation method for solving time fractional gas dynamics equation. (English) Zbl 1383.76367

Summary: In the present manuscript, a cubic B-spline finite element collocation method has been used to obtain numerical solutions of the nonlinear time fractional gas dynamics equation. While the Caputo form is used for the time fractional derivative appearing in the equation, the \(L1\) discretization formula is applied to the equation in terms of time. It has been seen that the results of the present study are in agreement with the those of exact solution. Therefore, the present method can be used as an alternative and efficient one to find out the numerical solutions of both linear and nonlinear fractional differential equations available in the literature. In order to control the accuracy and efficiency of the present method, the error norms \(L_2\) and \(L_\infty\) have been calculated. It is evident that the newly obtained numerical solutions by the present method can be computed easily with the implementation and effectiveness of the approach used in the article.


76M22 Spectral methods applied to problems in fluid mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics (general theory)
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[1] [1] K. B. Oldham and J. Spanier, The fractional calculus, Academic, New York, 1974. · Zbl 0292.26011
[2] J. Singh, D. Kumar and A. Kilicman, Homotopy perturbation method for frac- tional gas dynamics equation using Sumudu transform, Abstr. Appl. Anal. (2013), http://dx.doi.org/10.1155/2013/934060, Article ID 934060, 8 pp. · Zbl 1262.76082
[3] L. Debnath, Fractional integral and fractional differential equations in uid mechan- ics, Fract. Calc. Appl. Anal. 6 (2003) 119-155.
[4] S. Monami and Z. Odibat, Analytical approach to linear fractional partial differential equations arising in uid mechanics, Phys. Lett. A 355 (2006) 271-279. · Zbl 1378.76084
[5] D.L. Logan, A first course in the finite element method (Fourth Edition), Thomson, 2007.
[6] A. A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006. · Zbl 1092.45003
[7] S. S. Ray, Exact solutions for time-fractional diffusion-wave equations by decomposi- tion method, Phys. Scr. 75 (2007) 53-61. · Zbl 1197.35147
[8] O.P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlin. Dynam. 29 (2002) 145-155. · Zbl 1009.65085
[9] H. Jafari and S.Momani, Solving fractional diffusion and waves equations by modifiy- ing homotopy perturbation method, Phys. Lett. A 370 (2007) 388-396. · Zbl 1209.65111
[10] A. Esen, Y. Ucar, N. Yagmurlu and O. Tasbozan, A Galerkin finite element method to solve fractional diffusion and fractional diffusion-wave equations, Math. Model. and Anal. 18 (2013) 260-273. · Zbl 1266.65026
[11] A. Esen, O. Tasbozan, Y. Ucar and N.M. Yagmurlu, A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations, Tbilisi Math- ematical Journal 8 (2015) 181-193. · Zbl 1342.80013
[12] A. Mohebbi, A. Mostafa and M. Dehghan, The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrodinger equation arising in quantum mechanics, Eng. Anal. with Bound. Elem. 37 (2013) 475-485. · Zbl 1352.65397
[13] V. R. Hosseini, W. Chen and Z. Avazzade, Numerical solution of fractional telegraph equation by using radial basis functions, Eng. Anal. with Bound. Elem. 38 (2014) 31-39. · Zbl 1287.65085
[14] L.Wei, H. Dai, D. Zhang and Z. Si, Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation, Calcolo 51 (2014) 175-192. · Zbl 1311.35331
[15] J. Q. Murillo and S.B. Yuste, An explicit difference method for solving fractional diffusion and diffusion-wave equations in the Caputo form, J. Comput. Nonlinear Dynam. 6 (2011) 021014.
[16] S. Das and R. Kumar, Approximate analytical solutions of fractional gas dynamic equations, Appl. Math. and Comput. 217 (2011) 9905-9915. · Zbl 1387.35606
[17] T-P. Liu, Nonlinear waves in mechanics and gas dynamics, Defense Technical Infor- mation Center Accession Number: ADA 238340 (1990).
[18] M. Rasulov and T. Karaguler, Finite difference schemes for solving system equations of gas dynamic ina class of discontinuous functions, Appl. Math. and Comput. 143 (2003) 45-164. · Zbl 1109.76364
[19] I. Podlubny, Fractional differential dquations, Academic Press, San Diego, 1999. · Zbl 0924.34008
[20] P. M. Prenter, Splines and variasyonel methods, New York, John Wiley, 1975. · Zbl 0344.65044
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