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A HIGHER-ORDER APPROACH FOR TIME-FRACTIONAL GENERALIZED BURGERS’ EQUATION

KOMAL TANEJA

Department of Mathematics, Birla Institute of Technology and Science, Pilani, Rajasthan 333031, India

E-mail Address: komaltaneja1995@gmail.com

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KOMAL DESWAL

https://orcid.org/0000-0002-1896-4014

Department of Mathematics, Birla Institute of Technology and Science, Pilani, Rajasthan 333031, India

E-mail Address: komaldeswal94@gmail.com

Corresponding author.

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DEVENDRA KUMAR

Department of Mathematics, Birla Institute of Technology and Science, Pilani, Rajasthan 333031, India

E-mail Address: dkumar@pilani.bits-pilani.ac.in

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, and

DUMITRU BALEANU

Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, TR-06530 Ankara, Turkey

Institute of Space Science, R-077125 Magurle-Bucharest, Romania

E-mail Address: dumitru.baleanu@gmail.com

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https://doi.org/10.1142/S0218348X23500676Cited by:1 (Source: Crossref)

Abstract

A fast higher-order scheme is established for solving inhomogeneous time-fractional generalized Burgers’ equation. The time-fractional operator is taken as the modified operator with the Mittag-Leffler kernel. Through stability analysis, it has been demonstrated that the proposed numerical approach is unconditionally stable. The convergence of the numerical method is analyzed theoretically using von Neumann’s method. It has been proved that the proposed numerical method is fourth-order convergent in space and second-order convergent in time in the $L_{2}$ -norm. The scheme’s proficiency and effectiveness are examined through two numerical experiments to validate the theoretical estimates. The tabular and graphical representations of numerical results confirm the high accuracy and versatility of the scheme.

Keywords:

References

1. H. Bateman , Some recent researches in motion of fluids, Mon. Weather Rev. 43 (1915) 163–170. Crossref, Google Scholar
2. J. M. Burgers , A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1 (1948) 171–199. Crossref, Google Scholar
3. I. M. Davies, A. Truman and H. Zhao , Stochastic heat and Burgers equations and their singularities. I. Geometrical properties, J. Math. Phys. 43 (2002) 3293–3328. Crossref, Web of Science, Google Scholar
4. S. Cifani and E. R. Jakobsen , Entropy solution theory for fractional degenerate convection-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011) 413–441. Crossref, Web of Science, Google Scholar
5. J. H. He , Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg. 167 (1998) 57–68. Crossref, Web of Science, Google Scholar
6. B. Mathieu, P. Melchior, A. Oustaloup and C. Ceyral , Fractional differentiation for edge detection, Signal Process. 83 (2003) 2421–2432. Crossref, Web of Science, Google Scholar
7. M. Caputo and C. Cametti , Diffusion with memory in two cases of biological interest, J. Theor. Biol. 254 (2008) 697–703. Crossref, Web of Science, Google Scholar
8. R. Hilfer , Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000). Link, Google Scholar
9. R. L. Magin , Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng. 32 (2004) 1–104. Crossref, Google Scholar
10. J. J. Trujillo, E. Scalas, K. Diethelm and D. Baleanu , Fractional Calculus: Models and Numerical Methods (World Scientific, 2016). Google Scholar
11. J. H. He, G. M. Moatimid and M. H. Zekry , Forced nonlinear oscillator in a fractal space, Facta Univ. Ser. Mech. Eng. 20 (2022) 1–20. Web of Science, Google Scholar
12. Y. T. Zuo and H. J. Liu , Fractal approach to mechanical and electrical properties of graphene/composites, Facta Univ. Ser. Mech. Eng. 19 (2021) 271–284. Web of Science, Google Scholar
13. X. Li, Z. Liu and J. H. He , A fractal two-phase flow model for the fiber motion in a polymer filling process, Fractals 28 (2020) 2050093. Link, Web of Science, Google Scholar
14. D. D. Fei, F. J. Liu, Q. N. Cui and J. H. He , Fractal approach to heat transfer in silkworm cocoon hierarchy, Therm. Sci. 17 (2013) 1546–1548. Crossref, Web of Science, Google Scholar
15. F. Liu, T. Zhang, C. H. He and D. Tian , Thermal oscillation arising in a heat shock of a porous hierarchy and its application, Facta Univ. Ser. Mech. Eng. 20 (2022) 633–645. Web of Science, Google Scholar
16. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo , Theory and Applications of Fractional Differential Equations, Vol. 204 (Elsevier, 2006). Google Scholar
17. K. S. Miller and B. Ross , An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993). Google Scholar
18. I. Podlubny , Fractional differential equations, Math. Sci. Eng. 198 (1999) 41–119. Google Scholar
19. S. G. Samko, A. A. Kilbas and O. I. Marichev , Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, Switzerland, 1993). Google Scholar
20. E. F. D. Goufo , Application of the Caputo–Fabrizio fractional derivative without singular kernel to Korteweg–de Vries–Burgers equation, Math. Model. Anal. 21 (2016) 188–198. Crossref, Web of Science, Google Scholar
21. M. Al-Refai and D. Baleanu , On an extension of the operator with Mittag-Leffler kernel, Fractals 30 (2022) 2240129. Link, Web of Science, Google Scholar
22. W. A. Woyczyński , Lévy processes in the physical sciences, in Lévy Processes (Birkhäuser, Boston, MA, 2001), pp. 241–266. Crossref, Google Scholar
23. T. Funaki and W. A. Woyczyński , Interacting particle approximation for fractal Burgers equation, in Stochastic Processes and Related Topics (Birkhäuser, Boston, MA, 1998), pp. 141–166. Crossref, Google Scholar
24. P. Clavin , Instabilities and nonlinear patterns of overdriven detonations in gases, in Nonlinear PDE’s in Condensed Matter and Reactive Flows, eds. H. Berestycki and Y. Pomeau , NATO Science Series, Vol. 569 (Springer, 2002), pp. 49–97. Crossref, Google Scholar
25. N. Sugimoto , Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves, J. Fluid Mech. 225 (1991) 631–653. Crossref, Web of Science, Google Scholar
26. J. J. Keller , Propogation of simple non-linear waves in gas filled tubes with friction, Z. Angew. Math. Phys. 32 (1981) 170–181. Crossref, Web of Science, Google Scholar
27. X. Li, D. Wang and T. Saeed , Multi-scale numerical approach to the polymer filling process in the weld line region, Facta Univ. Ser. Mech. Eng. 20 (2022) 363–380. Crossref, Web of Science, Google Scholar
28. Q. Wang , Numerical solutions for fractional KdV–Burgers equation by Adomian decomposition method, Appl. Math. Comput. 182 (2006) 1048–1055. Web of Science, Google Scholar
29. M. Dehghan, J. Manafian and A. Saadatmandi , Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differential Equations 26 (2010) 448–479. Crossref, Web of Science, Google Scholar
30. L. Song and H. Q. Zhang , Application of homotopy analysis method to fractional KdV–Burgers–Kuramoto equation, Phys. Lett. A 367 (2007) 88–94. Crossref, Web of Science, Google Scholar
31. T. S. El-Danaf and A. R. Hadhoud , Parametric spline functions for the solution of the one time fractional Burgers’ equation, Appl. Math. Model. 36 (2012) 4557–4564. Crossref, Web of Science, Google Scholar
32. M. Inc , The approximate and exact solutions of the space-and time-fractional Burgers equations with initial conditions by variational iteration method, J. Math. Anal. Appl. 345 (2008) 476–484. Crossref, Web of Science, Google Scholar
33. D. Li, C. Zhang and M. Ran , A linear finite difference scheme for generalized time fractional Burgers equation, Appl. Math. Model. 40 (2016) 6069–6081. Crossref, Web of Science, Google Scholar
34. S. Yadav and R. K. Pandey , Numerical approximation of fractional Burgers equation with Atangana–Baleanu derivative in Caputo sense, Chaos Solitons Fractals 133 (2020) 109630. Crossref, Web of Science, Google Scholar
35. K. L. Wang and C. H. He , A remark on Wang’s fractal variational principle, Fractals 27 (2019) 1950134. Link, Web of Science, Google Scholar
36. K. S. Patel and M. Mehra , Fourth order compact scheme for space fractional advection-diffusion reaction equations with variable coefficients, J. Comput. Appl. Math. 380 (2020) 112963. Crossref, Web of Science, Google Scholar
37. P. Roul and V. M. K. P. Goura , A compact finite difference scheme for fractional black-scholes option pricing model, Appl. Numer. Math. 166 (2021) 40–60. Crossref, Web of Science, Google Scholar
38. A. Atangana and D. Baleanu , New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci. 20 (2016) 763–769. Crossref, Web of Science, Google Scholar
39. M. Caputo and M. Fabrizio , A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2015) 73–85. Google Scholar
40. N. Anjum and Q. T. Ain , Application of He’s fractional derivative and fractional complex transform for time fractional Camassa–Holm equation, Therm. Sci. 24 (2020) 3023–3030. Crossref, Web of Science, Google Scholar
41. J. H. He and Y. O. El-Dib , A tutorial introduction to the two-scale fractal calculus and its application to the fractal Zhiber–Shabat oscillator, Fractals 29 (2021) 2150268. Link, Web of Science, Google Scholar
42. H. Sun and Z. Sun , On two linearized difference schemes for Burgers’ equation, Int. J. Comput. Math. 92 (2015) 1160–1179. Crossref, Web of Science, Google Scholar
43. R. E. Bellman and R. Kalaba , Quasilinearization and Nonlinear Boundary-Value Problems (American Elsevier Publishing, New York, 1965). Google Scholar
44. R. Chawla, K. Deswal and D. Kumar , A novel finite difference based numerical approach for modified Atangana–Baleanu Caputo derivative, AIMS Math. 7 (2022) 17252–17268. Crossref, Web of Science, Google Scholar