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Numerical approximation of inhomogeneous time fractional Burgers–Huxley equation with B-spline functions and Caputo derivative

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Abstract

A prototype model used to explain the relationship between mechanisms of reaction, convection effects, and transportation of diffusion is the generalized Burgers–Huxley equation. This study presents numerical solution of non-linear inhomogeneous time fractional Burgers–Huxley equation using cubic B-spline collocation method. For this purpose, Caputo derivative is used for the temporal derivative which is discretized by L1 formula and spatial derivative is interpolated with the help of B-spline basis functions, so the dependent variable is continuous throughout the solution range. The validity of the proposed scheme is examined by solving four test problems with different initial-boundary conditions. The algorithm for the execution of scheme is also presented. The effect of non-integer parameter \(\alpha \) and time on dependent variable is studied. Moreover, convergence and stability of the proposed scheme is analyzed, and proved that scheme is unconditionally stable. The accuracy is checked by error norms. Based on obtained results we can say that the proposed scheme is a good addition to the existing schemes for such real-life problems.

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References

  1. Solomon TH, Weeks ER, Swinney HL (1993) Observation of anomalous diffusion and Levy flights in a two-dimensional rotating flow. Phys Rev Lett 71(24):3975–3978

    Article  Google Scholar 

  2. Bhalekar S, Daftardar-Gejji V, Baleanu D, Magin R (2011) Fractional Bloch equation with delay. Comput Math Appl 61(5):1355–1365

    Article  MathSciNet  Google Scholar 

  3. Magin RL (2006) Fractional calculus in bioengineering. Begell House, Connecticut

    Google Scholar 

  4. Magin RL, Abdullah O, Baleanu D, Zhou XJ (2008) Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation. J Magn Reson 190(2):255–270

    Article  Google Scholar 

  5. Satsuma J (1987) Topics in soliton theory and exactly solvable nonlinear equations. In: Ablowitz M, Fuchssteiner B, Kruskal M (eds) World Scientific, Singapore, pp 255–262

  6. Inan B (2017) Finite difference methods for the generalized Huxley and Burgers–Huxley equations. Kuwait J Sci 44(3)

  7. Freihet AA, Zuriqat M (2019) Analytical solution of fractional Burgers–Huxley equations via residual power series method. Lobachevskii J Math 40(2):174–182

    Article  MathSciNet  Google Scholar 

  8. Wang XY, Zhu ZS, Lu YK (1990) Solitary wave solutions of the generalized Burgers–Huxley equation. J Phys A Math Gen 23:271–274

    Article  Google Scholar 

  9. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544

    Article  Google Scholar 

  10. Fitzhugh R (1969) Mathematical models of excitation and propagation in nerve. In: Schwan HP (ed) Biological engineering. McGraw-Hill, New York, pp 1–85

    Google Scholar 

  11. Lakestani M, Dehghan M (2013) Four techniques based on the B-spline expansion and the collocation approach for the numerical solution of the Lane–Emden equation. Math Methods Appl Sci 36(16):2243–2253

    Article  MathSciNet  Google Scholar 

  12. Ali KK, Raslan KR, El-Danaf TS (2015) Non-polynomial spline method for solving coupled burgers equations. Comput Methods Differ Equ 3(3):218–230

    MathSciNet  MATH  Google Scholar 

  13. Roul P, Goura VP (2020) A high order numerical method and its convergence for time-fractional fourth order partial differential equations. Appl Math Comput 366:124727

    MathSciNet  MATH  Google Scholar 

  14. Manafian J, Lakestani M (2017) A new analytical approach to solve some of the fractional-order partial differential equations. Indian J Phys 91(3):243–258

    Article  Google Scholar 

  15. Alinia N, Zarebnia M (2019) A numerical algorithm based on a new kind of tension B-spline function for solving Burgers–Huxley equation. Numer Algorithms 1–22

  16. Zibaei S, Zeinadini M, Namjoo M (2016) Numerical solutions of Burgers–Huxley equation by exact finite difference and NSFD schemes. J Differ Equ Appl 22(8):1098–1113

    Article  MathSciNet  Google Scholar 

  17. Zhou S, Cheng X (2011) A linearly semi-implicit compact scheme for the Burgers–Huxley equation. Int J Comput Math 88:795–804

    Article  MathSciNet  Google Scholar 

  18. Dehghan M, Saray BN, Lakestani M (2012) Three methods based on the interpolation scaling functions and the mixed collocation finite difference schemes for the numerical solution of the nonlinear generalized Burgers–Huxley equation. Math Comput Model 55:1129–1142

    Article  MathSciNet  Google Scholar 

  19. Gupta V, Kadalbajoo MK (2011) A singular perturbation approach to solve Burgers–Huxley equation via monotone finite difference scheme on layer adaptive mesh. Commun Nonlinear Sci Numer Simul 16:1825–1844

    Article  MathSciNet  Google Scholar 

  20. Mohanty RK, Dai W, Liu D (2015) Operator compact method of accuracy two in time and four in space for the solution of time dependent Burgers-Huxley equation. Numer Algorithms 70(3):591–605

    Article  MathSciNet  Google Scholar 

  21. Bazgir H, Ghazanfari B (2019) Spectral solution of fractional fourth order partial integro-differential equations. Comput Methods Differ Equ 7(2):289–301

    MathSciNet  MATH  Google Scholar 

  22. Aghili A (2017) Solution to time fractional generalized KdV of order 2q+ 1 and system of space fractional PDEs. Comput Methods Differ Equ 5(3):246–255

    MathSciNet  MATH  Google Scholar 

  23. Ashyralyev A, Hicdurmaz BETÜL (2018) A stable second order of accuracy difference scheme for a fractional Schrodinger differential equation

  24. Yusuf A, Aliyu AI, Baleanu D (2018) Lie symmetry analysis and explicit solutions for the time fractional generalized Burgers–Huxley equation. Opt Quantum Electron 50(2):94

    Article  Google Scholar 

  25. Mustafa Inc, Partohaghighi M, Akinlar MA, Agarwal P, Chu Y-M (2020) New solutions of fractional-order Burger–Huxley equation. Results Phys 18:103290

  26. Ismail HNA, Raslan K, Rabboh AAA (2004) Adomian decomposition method for Burgers-Huxley and Burgers-Fisher equations. Appl Math Comput 159:291–301

    MathSciNet  MATH  Google Scholar 

  27. Deng X (2008) Traveling wave solutions for the generalized Burgers–Huxley equation. Appl Math Comput 204:733–737

    MathSciNet  MATH  Google Scholar 

  28. Majeed A, Kamran M, Iqbal MK, Baleanu D (2020) Solving time fractional Burgers and Fishers equations using cubic B spline approximation method. Adv Differ Equ 2020(1):1–15

    Article  MathSciNet  Google Scholar 

  29. Koch PE, Lyche T (1991) Construction of exponential tension B-splines of arbitrary order. Academic, New York, pp 255–258

    MATH  Google Scholar 

  30. Lu YG, Wang GZ, Yang XN (2002) Uniform hyperbolic polynomial B-spline curves. Comput Aided Geom Des 19:379–393

    Article  MathSciNet  Google Scholar 

  31. Mainar E, Pena JM (2002) A basis of C-Bezier splines with optimal properties. Comput Aided Geom Des 19:161–175

    Article  Google Scholar 

  32. Chen QY, Wang GZ (2003) A class of Bezier-like curves. Comput Aided Geom Des 20:29–39

    Article  Google Scholar 

  33. Majeed A, Abbas M, Miura KT, Kamran M, Nazir T (2020) Surface modeling from 2D contours with an application to craniofacial fracture construction. Mathematics 8(8):1246

    Article  Google Scholar 

  34. Podlubny I (1999) Fractional differential equations. Academic, San Diego

    MATH  Google Scholar 

  35. Majeed A, Kamran M, Rafique M (2020) An approximation to the solution of time fractional modified Burgers equation using extended cubic B-spline method. Comput Appl Math 39(4):1–21

    Article  MathSciNet  Google Scholar 

  36. Rubin SG, Graves RA (1975) Cubic spline approximation for problems in fluid mechanics. Nasa TR R-436, Washington DC

    Google Scholar 

  37. Dag I, Irk D, Saka B (2005) A numerical solution of Burgers equation using cubic B-splines. Appl Math Comput 163:199–211

    MathSciNet  MATH  Google Scholar 

  38. El-Danaf TS, Hadhoud AR (2012) Parametric spline functions for the solution of the one time fractional burger equation. Appl Math Model 36:4557–4564

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Division of Science and Technology, Department of Mathematics, University of Education Lahore, Lahore, Pakistan

    Abdul Majeed, Mohsin Kamran & Noreen Asghar

  2. Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, 06530, Ankara, Turkey

    Dumitru Baleanu

  3. Institute of Space Sciences, Magurele-Bucharest, Romania

    Dumitru Baleanu

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  1. Abdul Majeed
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  2. Mohsin Kamran
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  3. Noreen Asghar
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  4. Dumitru Baleanu
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Correspondence to Abdul Majeed.

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Majeed, A., Kamran, M., Asghar, N. et al. Numerical approximation of inhomogeneous time fractional Burgers–Huxley equation with B-spline functions and Caputo derivative. Engineering with Computers 38 (Suppl 2), 885–900 (2022). https://doi.org/10.1007/s00366-020-01261-y

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  • DOI: https://doi.org/10.1007/s00366-020-01261-y

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