Skip to main content
SpringerLink
Log in

An approximation to the solution of time fractional modified Burgers’ equation using extended cubic B-spline method

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

This paper aims to investigate numerical solution of time fractional modified Burgers’ equation via Caputo fractional derivative. Extended cubic B-spline collocation scheme which reduces the nonlinear equation to a system of linear equation in the matrix form has been used for this investigation. The nonlinear part in fractional partial differential equation has been linearized by modified form of the existing method. The validity of proposed scheme has been examined on three test problems and effect of viscosity \(\nu \) and \(\alpha \ \epsilon \ [0, 1]\) variation displayed in 2D and 3D graphics. Moreover, the working of proposed scheme has also been explained through algorithm and stability of proposed scheme has been analyzed by von Neumann scheme and has proved to be unconditionally stable. To quantify the accuracy of suggested scheme, error norms have been computed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Aguilar J-P, Korbel J, Luchko Y (2019) Applications of the fractional diffusion equation to option pricing and risk calculations. Mathematics 7(9):796

    Google Scholar 

  • Aksan EN (2006) Quadratic B-spline finite element method for numerical solution of the Burgers’ equation. Appl Math Comput 174:884–96

    MathSciNet  MATH  Google Scholar 

  • Almeida R, Torres DFM (2011) Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun Nonlinear Sci Numer Simul 16(3):1490–1500

    MathSciNet  MATH  Google Scholar 

  • Asgari Z, Hosseini SM (2018) Efficient numerical schemes for the solution of generalized time fractional Burgers type equations. Numer Algorithms 77(3):763–792

    MathSciNet  MATH  Google Scholar 

  • Bonkile MP, Awasthi A, Lakshmi C, Mukundan V, Aswin VS (2018) A systematic literature review of Burgers’ equation with recent advances. Pramana 90(6):69

    Google Scholar 

  • Bulut H, Baskonus HM, Pandir Y (2013) The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation. In: Abstract and applied analysis, vol 2013. Hindawi

  • Burgers JM (1995) Mathematical example illustrating relations occurring in the theory of turbulent fluid motion. Selected Papers of JM Burgers. Springer, Dordrecht, pp 281–334

  • Çelik C, Duman M (2012) Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J Comput Phys 231(4):1743–1750

    MathSciNet  MATH  Google Scholar 

  • Chen W, Ye L, Sun H (2010) Fractional diffusion equations by the Kansa method. Comput Math Appl 59(5):1614–1620

    MathSciNet  MATH  Google Scholar 

  • Dag I, Irk D, Saka B (2005) A numerical solution of Burgers equation using cubic B-splines. Appl Math Comput 163(1):199–211

    MathSciNet  MATH  Google Scholar 

  • Das S (2009) Analytical solution of a fractional diffusion equation by variational iteration method. Comput Math Appl 57(3):483–487

    MathSciNet  MATH  Google Scholar 

  • Djordjevica VD, Atanackovic TM (2008) Similarity solutions to the nonlinear heat conduction and Burgers/Korteweg de Vries fractional equations. J Comput Appl Math 222(2):701–714

    MathSciNet  Google Scholar 

  • Dogan A, Galerkin A (2004) Finite element approach to Burgers’ equation. Appl Math Comput 157:331–346

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Esen A, Tasbozan O (2015) Numerical solution of time fractional Burgers equation. Acta Univ Sapientiae Math 7(2):167–185

    MathSciNet  MATH  Google Scholar 

  • Esipov SE (1995) Coupled Burgers’ equations: a model of polydispersive sedimentation. Phys Rev 52:3711–18

    Google Scholar 

  • Fletcher CAJ (1983) Generating exact solutions of the two-dimensional Burgers’ equation. Int J Numer Methods Fluids 3:213–216

    MATH  Google Scholar 

  • Gorenflo R, Luchko Y, Yamamoto M (2015) Time-fractional diffusion equation in the fractional Sobolev spaces. Fract Calc Appl Anal 18(3):799–820

    MathSciNet  MATH  Google Scholar 

  • Gorgulu MZ, Dag I, Irk D (2016) Galerkin method for the numerical solution of the Burgers’ equation by using exponential B-splines. arXiv preprint. arXiv:1604.04266

  • Han XL, Liu SJ (2003) J Comput Aided Des Comput Graph 15:576–578

    Google Scholar 

  • Harris PA, Garra R (2013) Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace method. arXiv preprint. arXiv:1306.1942

  • Hassan QMU, Mohyud-Din ST (2013) Exp-function method using modified Riemann–Liouville derivative for Burger’s equations of fractional-order. QSci Connect 2013(1):19

    Google Scholar 

  • Hassani H, Naraghirad E (2019) A new computational method based on optimization scheme for solving variable-order time fractional Burgers’ equation. Math Comput Simul 162:1–17

    MathSciNet  Google Scholar 

  • Hassanien IA, Salama AA, Hosham HA (2005) Fourth-order finite difference method for solving Burgers’ equation. Appl Math Comput 170:781–800

    MathSciNet  MATH  Google Scholar 

  • Huda MA, Akbar MA, Shanta SS (2018) The new types of wave solutions of the Burger’s equation and the Benjamin–Bona–Mahony equation. J Ocean Eng Sci 3(1):1–10

    Google Scholar 

  • Inc M (2008) The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method. Math Anal Appl 345:476484

    MathSciNet  Google Scholar 

  • Jiang Z, Wang R (2010) An improved numerical solution of Burgers’ equation by cubic B-spline quasi-interpolation. J Inf Comput Sci 7(5):1013–21

    Google Scholar 

  • Kadalbajoo MK, Sharma KK, Awasthi A (2005) A parameter-uniform implicit difference scheme for solving time-dependent Burgers’ equation. Appl Math Comput 170:1365–93

    MathSciNet  MATH  Google Scholar 

  • Kahn NA, Ara A, Mahmood A (2012) Numerical solutions of time-fractional Burgers equations. Int J Numer Methods Heat Fluid Flow

  • Karakoç SBG, Bashan A, Geyikli T (2014) Two different methods for numerical solution of the modified Burgers’ equation. Sci World J 2014

  • Kurt A, Çenesiz Y, Tasbozan O (2015) On the solution of Burgers’ equation with the new fractional derivative. Open Phys 13(1)

  • Langlands TAM (2006) Solution of a modified fractional diffusion equation. Phys A Stat Mech Appl 367:136–144

    MathSciNet  Google Scholar 

  • Lin Y, Xu C (2007) Finite difference/spectral approximations for the time-fractional diffusion equation. J Comput Phys 225(2):1533–1552

    MathSciNet  MATH  Google Scholar 

  • Lin R, Liu F, Anh V, Turner I (2009) Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl Math Comput 212(2):435–445

    MathSciNet  MATH  Google Scholar 

  • Liu C-S (2010) The fictitious time integration method to solve the space- and time-fractional Burgers equations. Comput Mater Contin 15(3):221–240

    Google Scholar 

  • Liu JC, Hou GL (2011) Numerical solutions of the space- and time-fractional coupled burgers equations by generalized differential transform method. Appl Math Comput 217:70017008

    MathSciNet  Google Scholar 

  • Majeed A, Piah ARM, Yahya ZR, Abdullah JY, Rafique M (2018) Construction of occipital bone fracture using B-spline curves. Comput Appl Math 37(3):2877–2896

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  • Mao Z, Karniadakis GE (2017) Fractional Burgers equation with nonlinear non-locality: spectral vanishing viscosity and local discontinuous Galerkin methods. J Comput Phys 336:143–163

    MathSciNet  MATH  Google Scholar 

  • Mittal RC, Jain RK (2012) Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method. Appl Math Comput 218:7839–7855

    MathSciNet  MATH  Google Scholar 

  • Mohamed NA (2019) Solving one-and two-dimensional unsteady Burgers’ equation using fully implicit finite difference schemes. Arab J Basic Appl Sci 26(1):254–268

    Google Scholar 

  • Momani S (2006) Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. Chaos Solitons Fractals 28:930937

    MathSciNet  Google Scholar 

  • Ozis T, Esen A, Kutluay S (2005) Numerical solution of Burgers’ equation by quadratic B-spline finite elements. Appl Math Comput 165:237–49

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Rawashdeh MS (2017) A reliable method for the space-time fractional Burgers and time-fractional Cahn–Allen equations via the FRDTM. Adv Differ Equ 2017(1):99

    MathSciNet  MATH  Google Scholar 

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

  • Ruiz-Medina MD, Angulo JM, Anh VV (2001) Scaling limit solution of a fractional burgers equation. Stoch Process Appl 93:285300

    MathSciNet  MATH  Google Scholar 

  • Saad KM, Al-Sharif EHF (2017) Analytical study for time and time-space fractional Burgers’ equation. Adv Differ Equ 2017(1):300

    MathSciNet  MATH  Google Scholar 

  • Saka B, Dag I (2007) Quartic B-spline collocation method to the numerical solutions of the Burgers’ equation. Chaos Solitons Fractals 32:1125–37

    MathSciNet  MATH  Google Scholar 

  • Siddiqi SS, Arshed S (2015) Numerical solution of time-fractional fourth-order partial differential equations. Int J Comput Math 92(7):1496–1518

    MathSciNet  MATH  Google Scholar 

  • Song L, Zhang HQ (2007) Application of homotopy analysis method to fractional KdV–Burgers–Kuramoto equation. Phys Lett A 367(1–2):8894

    MathSciNet  Google Scholar 

  • Srivastava VK, Awasthi MK, Singh S (2013) An implicit logarithmic finite difference technique for two dimensional coupled viscous Burgers’ equation. AIP Adv 3:122105

    Google Scholar 

  • Sugimoto N (1989) Generalized Burgers equation and fractional calculus. In: Nonlinear wave motion, pp 162–179

  • Sungnul S, Jitsom B, Punpocha M (2018) Numerical solutions of the modified Burger’s equation using FTCS implicit scheme. IAENG Int J Appl Math 48(1):53–61

    MathSciNet  Google Scholar 

  • Syam MI, Obayda DA, Alshamsi W, Al-Wahashi N, Alshehhi M (2019) Generalized solutions of the fractional Burger’s equation. Results Phys 15:102525

    Google Scholar 

  • Tadjeran C, Meerschaert MM, Scheffler H-P (2006) A second-order accurate numerical approximation for the fractional diffusion equation. J Comput Phys 213(1):205–213

    MathSciNet  MATH  Google Scholar 

  • Tamsir M, Dhiman N, Srivastava VK (2016) Extended modified cubic B-spline algorithm for nonlinear Burgers’ equation. Beni-Suef Univ J Basic Appl Sci 5(3):244–254

    Google Scholar 

  • Tasbozan O, Esen A, Yagmurlu NM, Ucar Y (2013) A numerical solution to fractional diffusion equation for force-free case. In: Abstract and applied analysis, vol 2013. Hindawi

  • Tasbozan O, Esen A, Ucar Y, Yagmurlu NM (2015) A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations. Tbilisi Math J 8:181–193

    MathSciNet  MATH  Google Scholar 

  • Trigeassou J-C, Maamri N (2019) Analysis, modeling and stability of fractional order differential systems 2: the infinite state approach. Wiley, New York

    MATH  Google Scholar 

  • Trigeassou JC, Maamri N, Oustaloup A (2013) The infinite state approach: origin and necessity. Comput Math Appl 66(5):892–907

    MathSciNet  MATH  Google Scholar 

  • Ucar Y, Yagmurlu NM, Tasbozan O (2017) Numerical solutions of the modified Burgers’ equation by finite difference methods. J Appl Math Stat Inform 13(1):19–30

    MathSciNet  MATH  Google Scholar 

  • Wang Q (2006) Numerical solutions for KdV-Burgers equation by Adomian decomposition method. Appl Math Comput 182(2):10481055

    MathSciNet  Google Scholar 

  • Wang G-W, Kara AH (2018) Group analysis, fractional explicit solutions and conservation laws of time fractional generalized burgers equation. Commun Theor Phys 69(1):5

    MathSciNet  MATH  Google Scholar 

  • Wang Z, Vong S (2014) Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J Comput Phys 277:1–15

    MathSciNet  MATH  Google Scholar 

  • Xie SS, Heo S, Kim S, Woo G, Yi S (2008) Numerical solution of one-dimensional Burgers’ equation using reproducing kernel function. J Comput Appl Math 214:417–34

    MathSciNet  MATH  Google Scholar 

  • Xue G, Zhang L (2013) A new finite difference scheme for generalized Rosenau-Burgers equation. Appl Math Comput 222:490496

    MathSciNet  Google Scholar 

  • Yang X-J (2016) Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems. arXiv preprint. arXiv:1612.03202

  • Yang X-J (2017a) New general fractional-order rheological models with kernels of Mittag–Leffler functions. Rom Rep Phys 69(4):118

    Google Scholar 

  • Yang (2017b) New rheological problems involving general fractional derivatives with nonsingular power-law kernels

  • Yang XJ (2019) General fractional derivatives: theory, methods and applications. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Yang X-J, Gao F (2017) A new technology for solving diffusion and heat equations. Therm Sci 21(1 Part A):133–140

  • Yang X-J, Machado JAT (2017) A new fractional operator of variable order: application in the description of anomalous diffusion. Phys A Stat Mech Appl 481:276–283

    MathSciNet  Google Scholar 

  • Yang X-J, Gao F, Srivastava HM (2017) New rheological models within local fractional derivative. Rom Rep Phys 69(3):113

    Google Scholar 

  • Yang X-J, Gao F, Ju Y, Zhou H-W (2018) Fundamental solutions of the general fractional-order diffusion equations. Math Methods Appl Sci 41(18):9312–9320

    MathSciNet  MATH  Google Scholar 

  • Yang X-J, Feng Y-Y, Cattani C, Inc M (2019) Fundamental solutions of anomalous diffusion equations with the decay exponential kernel. Math Methods Appl Sci 42(11):4054–4060

    MathSciNet  MATH  Google Scholar 

  • Yang X-J, Gao F, Ju Y (2020) General fractional derivatives with applications in viscoelasticity. Academic Press, Cambridge

    MATH  Google Scholar 

  • Yaseen M, Abbas M (2020) An efficient computational technique based on cubic trigonometric B-splines for time fractional Burgers’ equation. Int J Comput Math 97(3):725–738

    MathSciNet  Google Scholar 

  • Yokus A, Kaya D (2017) Numerical and exact solutions for time fractional Burgers’ equation. J Nonlinear Sci Appl 10(7):3419–3428

    MathSciNet  MATH  Google Scholar 

  • Yuste SB (2006) Weighted average finite difference methods for fractional diffusion equations. J Comput Phys 216(1):264–274

    MathSciNet  MATH  Google Scholar 

  • Zafar ZU, Ahmed MO (2013) Solution of Burger’s equation with the help of Laplace decomposition method. Pak J Eng Appl Sci 12(1):39–42

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Abdul Majeed & Mohsin Kamran

  2. Faculty of Islamabad Business School, Islamabad, Pakistan

    Muhammad Rafique

Authors
  1. Abdul Majeed
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohsin Kamran
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Muhammad Rafique
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abdul Majeed.

Additional information

Communicated by José Tenreiro Machado.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Majeed, A., Kamran, M. & Rafique, M. An approximation to the solution of time fractional modified Burgers’ equation using extended cubic B-spline method. Comp. Appl. Math. 39, 257 (2020). https://doi.org/10.1007/s40314-020-01307-3

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s40314-020-01307-3

Keywords

Mathematics Subject Classification

Search

Navigation