Journal of Applied and Computational Mechanics
A Novel Approach to Compute the Numerical Solution of Variable Coefficient Fractional Burgers' Equation with Delay
Document Type : Research Paper
Authors
1 Department of Mathematics, Indian Institute of Technology Patna, Patna–801106, Bihar, India
2 Department of Mathematics, Texas A&M University-Kingville, Kingsville, TX, USA
Abstract
In this article, we come up with a novel numerical scheme based on Haar wavelet (HW) along with nonstandard finite difference (NSFD) scheme to solve time-fractional Burgers’ equation with variable diffusion coefficient and time delay. In the solution process, we discretize the fractional time derivative by NSFD formula and spatial derivative by HWs series expansion. We use the quasilinearisation process to linearize the nonlinear term. Also, the convergence of the scheme is discussed. The efficiency and correctness of the proposed scheme are assessed by L∞-error and L2 -error norms.
Keywords
- Fractional Burgers’ equation
- Nonstandard finite difference
- Haar wavelets
- Variable Coefficient
- Time delay
Main Subjects
[1] Miller, K. S., Ross, B., An introduction to the fractional calculus and fractional differential equations, Wiley, 1993.
[2] Oldham, K., Spanier, J., The fractional calculus theory and applications of differentiation and integration to arbitrary order, Elsevier, 1974.
[3] Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
[4] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and applications of fractional differential equations, Elsevier, 2006.
[5] Agarwal, P., Agarwal, R. P., Ruzhansky, M., Special Functions and Analysis of Differential Equations, CRC Press, 2020.
[6] Adomian, G., Rach, R., Nonlinear stochastic differential delay equations, Journal of Mathematical Analysis and Applications, 91(1), 1983, 94–101.
[7] Gurtin, M. E., MacCamy, R. C., On the diffusion of biological populations, Mathematical biosciences, 33(1-2), 1977, 35–49.
[8] Gurney, W. S. C. Blythe, S. P., Nisbet, R. M., Nicholson’s blowflies revisited Nature, 287(5777), 1980, 17–21.
[9] Polyanin, A. D., Sorokin, V. G., Vyazmin, A. V., Reaction-diffusion models with delay: some properties, equations, problems, and solutions,Theoretical Foundations of Chemical Engineering, 52(1), 2018, 334–348.
[10] Gyllenberg, M. Heijmans, H. J., An abstract delay-differential equation modelling size dependent cell growth and division, SIAM journal on mathematical analysis, 18(1), 1987, 74–88.
[11] Nicholson, A. J., An outline of the dynamics of animal populations, Australian journal of Zoology, 2(1), 1954, 9–65.
[12] Zhang, Q., Zhang, C., A new linearized compact multisplitting scheme for the nonlinear convection–reaction–diffusion equations with delay, Communications in Nonlinear Science and Numerical Simulation, 12(18), 2013, 3278–3288.
[13] Pao, C. V., Monotone iterations for numerical solutions of reaction-diffusion-convection equations with time delay, Numerical Methods for Partial Differential Equations: An International Journal, 14(3), 1998, 339–351.
[14] Zhang, G., Xiao, A., Zhou, J., Implicit–explicit multistep finite-element methods for nonlinear convection-diffusion-reaction equations with time delay, International Journal of Computer Mathematics, 95(12), 2018, 2496–2510.
[15] Xie, J., Deng, D., Zheng, H., A Compact Difference Scheme for One-dimensional Nonlinear Delay Reaction-diffusion Equations with Variable Coefficient, International Journal of Applied Mathematics, 47(1), 2017, 14–19.
[16] Zhang, Q., Chen, M., Xu, Y., Xu, D., Compact θ-method for the generalized delay diffusion equation, Applied Mathematics and Computation, 316, 2018, 357–369.
[17] Davis, L. C., Modifications of the optimal velocity traffic model to include delay due to driver reaction time, Physica A: Statistical Mechanics and its Applications, 319, 2003, 557–567.
[18] Kuang, Y., Delay differential equations: with applications in population dynamics, Academic press, 1993.
[19] Ran, M., He, Y., Linearized Crank–Nicolson method for solving the nonlinear fractional diffusion equation with multi-delay, International Journal of Computer Mathematics, 95(12), 2018, 2458–2470.
[20] Liang, D., Jianhong, W., Zhang, F., Modelling population growth with delayed nonlocal reaction in 2-dimensions, Mathematical Biosciences & Engineering, 2(1), 2005, 111–132.
[21] Wang, P., Keng, C., Asymptotic stability of a time-delayed diffusion system,Journal of Applied Mechanics, 30(4), 1963, 500-504 .
[22] Murray, J., D., Spatial structures in predator-prey communities—a nonlinear time delay diffusional model, Mathematical Biosciences, 31(1-2), 1976, 73–85.
[23] Ling, Z., Lin, Z., Traveling wavefront in a hematopoiesis model with time delay, Applied Mathematics Letters, 23(4), 2010, 426–431.
[24] Agarwal, R., Jain, S., Agarwal, R. P., Mathematical modeling and analysis of dynamics of cytosolic calcium ion in astrocytes using fractional calculus, Journal of Fractional Calculus and Applications, 9(2), 2018, 1–12.
[25] Lu, D., C., Hong, B., J., Tian, L., Backlund transformation and n-soliton-like solutions to the combined KdV-Burgers’ equation with variable coefficients, International Journal of Nonlinear Science, 2(1), 2006, 3–10.
[26] Cui, M., Geng, F., A computational method for solving one-dimensional variable-coefficient Burgers’ equation, Applied Mathematics and Computation, 188(2), 2007, 1389–1401.
[27] Zhao, X., Xu, Q., Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient, Applied Mathematical Modelling, 38(15-16), 2014, 3848–3859.
[28] Cui, M., Compact exponential scheme for the time fractional convection–diffusion reaction equation with variable coefficients, Journal of Computational Physics, 280, 2015, 143–163.
[29] Abdel-Gawad, H. I., Tantawy, M., Baleanu, D., Fractional KdV and Boussenisq-Burgers’ equations, reduction to PDE and stability approaches, Mathematical Methods in the Applied Sciences, 43(7), 2020, 4125–4135.
[30] Ouyang, Z., Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay, Computers & Mathematics with Applications, 61(4), 2011, 860–870.
[31] Rihan, F. A., Computational methods for delay parabolic and time-fractional partial differential equations, Numerical Methods for Partial Differential Equations, 26(6), 2010, 1556–1571.
[32] Pimenov, V. G., Hendy, A. S., A numerical solution for a class of time fractional diffusion equations with delay, International Journal of applied mathematics and computer science, 27(3), 2017, 477–488.
[33] Zhang, Q., Ran, M., Xu, D., Analysis of the compact difference scheme for the semilinear fractional partial differential equation with time delay, Applicable Analysis, 96(11), 2017, 1867–1884.
[34] Mohebbi, A., Finite difference and spectral collocation methods for the solution of semilinear time fractional convection-reaction-diffusion equations with time delay, Journal of Applied Mathematics and Computing, 61(1-2), 2019, 635–656.
[35] Sweilam, N. H., Al-Mekhlafi, S. M., Shatta, S. A., Baleanu, D., Numerical Study for Two Types Variable-Order Burgers’ Equations with Proportional Delay, Applied Numerical Mathematics, 156, 2020, 364–376.
[36] Jaradat, I., Alquran, M., Momani, S., Baleanu, D., Numerical schemes for studying biomathematics model inherited with memory-time and delay-time, Alexandria Engineering Journal, 59(5), 2020, 2969-2974.
[37] Wu, J. L., A wavelet operational method for solving fractional partial differential equations numerically, Applied Mathematics and Computation, 214(1), 2009, 31–40.
[38] Li, Y., Zhao, W., Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations,Applied Mathematics and Computation, 216(8), 2010, 2276–2285.
[39] Ur Rehman, M., Khan, R. A., Numerical solutions to initial and boundary value problems for linear fractional partial differential equations, Applied Mathematical Modelling, 37(7), 2013, 5233–5244.
[40] Yi, M., Huang, J., Wavelet operational matrix method for solving fractional differential equations with variable coefficients, Applied Mathematics and Computation, 230, 2014, 383–394.
[41] Verma, A. K., Kumar, N., Tiwari, D., Haar wavelets collocation method for a system of nonlinear singular differential equations, Engineering Computations, 2020.
[42] Heydari, M. H., Hooshmandasl, M. R., Ghaini, F. M., Cattani, C., Wavelets method for the time fractional diffusion-wave equation,Physics Letters A, 397(3), 2015, 71–76.
[43] Cattani, C., Haar wavelets based technique in evolution problems, Proceedings-Estonian Academy Of Sciences Physics Mathematics, 53(1), 2004, 45–63.
[44] Heydari, M. H., Hooshmandasl, M. R., Cattani, C., Hariharan, G., An optimization wavelet method for multi variable-order fractional differential equations, Fundamenta informaticae, 151(1-4), 2017, 255–273.
[45] Mickens, R. E., Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Numerical Methods for Partial Differential Equations: An International Journal, 23(3), 2007, 672–691.
[46] Mickens, R. E., Advances in the Applications of Nonstandard Finite Diffference Schemes, World Scientific, 2005.
[47] Zeinadini, M., Namjoo, M., A numerical method for discrete fractional–order chemostat model derived from nonstandard numerical scheme, Bulletin of the Iranian Mathematical Society, 43(5), 2017, 1165–1182.
[48] Zibaei, S., Namjoo, M., Solving fractional-order competitive Lotka-Volterra model by NSFD schemes, TWMS Journal of Applied and Engineering Mathematics, 6(2), 2016, 264.
[49] Zibaei S. and Namjoo M. A NSFD scheme for Lotka–Volterra food web model, Iranian Journal of Science and Technology (Sciences), 38(4), 2014, 399–414.
[50] Namjoo, M., Zibaei, S., Approximation of the Huxley equation with nonstandard finite-difference scheme, Iranian Journal of Numerical Analysis and Optimization, 9(1), 2019, 17–35.
[51] Verma, A. K., Kayenat, S., On the stability of Micken’s type NSFD schemes for generalized Burgers’ Fisher equation, Journal of Difference Equations and Applications, 25(12), 2019, 1706–1737.
[52] Verma, A. K., Kayenat, S., On the convergence of Mickens’ type nonstandard finite difference schemes on Lane-Emden type equations, Journal of Mathematical Chemistry, 56(6), 2018, 1667–1706.
[53] Amin, M., Abbas, M., Iqbal, M. K., Baleanu, D., Non-polynomial quintic spline for numerical solution of fourth-order time fractional partial differential equations, Advances in Difference Equations, 2019(1), 2019, 1–22.
[54] Ashpazzadeh, E., Han, B., Lakestani, M., Biorthogonal multiwavelets on the interval for numerical solutions of Burgers’ equation, Journal of Computational and Applied Mathematics, 317, 2017, 510–534.
[55] Bellman, R. E., Quasilinearization, R. K., Nonlinear Boundary Value Problems, American Elsevier Publishing Co., Inc., New York, 1965.
[56] Pervaiz, N., Aziz, I., Haar wavelet approximation for the solution of cubic nonlinear Schrodinger equations, Physica A: Statistical Mechanics and its Applications, 545, 2020, 123–738.
[57] Gu, W., Qin, H., Ran, M., Numerical investigations for a class of variable coefficient fractional Burgers’ equations with delay, IEEE Access, 7, 2019, 26892–26899.
[58] Verma, A. K., Tiwari, D., Higher resolution methods based on quasilinearization and Haar wavelets on Lane–Emden equations, International Journal of Wavelets, Multiresolution and Information Processing, 17(03), 2019, 1950005.
[59] Majak, J., Shvartsman, B. S., Kirs, M., Pohlak, M., Herranen, H., Convergence theorem for the Haar wavelet based discretization method, Composite Structures, 126, 2015, 227–232.
[2] Oldham, K., Spanier, J., The fractional calculus theory and applications of differentiation and integration to arbitrary order, Elsevier, 1974.
[3] Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
[4] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and applications of fractional differential equations, Elsevier, 2006.
[5] Agarwal, P., Agarwal, R. P., Ruzhansky, M., Special Functions and Analysis of Differential Equations, CRC Press, 2020.
[6] Adomian, G., Rach, R., Nonlinear stochastic differential delay equations, Journal of Mathematical Analysis and Applications, 91(1), 1983, 94–101.
[7] Gurtin, M. E., MacCamy, R. C., On the diffusion of biological populations, Mathematical biosciences, 33(1-2), 1977, 35–49.
[8] Gurney, W. S. C. Blythe, S. P., Nisbet, R. M., Nicholson’s blowflies revisited Nature, 287(5777), 1980, 17–21.
[9] Polyanin, A. D., Sorokin, V. G., Vyazmin, A. V., Reaction-diffusion models with delay: some properties, equations, problems, and solutions,Theoretical Foundations of Chemical Engineering, 52(1), 2018, 334–348.
[10] Gyllenberg, M. Heijmans, H. J., An abstract delay-differential equation modelling size dependent cell growth and division, SIAM journal on mathematical analysis, 18(1), 1987, 74–88.
[11] Nicholson, A. J., An outline of the dynamics of animal populations, Australian journal of Zoology, 2(1), 1954, 9–65.
[12] Zhang, Q., Zhang, C., A new linearized compact multisplitting scheme for the nonlinear convection–reaction–diffusion equations with delay, Communications in Nonlinear Science and Numerical Simulation, 12(18), 2013, 3278–3288.
[13] Pao, C. V., Monotone iterations for numerical solutions of reaction-diffusion-convection equations with time delay, Numerical Methods for Partial Differential Equations: An International Journal, 14(3), 1998, 339–351.
[14] Zhang, G., Xiao, A., Zhou, J., Implicit–explicit multistep finite-element methods for nonlinear convection-diffusion-reaction equations with time delay, International Journal of Computer Mathematics, 95(12), 2018, 2496–2510.
[15] Xie, J., Deng, D., Zheng, H., A Compact Difference Scheme for One-dimensional Nonlinear Delay Reaction-diffusion Equations with Variable Coefficient, International Journal of Applied Mathematics, 47(1), 2017, 14–19.
[16] Zhang, Q., Chen, M., Xu, Y., Xu, D., Compact θ-method for the generalized delay diffusion equation, Applied Mathematics and Computation, 316, 2018, 357–369.
[17] Davis, L. C., Modifications of the optimal velocity traffic model to include delay due to driver reaction time, Physica A: Statistical Mechanics and its Applications, 319, 2003, 557–567.
[18] Kuang, Y., Delay differential equations: with applications in population dynamics, Academic press, 1993.
[19] Ran, M., He, Y., Linearized Crank–Nicolson method for solving the nonlinear fractional diffusion equation with multi-delay, International Journal of Computer Mathematics, 95(12), 2018, 2458–2470.
[20] Liang, D., Jianhong, W., Zhang, F., Modelling population growth with delayed nonlocal reaction in 2-dimensions, Mathematical Biosciences & Engineering, 2(1), 2005, 111–132.
[21] Wang, P., Keng, C., Asymptotic stability of a time-delayed diffusion system,Journal of Applied Mechanics, 30(4), 1963, 500-504 .
[22] Murray, J., D., Spatial structures in predator-prey communities—a nonlinear time delay diffusional model, Mathematical Biosciences, 31(1-2), 1976, 73–85.
[23] Ling, Z., Lin, Z., Traveling wavefront in a hematopoiesis model with time delay, Applied Mathematics Letters, 23(4), 2010, 426–431.
[24] Agarwal, R., Jain, S., Agarwal, R. P., Mathematical modeling and analysis of dynamics of cytosolic calcium ion in astrocytes using fractional calculus, Journal of Fractional Calculus and Applications, 9(2), 2018, 1–12.
[25] Lu, D., C., Hong, B., J., Tian, L., Backlund transformation and n-soliton-like solutions to the combined KdV-Burgers’ equation with variable coefficients, International Journal of Nonlinear Science, 2(1), 2006, 3–10.
[26] Cui, M., Geng, F., A computational method for solving one-dimensional variable-coefficient Burgers’ equation, Applied Mathematics and Computation, 188(2), 2007, 1389–1401.
[27] Zhao, X., Xu, Q., Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient, Applied Mathematical Modelling, 38(15-16), 2014, 3848–3859.
[28] Cui, M., Compact exponential scheme for the time fractional convection–diffusion reaction equation with variable coefficients, Journal of Computational Physics, 280, 2015, 143–163.
[29] Abdel-Gawad, H. I., Tantawy, M., Baleanu, D., Fractional KdV and Boussenisq-Burgers’ equations, reduction to PDE and stability approaches, Mathematical Methods in the Applied Sciences, 43(7), 2020, 4125–4135.
[30] Ouyang, Z., Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay, Computers & Mathematics with Applications, 61(4), 2011, 860–870.
[31] Rihan, F. A., Computational methods for delay parabolic and time-fractional partial differential equations, Numerical Methods for Partial Differential Equations, 26(6), 2010, 1556–1571.
[32] Pimenov, V. G., Hendy, A. S., A numerical solution for a class of time fractional diffusion equations with delay, International Journal of applied mathematics and computer science, 27(3), 2017, 477–488.
[33] Zhang, Q., Ran, M., Xu, D., Analysis of the compact difference scheme for the semilinear fractional partial differential equation with time delay, Applicable Analysis, 96(11), 2017, 1867–1884.
[34] Mohebbi, A., Finite difference and spectral collocation methods for the solution of semilinear time fractional convection-reaction-diffusion equations with time delay, Journal of Applied Mathematics and Computing, 61(1-2), 2019, 635–656.
[35] Sweilam, N. H., Al-Mekhlafi, S. M., Shatta, S. A., Baleanu, D., Numerical Study for Two Types Variable-Order Burgers’ Equations with Proportional Delay, Applied Numerical Mathematics, 156, 2020, 364–376.
[36] Jaradat, I., Alquran, M., Momani, S., Baleanu, D., Numerical schemes for studying biomathematics model inherited with memory-time and delay-time, Alexandria Engineering Journal, 59(5), 2020, 2969-2974.
[37] Wu, J. L., A wavelet operational method for solving fractional partial differential equations numerically, Applied Mathematics and Computation, 214(1), 2009, 31–40.
[38] Li, Y., Zhao, W., Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations,Applied Mathematics and Computation, 216(8), 2010, 2276–2285.
[39] Ur Rehman, M., Khan, R. A., Numerical solutions to initial and boundary value problems for linear fractional partial differential equations, Applied Mathematical Modelling, 37(7), 2013, 5233–5244.
[40] Yi, M., Huang, J., Wavelet operational matrix method for solving fractional differential equations with variable coefficients, Applied Mathematics and Computation, 230, 2014, 383–394.
[41] Verma, A. K., Kumar, N., Tiwari, D., Haar wavelets collocation method for a system of nonlinear singular differential equations, Engineering Computations, 2020.
[42] Heydari, M. H., Hooshmandasl, M. R., Ghaini, F. M., Cattani, C., Wavelets method for the time fractional diffusion-wave equation,Physics Letters A, 397(3), 2015, 71–76.
[43] Cattani, C., Haar wavelets based technique in evolution problems, Proceedings-Estonian Academy Of Sciences Physics Mathematics, 53(1), 2004, 45–63.
[44] Heydari, M. H., Hooshmandasl, M. R., Cattani, C., Hariharan, G., An optimization wavelet method for multi variable-order fractional differential equations, Fundamenta informaticae, 151(1-4), 2017, 255–273.
[45] Mickens, R. E., Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Numerical Methods for Partial Differential Equations: An International Journal, 23(3), 2007, 672–691.
[46] Mickens, R. E., Advances in the Applications of Nonstandard Finite Diffference Schemes, World Scientific, 2005.
[47] Zeinadini, M., Namjoo, M., A numerical method for discrete fractional–order chemostat model derived from nonstandard numerical scheme, Bulletin of the Iranian Mathematical Society, 43(5), 2017, 1165–1182.
[48] Zibaei, S., Namjoo, M., Solving fractional-order competitive Lotka-Volterra model by NSFD schemes, TWMS Journal of Applied and Engineering Mathematics, 6(2), 2016, 264.
[49] Zibaei S. and Namjoo M. A NSFD scheme for Lotka–Volterra food web model, Iranian Journal of Science and Technology (Sciences), 38(4), 2014, 399–414.
[50] Namjoo, M., Zibaei, S., Approximation of the Huxley equation with nonstandard finite-difference scheme, Iranian Journal of Numerical Analysis and Optimization, 9(1), 2019, 17–35.
[51] Verma, A. K., Kayenat, S., On the stability of Micken’s type NSFD schemes for generalized Burgers’ Fisher equation, Journal of Difference Equations and Applications, 25(12), 2019, 1706–1737.
[52] Verma, A. K., Kayenat, S., On the convergence of Mickens’ type nonstandard finite difference schemes on Lane-Emden type equations, Journal of Mathematical Chemistry, 56(6), 2018, 1667–1706.
[53] Amin, M., Abbas, M., Iqbal, M. K., Baleanu, D., Non-polynomial quintic spline for numerical solution of fourth-order time fractional partial differential equations, Advances in Difference Equations, 2019(1), 2019, 1–22.
[54] Ashpazzadeh, E., Han, B., Lakestani, M., Biorthogonal multiwavelets on the interval for numerical solutions of Burgers’ equation, Journal of Computational and Applied Mathematics, 317, 2017, 510–534.
[55] Bellman, R. E., Quasilinearization, R. K., Nonlinear Boundary Value Problems, American Elsevier Publishing Co., Inc., New York, 1965.
[56] Pervaiz, N., Aziz, I., Haar wavelet approximation for the solution of cubic nonlinear Schrodinger equations, Physica A: Statistical Mechanics and its Applications, 545, 2020, 123–738.
[57] Gu, W., Qin, H., Ran, M., Numerical investigations for a class of variable coefficient fractional Burgers’ equations with delay, IEEE Access, 7, 2019, 26892–26899.
[58] Verma, A. K., Tiwari, D., Higher resolution methods based on quasilinearization and Haar wavelets on Lane–Emden equations, International Journal of Wavelets, Multiresolution and Information Processing, 17(03), 2019, 1950005.
[59] Majak, J., Shvartsman, B. S., Kirs, M., Pohlak, M., Herranen, H., Convergence theorem for the Haar wavelet based discretization method, Composite Structures, 126, 2015, 227–232.
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