A numerical scheme to simulate the distributed-order time 2D Benjamin Bona Mahony Burgers equation with fractional-order space
Abstract
In this study, a new class of the Benjamin Bona Mahony Burgers equation is introduced, which considers the distributedorder in the time variable and fractional-order space in the Caputo form in the 2D case. The 2D-modified orthonormal normalized shifted Ultraspherical polynomials are derived from 1Dmodified orthonormal normalized shifted Ultraspherical polynomials and 2D-modified orthonormal normalized shifted Ultraspherical polynomials and the orthonormal normalized shifted Ultraspherical polynomials are applied to approximate of the space and time variables, respectively. Moreover, the convergence analysis of these basis functions is investigated. Due to the time variable being in the distributed-order mode and the space variable being in the fractional-order case, to apply the desired numerical algorithm for this type of equation, operational matrices of ordinary, fractional and distributed-order derivatives are computed. In the proposed method, once the unknown function is approximated using the mentioned polynomial, the matrix form of the residual function is derived and then a system of algebraic equations is adopted by applying the collocation approach. An approximate solution is extracted for the original problem by solving constructed equation system. Several examples are examined to demonstrate the accuracy and capability of the method.
Keyword : distributed-order time, fractional-order space, 2D Benjamin Bona Mahony Burgers equation, ultraspherical polynomials, Caputo fractional derivative
How to Cite
Azin, H., Baghani, O., & Habibirad, A. (2025). A numerical scheme to simulate the distributed-order time 2D Benjamin Bona Mahony Burgers equation with fractional-order space. Mathematical Modelling and Analysis, 30(2), 277–298. https://doi.org/10.3846/mma.2025.20964
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
S. Abbasbandy and A. Shirzadi. The first integral method for modified Benjamin–Bona–Mahony equation. Communications in Nonlinear Science and Numerical Simulation, 15(7):1759–1764, 2010. https://doi.org/10.1016/j.cnsns.2009.08.003
W.M. Abd-Elhameed and H.M. Ahmed. Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of Chebyshev polynomials. AIMS Mathematics, 9(1):2137–2166, 2024. https://doi.org/10.3934/math.2024107
M. Abdelhakem, D. Mahmoud, D. Baleanu and M. El-kady. Shifted ultraspherical pseudo-Galerkin method for approximating the solutions of some types of ordinary fractional problems. Advances in Difference Equations, 2021(110):1– 18, 2021. https://doi.org/10.1186/s13662-021-03247-6
H.M. Ahmed, R.M. Hafez and W.M. Abd-Elhameed. A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighthkind Chebyshev operational matrices. Physica Scripta, 99(4):045250, 2024. https://doi.org/10.1088/1402-4896/ad3482
A. Alsaedi, J.J. Nieto and V. Venktesh. Fractional electrical circuits. Advances in Mechanical Engineering, 7(12), 2015. https://doi.org/10.1177/1687814015618127
H. Azin, F. Mohammadi and J.A.T. Machado. A piecewise spectralcollocation method for solving fractional Riccati differential equation in large domains. Computational and Applied Mathematics, 38(96):1–13, 2019. https://doi.org/10.1007/s40314-019-0860-2
D. Baleanu, J.A.T. Machado and A.C.J. Luo. Fractional dynamics and control. Springer Science & Business Media, 2012. https://doi.org/10.1007/978-1-4614-0457-6
T.B. Benjamin, J.L. Bona and J.J. Mahony. Model equations for long waves in nonlinear dispersive systems. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 272(1220):47–78, 1972. https://doi.org/10.1098/rsta.1972.0032
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang. Spectral methods: fundamentals in single domains. Springer Science & Business Media, 2007. https://doi.org/10.1007/978-3-540-30726-6
S. Das and I. Pan. Fractional order signal processing: introductory concepts and applications. Springer Science & Business Media, 2011.
M. Dehghan, M. Abbaszadeh and A. Mohebbi. The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation via the meshless method of radial basis functions. Computers & Mathematics with Applications, 68(3):212–237, 2014. https://doi.org/10.1016/j.camwa.2014.05.019
M. Dehghan, M. Abbaszadeh and A. Mohebbi. The use of interpolating elementfree Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate. Journal of Computational and Applied Mathematics, 286:211– 231, 2015. https://doi.org/10.1016/j.cam.2015.03.012
M. Dehghan, N. Shafieeabyaneh and M. Abbaszadeh. Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the legendre spectral element method. Numerical Methods for Partial Differential Equations, 37(1):360–382, 2021. https://doi.org/10.1002/num.22531
D.B. Dhaigude, G.A. Birajdar and V.R. Nikam. Adomain decomposition method for fractional Benjamin–Bona–Mahony–Burgers’ equations. Int. J. Appl. Math. Mech, 8(12):42–51, 2012.
E.H. Doha and W.M. Abd-Elhameed. Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials. SIAM Journal on Scientific Computing, 24(2):548–571, 2002. https://doi.org/10.1137/S1064827500378933
E.H. Doha and W.M. Abd-Elhameed. Efficient spectral ultraspherical-dualPetrov–Galerkin algorithms for the direct solution of (2n + 1)th-order linear differential equations. Mathematics and Computers in Simulation, 79(11):3221– 3242, 2009. https://doi.org/10.1016/j.matcom.2009.03.011
M. Erfanian, H. Zeidabadi and O. Baghani. Solving an inverse problem for a time-fractional advection-diffusion equation with variable coefficients by rationalized Haar wavelet method. Journal of Computational Science, 64:101869, 2022. https://doi.org/10.1016/j.jocs.2022.101869
A.-J. Guel-Cortez, C.-F. Méndez-Barrios, E. Kim and M. Sen. Fractionalorder controllers for irrational systems. IET Control Theory & Applications, 15(7):965–977, 2021. https://doi.org/10.1049/cth2.12095
A. Habibirad, E. Hesameddini and Y. Shekari. A suitable hybrid meshless method for the numerical solution of time-fractional fourth-order reaction–diffusion model in the multi-dimensional case. Engineering Analysis with Boundary Elements, 145:149–160, 2022. https://doi.org/10.1016/j.enganabound.2022.09.007
M.H. Heydari, M. Razzaghi and D. Baleanu. A numerical method based on the piecewise Jacobi functions for distributed-order fractional Schro¨dinger equation. Communications in Nonlinear Science and Numerical Simulation, 116:106873, 2023. https://doi.org/10.1016/j.cnsns.2022.106873
K. Omrani and M. Ayadi. Finite difference discretization of the Benjamin–Bona–Mahony–Burgers equation. Numerical Methods for Partial Differential Equations: An International Journal, 24(1):239–248, 2008. https://doi.org/10.1002/num.20256
I. Podlubny. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, 1998.
N. Vieira, M.M. Rodrigues and M. Ferreira. Time-fractional telegraph equation of distributed order in higher dimensions. Communications in Nonlinear Science and Numerical Simulation, 102:105925, 2021. https://doi.org/10.1016/j.cnsns.2021.105925
Y.H. Youssri. Orthonormal ultraspherical operational matrix algorithm for fractal–fractional Riccati equation with generalized Caputo derivative. Fractal and Fractional, 5(3):100, 2021. https://doi.org/10.3390/fractalfract5030100
Y.H. Youssri, W.M. Abd-Elhameed and E.H. Doha. Ultraspherical wavelets method for solving Lane–Emden type equations. Rom. J. Phys, 60(9-10):1298– 1314, 2015.
M. Yousuf, K.M. Furati and A.Q.M. Khaliq. A fourth-order timestepping method for two-dimensional, distributed-order, space-fractional, inhomogeneous parabolic equations. Fractal and Fractional, 6(10):592, 2022. https://doi.org/10.3390/fractalfract6100592
M. Zarebnia and R. Parvaz. On the numerical treatment and analysis of Benjamin–Bona–Mahony–Burgers equation. Applied Mathematics and Computation, 284:79–88, 2016. https://doi.org/10.1016/j.amc.2016.02.037
H. Zhang, F. Liu, X. Jiang and I. Turner. Spectral method for the twodimensional time distributed-order diffusion-wave equation on a semi-infinite domain. Journal of Computational and Applied Mathematics, 399:113712, 2022. https://doi.org/10.1016/j.cam.2021.113712
W.M. Abd-Elhameed and H.M. Ahmed. Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of Chebyshev polynomials. AIMS Mathematics, 9(1):2137–2166, 2024. https://doi.org/10.3934/math.2024107
M. Abdelhakem, D. Mahmoud, D. Baleanu and M. El-kady. Shifted ultraspherical pseudo-Galerkin method for approximating the solutions of some types of ordinary fractional problems. Advances in Difference Equations, 2021(110):1– 18, 2021. https://doi.org/10.1186/s13662-021-03247-6
H.M. Ahmed, R.M. Hafez and W.M. Abd-Elhameed. A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighthkind Chebyshev operational matrices. Physica Scripta, 99(4):045250, 2024. https://doi.org/10.1088/1402-4896/ad3482
A. Alsaedi, J.J. Nieto and V. Venktesh. Fractional electrical circuits. Advances in Mechanical Engineering, 7(12), 2015. https://doi.org/10.1177/1687814015618127
H. Azin, F. Mohammadi and J.A.T. Machado. A piecewise spectralcollocation method for solving fractional Riccati differential equation in large domains. Computational and Applied Mathematics, 38(96):1–13, 2019. https://doi.org/10.1007/s40314-019-0860-2
D. Baleanu, J.A.T. Machado and A.C.J. Luo. Fractional dynamics and control. Springer Science & Business Media, 2012. https://doi.org/10.1007/978-1-4614-0457-6
T.B. Benjamin, J.L. Bona and J.J. Mahony. Model equations for long waves in nonlinear dispersive systems. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 272(1220):47–78, 1972. https://doi.org/10.1098/rsta.1972.0032
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang. Spectral methods: fundamentals in single domains. Springer Science & Business Media, 2007. https://doi.org/10.1007/978-3-540-30726-6
S. Das and I. Pan. Fractional order signal processing: introductory concepts and applications. Springer Science & Business Media, 2011.
M. Dehghan, M. Abbaszadeh and A. Mohebbi. The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation via the meshless method of radial basis functions. Computers & Mathematics with Applications, 68(3):212–237, 2014. https://doi.org/10.1016/j.camwa.2014.05.019
M. Dehghan, M. Abbaszadeh and A. Mohebbi. The use of interpolating elementfree Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate. Journal of Computational and Applied Mathematics, 286:211– 231, 2015. https://doi.org/10.1016/j.cam.2015.03.012
M. Dehghan, N. Shafieeabyaneh and M. Abbaszadeh. Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the legendre spectral element method. Numerical Methods for Partial Differential Equations, 37(1):360–382, 2021. https://doi.org/10.1002/num.22531
D.B. Dhaigude, G.A. Birajdar and V.R. Nikam. Adomain decomposition method for fractional Benjamin–Bona–Mahony–Burgers’ equations. Int. J. Appl. Math. Mech, 8(12):42–51, 2012.
E.H. Doha and W.M. Abd-Elhameed. Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials. SIAM Journal on Scientific Computing, 24(2):548–571, 2002. https://doi.org/10.1137/S1064827500378933
E.H. Doha and W.M. Abd-Elhameed. Efficient spectral ultraspherical-dualPetrov–Galerkin algorithms for the direct solution of (2n + 1)th-order linear differential equations. Mathematics and Computers in Simulation, 79(11):3221– 3242, 2009. https://doi.org/10.1016/j.matcom.2009.03.011
M. Erfanian, H. Zeidabadi and O. Baghani. Solving an inverse problem for a time-fractional advection-diffusion equation with variable coefficients by rationalized Haar wavelet method. Journal of Computational Science, 64:101869, 2022. https://doi.org/10.1016/j.jocs.2022.101869
A.-J. Guel-Cortez, C.-F. Méndez-Barrios, E. Kim and M. Sen. Fractionalorder controllers for irrational systems. IET Control Theory & Applications, 15(7):965–977, 2021. https://doi.org/10.1049/cth2.12095
A. Habibirad, E. Hesameddini and Y. Shekari. A suitable hybrid meshless method for the numerical solution of time-fractional fourth-order reaction–diffusion model in the multi-dimensional case. Engineering Analysis with Boundary Elements, 145:149–160, 2022. https://doi.org/10.1016/j.enganabound.2022.09.007
M.H. Heydari, M. Razzaghi and D. Baleanu. A numerical method based on the piecewise Jacobi functions for distributed-order fractional Schro¨dinger equation. Communications in Nonlinear Science and Numerical Simulation, 116:106873, 2023. https://doi.org/10.1016/j.cnsns.2022.106873
K. Omrani and M. Ayadi. Finite difference discretization of the Benjamin–Bona–Mahony–Burgers equation. Numerical Methods for Partial Differential Equations: An International Journal, 24(1):239–248, 2008. https://doi.org/10.1002/num.20256
I. Podlubny. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, 1998.
N. Vieira, M.M. Rodrigues and M. Ferreira. Time-fractional telegraph equation of distributed order in higher dimensions. Communications in Nonlinear Science and Numerical Simulation, 102:105925, 2021. https://doi.org/10.1016/j.cnsns.2021.105925
Y.H. Youssri. Orthonormal ultraspherical operational matrix algorithm for fractal–fractional Riccati equation with generalized Caputo derivative. Fractal and Fractional, 5(3):100, 2021. https://doi.org/10.3390/fractalfract5030100
Y.H. Youssri, W.M. Abd-Elhameed and E.H. Doha. Ultraspherical wavelets method for solving Lane–Emden type equations. Rom. J. Phys, 60(9-10):1298– 1314, 2015.
M. Yousuf, K.M. Furati and A.Q.M. Khaliq. A fourth-order timestepping method for two-dimensional, distributed-order, space-fractional, inhomogeneous parabolic equations. Fractal and Fractional, 6(10):592, 2022. https://doi.org/10.3390/fractalfract6100592
M. Zarebnia and R. Parvaz. On the numerical treatment and analysis of Benjamin–Bona–Mahony–Burgers equation. Applied Mathematics and Computation, 284:79–88, 2016. https://doi.org/10.1016/j.amc.2016.02.037
H. Zhang, F. Liu, X. Jiang and I. Turner. Spectral method for the twodimensional time distributed-order diffusion-wave equation on a semi-infinite domain. Journal of Computational and Applied Mathematics, 399:113712, 2022. https://doi.org/10.1016/j.cam.2021.113712