Collocation method with Lagrange polynomials for variable-order time-fractional advection-diffusion problems. (English) Zbl 1536.65117
Summary: In this study, we proposed a numerical technique to solve a class of variable-order time-fractional advection-diffusion equations (VOTFADEs) by applying an operational matrix of differentiation based on fractional-order Lagrange polynomials (FOLPs). The variable-order fractional derivative is assumed to be Caputo’s derivative. Using the operational matrix and collocation method, the advection-diffusion equation can be reduced to an algebraic system of equations that can be solved using Newton’s iterative method. Error analysis also has been carried out for the proposed method. The current approach is simple to use and computer oriented and provides highly accurate approximate solutions. The effectiveness and accuracy of the proposed method are demonstrated using a few numerical examples.
© 2023 John Wiley & Sons Ltd.
© 2023 John Wiley & Sons Ltd.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Keywords:
Caputo fractional derivative; fractional advection-diffusion equations; fractional-order Lagrange polynomials; operational matrix of derivativeReferences:
| [1] | V. S.Ertürk and S.Momani, Solving systems of fractional differential equations using differential transform method, J. Comput. Appl. Math.215 (2008), 142-151. · Zbl 1141.65088 |
| [2] | H.Jafari, S. A.Yousefi, M. A.Firoozjaee, S.Momani, and C. M.Khalique, Application of Legendre wavelets for solving fractional differential equations, Comput. Math. Appl.62 (2011), 1038-1045. · Zbl 1228.65253 |
| [3] | V.Daftardar‐Gejji and H.Jafari, Solving a multi‐order fractional differential equation using Adomian decomposition, Appl. Math. Comput.189 (2007), 541-548. · Zbl 1122.65411 |
| [4] | J.Ma, J.Liu, and Z.Zhou, Convergence analysis of moving finite element methods for space fractional differential equations, J. Comput. Appl. Math.255 (2014), 661-670. · Zbl 1291.65303 |
| [5] | H.Jafari, H.Tajadodi, and D.Baleanu, A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials, Fract. Calc. Appl.16 (2013), 109-122. · Zbl 1312.34016 |
| [6] | S.Kumar and V.Gupta, An application of variational iteration method for solving fuzzy time‐fractional diffusion equations, Neural Comput. & Applic.33 (2021), 17659-17668. |
| [7] | W.Deng, S.Du, and Y.Wu, High order finite difference WENO schemes for fractional differential equations, Appl. Math. Lett.26 (2013), 362-366. · Zbl 1259.65128 |
| [8] | Y.Wang and Q.Fan, The second kind Chebyshev wavelet method for solving fractional differential equations, Appl. Math. Comput.218 (2012), 8592-8601. · Zbl 1245.65090 |
| [9] | E.Keshavarz, Y.Ordokhani, and M.Razzaghi, Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Model.38 (2014), 6038-6051. · Zbl 1429.65170 |
| [10] | S.Kumar and V.Gupta, An approach based on fractional‐order Lagrange polynomials for the numerical approximation of fractional order non‐linear Volterra‐Fredholm integro‐differential equations, J. Appl. Math. Comput.69 (2023), 251-272. · Zbl 07676658 |
| [11] | S.Kumar, V.Gupta, and G. F.Gómez‐Aguilar, An efficient operational matrix technique to solve the fractional order non‐local boundary value problems, J. Math. Chem.60 (2022), 1463-1479. · Zbl 1497.92373 |
| [12] | S. G.Samko and B.Ross, Integration and differentiation to a variable fractional order, Integr. Transform. Spec. Funct.1 (1993), 277-300. · Zbl 0820.26003 |
| [13] | S. G.Samko, A. A.Kilbas, and O. I.Marichev, Fractional integrals and derivatives, vol. 1, Gordon and Breach Science Publishers, Yverdon‐les‐Bains, Switzerland, 1993. · Zbl 0818.26003 |
| [14] | H. T. C.Pedro, M. H.Kobayashi, J. M. C.Pereira, and C. F. M.Coimbra, Variable order modeling of diffusive‐convective effects in the oscillatory flow past a sphere, IFAC Proc. vol. 39 (2006), 454-459. |
| [15] | L. E. S.Ramirez and C. F. M.Coimbra, On the selection and meaning of variable order operators for dynamic modeling, Int. J. Differ. Equ.2010 (2010), 1-6. · Zbl 1207.34011 |
| [16] | L. E. S.Ramirez and C. F. M.Coimbra, On the variable order dynamics of the nonlinear wake caused by a sedimenting particle, Phys. D Nonlinear Phenom.240 (2011), 1111-1118. · Zbl 1219.76054 |
| [17] | C. F. M.Coimbra, Mechanics with variable‐order differential operators, Ann. Phys.12 (2003), 692-703. · Zbl 1103.26301 |
| [18] | J.Orosco and C. F. M.Coimbra, On the control and stability of variable‐order mechanical systems, Nonlinear Dyn.86 (2016), 695-710. |
| [19] | H.Sun, Y.Chen, and W.Chen, Random‐order fractional differential equation models, Signal Process.91 (2011), 525-530. · Zbl 1203.94056 |
| [20] | L. E. S.Ramirez and C. F. M.Coimbra, A variable order constitutive relation for viscoelasticity, Ann. der Phys.16 (2007), 543-552. · Zbl 1159.74008 |
| [21] | H. G.Sun, W.Chen, and Y. Q.Chen, Variable order fractional differential operators in anomalous diffusion modeling, Phys. A388 (2009), 4586-4592. |
| [22] | H. G.Sun, W.Chen, H.Wei, and Y. Q.Chen, A comparative study of constant‐order and variable‐order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top.193 (2011), no. 1, 185-192. |
| [23] | A.Razminia, A. F.Dizaji, and V. J.Majd, Solution existence for non‐autonomous variable‐order fractional differential equations, Math. Comput. Modell.55 (2011), 1106-1117. · Zbl 1255.34008 |
| [24] | S.Zhang, Existence result of solutions to differential equations of variable‐order with nonlinear boundary value conditions, Commun. Nonlinear Sci. Numer. Simul.18 (2013), 3289-3297. · Zbl 1344.34022 |
| [25] | S.Zhang and X.Xu, Unique existence of solution to initial value problem for fractional differential equation involving with fractional derivative of variable order, Chaos Solit. Fract.148 (2021), 111040. · Zbl 1485.34061 |
| [26] | R.Lin, F.Liu, V.Anh, and I.Turner, Stability and convergence of a new explicit finite‐difference approximation for the variable‐order nonlinear fractional diffusion equation, Appl. Math. Comput.212 (2009), 435-445. · Zbl 1171.65101 |
| [27] | Y.Chen, L.Liu, B.Li, and Y.Sun, Numerical solution for the variable order linear cable equation with Bernstein polynomials, Appl. Math. Comput.238 (2014), 329-341. · Zbl 1334.65167 |
| [28] | A. H.Bhrawy and M. A.Zaky, Numerical simulation for two‐dimensional variable‐order fractional nonlinear cable equation, Nonlinear Dyn.80 (2015), 101-116. · Zbl 1345.65060 |
| [29] | M. H.Heydari, M. R.Hooshmandasly, C.Cattani, and G.Hariharan, An optimization wavelet method for multi variable‐order fractional differential equations, Fundam. Inform.151 (2017), 255-273. · Zbl 1379.65046 |
| [30] | M. H.Heydari and Z.Avazzadeh, Legendre wavelets optimization method for variable‐order fractional Poisson equation, Chaos Solit. Fract.112 (2018), 180-190. · Zbl 1398.65316 |
| [31] | M. H.Heydari and Z.Avazzadeh, An operational matrix method for solving variable‐order fractional biharmonic equation, Comput. Appl. Math.37 (2018), 4397-4411. · Zbl 1402.65125 |
| [32] | M. H.Heydari, Z.Avazzadeh, and Y.Yang, A computational method for solving variable‐order fractional nonlinear diffusion‐wave equation, Appl. Math. Comput.352 (2019), 235-248. · Zbl 1429.65240 |
| [33] | S.Kumar, P.Pandey, and S.Das, Gegenbauer wavelet operational matrix method for solving variable‐order non‐linear reaction-diffusion and Galilei invariant advection-diffusion equations, Comput. Appl. Math.38 (2019), 1-22. · Zbl 1438.35433 |
| [34] | S.Sabermahani, Y.Ordokhani, and P. M.Lima, A novel Lagrange operational matrix and Tau‐Collocation method for solving variable‐order fractional differential equations, Iran. J. Sci. Technol. Trans. Sci.44 (2020), 127-135. |
| [35] | S.Sabermahani, Y.Ordokhani, and H.Hassani, General Lagrange scaling functions: application in general model of variable order fractional partial differential equations, Comp. Appl. Math.40 (2021), 269. · Zbl 1476.65319 |
| [36] | S.Sabermahani, Y.Ordokhani, and S.‐A.Yousefi, Fractional‐order lagrange polynomials: an application for solving delay fractional optimal control problems, Trans. Inst. Meas. Control41 (2019), no. 11, 2997-3009. |
| [37] | S.Sabermahani, Y.Ordokhani, and S. A.Yousefi, Fractional‐order general lagrange scaling functions and their applications, BIT Numer. Math.60 (2020), 101-128. · Zbl 1443.44003 |
| [38] | S.Sabermahani, Y.Ordokhani, and S. A.Yousefi, Numerical approach based on fractional‐order Lagrange polynomials for solving a class of fractional differential equations, Comput. Appl. Math.37 (2018), 3846-3868. · Zbl 1404.65078 |
| [39] | P.Zhuang, F.Liu, V.Anh, and I.Turner, Numerical methods for the variable‐order fractional advection‐diffusion equation with a nonlinear source term, SIAM J. Numer. Anal.47 (2009), no. 3, 1760-1761. · Zbl 1204.26013 |
| [40] | S.Shen, F.Liu, V.Anh, and J.Chen, A characteristic difference method for the variable‐order fractional advection‐diffusion equation, J. Appl. Math. Comput.42 (2013), 371-386. · Zbl 1296.65114 |
| [41] | A.Tayebi, Y.Shekari, and M. H.Heydari, A meshless method for solving two‐dimensional variable‐order time fractional advection-diffusion equation, J. Comput. Phys.340 (2017), 655-669. · Zbl 1380.65185 |
| [42] | S.Shen, F.Liu, J.Chen, I.Turner, and V.Anh, Numerical techniques for the variable order time fractional diffusion equation, Appl. Math. Comput.218 (2012), no. 22, 10861-10870. · Zbl 1280.65089 |
| [43] | N.Chen, J.Huang, Y.Wu, and Q.Xiao, Operational matrix method for the variable order time fractional diffusion equation using Legendre polynomials approximation, IAENG Int. J. Appl. Math.47 (2017), no. 3, 282-286. · Zbl 1512.65228 |
| [44] | M.Hajipour, A.Jajarmi, D.Baleanu, and H.Sun, On an accurate discretization of a variable‐order fractional reaction‐diffusion equation, Commun. Nonlinear Sci. Numer. Simul.69 (2019), 119-133. · Zbl 1509.65071 |
| [45] | C.‐M.Chen, F.Liu, V.Anh, and I.Turner, Numerical schemes with high spatial accuracy for a variable‐order anomalous subdiffusion equation, SIAM J. Sci. Comput.32 (2010), no. 4, 1740-1760. · Zbl 1217.26011 |
| [46] | J.Cao, Y.Qiu, and G.Song, A compact finite difference scheme for variable order subdiffusion equation, Commun. Nonlinear Sci. Numer. Simul.48 (2017), 140-149. · Zbl 1524.65323 |
| [47] | V.Gupta and M. K.Kadalbajoo, Qualitative analysis and numerical solution of Burgers’ equation via B‐spline collocation with implicit Euler method on piecewise uniform mesh, J. Numer. Math.24 (2016), 73-94. · Zbl 1339.65148 |
| [48] | S.Sahoo and V.Gupta, A robust uniformly convergent finite difference scheme for the time‐fractional singularly perturbed convection‐diffusion problem, Comput. Math. Appl.137 (2023), 126-146. · Zbl 1538.65263 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
