Operational matrices based on hybrid functions for solving general nonlinear two-dimensional fractional integro-differential equations. (English) Zbl 1463.65428
Summary: In the present paper, an attempt was made to develop a numerical method for solving a general form of two-dimensional nonlinear fractional integro-differential equations using operational matrices. Our approach is based on the hybrid of two-dimensional block-pulse functions and two-variable shifted Legendre polynomials. Error bound and convergence analysis of the proposed method are discussed. We prove that the order of convergence of our method is \(O( \frac{1}{2^{2M-1}N^MM!})\). The presented method is tested by seven test problems to demonstrate the accuracy and computational efficiency of the proposed method and to compare our results with other well-known methods. The comparison highlighted that the proposed method exhibits superior performance than the existing methods, even using a few numbers of bases.
MSC:
| 65R20 | Numerical methods for integral equations |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 45K05 | Integro-partial differential equations |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
two-dimensional nonlinear fractional integro-differential equations; two-variable hybrid functions; block-pulse functions; shifted Legendre polynomials; operational matrices; convergence analysisReferences:
| [1] | Abbas, S.; Benchohra, M., Fractional order integral equations of two independent variables, Appl Math Comput, 227, 755-761 (2014) · Zbl 1364.45005 |
| [2] | Abdelkawy MA, Lopes AM, Zaky MA (2019) Shifted fractional Jacobi spectral algorithm for solving distributed order time-fractional reaction-diffusion equations. Comput Appl Math. 10.1007/s40314-019-0845-1 · Zbl 1438.65244 |
| [3] | Akhavan, S.; Maleknejad, K., Improving Petrov-Galerkin elements via Chebyshev polynomials and solving Fredholm integral equation of the second kind by them, Appl Math Comput, 271, 352-364 (2015) · Zbl 1410.65485 |
| [4] | Atangana, A.; Kılıçman, A., Analytical solutions of the spacetime fractional derivative of advection dispersion equation, Math Probl Eng, 2013, 9 (2013) · Zbl 1299.35058 · doi:10.1155/2013/853127 |
| [5] | Atangana, A.; Kılıçman, A., A possible generalization of acoustic wave equation using the concept of perturbed derivative order, Math Probl Eng, 2013, 6 (2013) · Zbl 1299.76229 · doi:10.1155/2013/696597 |
| [6] | Atkinson, KE, The numerical solution of integral equations of the second kind (1997), Cambridge: Cambridge University Press, Cambridge · Zbl 0899.65077 |
| [7] | Avazzadeh, Z.; Hassani, H., Transcendental Bernstein series for solving reaction-diffusion equations with nonlocal boundary conditions through the optimization technique, Numer Methods Partial Differ Equ, 35, 6, 2258-2274 (2019) · Zbl 1431.65184 · doi:10.1002/num.22411 |
| [8] | Babaaghaie, A.; Maleknejad, K., Numerical solution of integro-differential equations of high order by wavelet basis, its algorithm and convergence analysis, J Comput Appl Math, 325, 125-133 (2017) · Zbl 1367.65195 · doi:10.1016/j.cam.2017.04.035 |
| [9] | Babaaghaie, A.; Maleknejad, K., Numerical solutions of nonlinear two-dimensional partial Volterra integro-differential equations by Haar wavelet, J Comput Appl Math, 317, 643-651 (2017) · Zbl 1357.65311 · doi:10.1016/j.cam.2016.12.012 |
| [10] | Babolian, E.; Biazar, J., Solution of a system of non-linear Volterra integral equations of the second kind, Far East J Math Sci, 2, 6, 935-945 (2000) · Zbl 0979.65123 |
| [11] | Benson, DA; Wheatcraft, SW; Meerschaert, MM, Application of a fractional advection-dispersion equation, Water Resour Res, 36, 6, 1403-1412 (2000) · doi:10.1029/2000WR900031 |
| [12] | Caputo, M., Linear models of dissipation whose Q is almost frequency independent-part II, Geophys J Int, 13, 5, 529-539 (1967) · doi:10.1111/j.1365-246X.1967.tb02303.x |
| [13] | Chen, CM; Liu, F.; Turner, I.; Anh, V., A Fourier method for the fractional diffusion equation describing sub-diffusion, J Comput Phys, 227, 2, 886-897 (2007) · Zbl 1165.65053 · doi:10.1016/j.jcp.2007.05.012 |
| [14] | Cheney, EW, Introduction to Approximation Theory (1966), New York: McGraw-Hill, New York · Zbl 0161.25202 |
| [15] | Cloot, A.; Botha, JF, A generalised groundwater flow equation using the concept of non-integer order derivatives, Water SA, 32, 1, 55-78 (2006) |
| [16] | Dahaghin, MS; Hassani, H., An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation, Nonlinear Dyn, 88, 3, 1587-1598 (2017) · Zbl 1380.35158 · doi:10.1007/s11071-017-3330-7 |
| [17] | Davis, P., Interpolation and approximation (1975), New York: Blaisdell, New York · Zbl 0329.41010 |
| [18] | Delkhosh M, Parand KA (2019) Hybrid numerical method to solve nonlinear parabolic partial differential equations of time-arbitrary order. Comput Appl Math. 10.1007/s40314-019-0840-6 · Zbl 1438.65247 |
| [19] | Esmaeilbeigi, M.; Mirzaee, F.; Moazami, D., A meshfree method for solving multidimensional linear Fredholm integral equations on the hypercube domains, Appl Math Comput, 298, 236-246 (2017) · Zbl 1411.65164 |
| [20] | Gasca, M.; Sauer, T., On the history of multivariate polynomial interpolation, J Comput Appl Math, 122, 1, 23-35 (2000) · Zbl 0968.65008 · doi:10.1016/S0377-0427(00)00353-8 |
| [21] | Golbabai A, Nikan O (2019) A computational method based on the moving least-squares approach for pricing double barrier options in a time-fractional Black-Scholes model. Comput Econ, pp 1-23 |
| [22] | Golbabai, A.; Nikan, O.; Nikazad, T., Numerical analysis of time fractional Black-Scholes European option pricing model arising in financial market, Comput Appl Math, 38, 4, 173 (2019) · Zbl 1463.91199 · doi:10.1007/s40314-019-0957-7 |
| [23] | Golbabai, A.; Nikan, O.; Nikazad, T., Numerical investigation of the time fractional mobile-immobile advection dispersion model arising from solute transport in porous media, Int J Appl Comput Math, 5, 3, 50 (2019) · Zbl 1411.76113 · doi:10.1007/s40819-019-0635-x |
| [24] | Hansen, PC, Discrete inverse problems: insight and algorithms. Fundamentals of algorithm (2010), Philadelphia: SIAM, Philadelphia · Zbl 1197.65054 |
| [25] | Hassani, H.; Naraghirad, E., A new computational method based on optimization scheme for solving variable-order time fractional Burgers’ equation, Math Comput Simul, 162, 1-17 (2019) · Zbl 1540.65412 · doi:10.1016/j.matcom.2019.01.002 |
| [26] | Hassani, H.; Tenreiro Machado, JA; Avazzadeh, Z., An effective numerical method for solving nonlinear variable-order fractional functional boundary value problems through optimization technique, Nonlinear Dyn, 97, 4, 2041-2054 (2019) · Zbl 1430.34005 · doi:10.1007/s11071-019-05095-2 |
| [27] | Hassani, H.; Tenreiro Machado, JA; Naraghirad, E., Generalized shifted Chebyshev polynomials for fractional optimal control problems, Commun Nonlinear Sci Numer Simul, 75, 50-61 (2019) · Zbl 1462.49014 · doi:10.1016/j.cnsns.2019.03.013 |
| [28] | Hassani, H.; Avazzadeh, Z.; Tenreiro Machado, JA, Numerical approach for solving variable-order space-time fractional telegraph equation using transcendental Bernstein series, Eng Comput (2019) · doi:10.1007/s00366-019-00736-x |
| [29] | Hesameddini, E.; Shahbazi, M., Two-dimensional shifted Legendre polynomials operational matrix method for solving the two-dimensional integral equations of fractional order, Appl Math Comput, 322, 40-54 (2018) · Zbl 1426.65211 |
| [30] | Jabari Sabeg, D.; Ezzati, R.; Maleknejad, K., A new operational matrix for solving two-dimensional nonlinear integral equations of fractional order, Cogent Math, 4, 1, 1347017 (2017) · Zbl 1426.65213 · doi:10.1080/23311835.2017.1347017 |
| [31] | Kılıçman, A.; Al Zhour, ZAA, Kronecker operational matrices for fractional calculus and some applications, Appl Math Comput, 187, 1, 250-265 (2007) · Zbl 1123.65063 |
| [32] | Kreyszig, E., Introductory functional analysis with applications (1989), New York: Wiley, New York · Zbl 0706.46001 |
| [33] | Linz, P., Analytical and Numerical Methods for Volterra Equations (1985), Philadelphia: SIAM, Philadelphia · Zbl 0566.65094 |
| [34] | Mainardi, F., Fractional calculus: some basic problems in continuumand statistical mechanics, Fractals Fract Calc Contin Mech, 378, 291-348 (1997) · Zbl 0917.73004 · doi:10.1007/978-3-7091-2664-6_7 |
| [35] | Maleknejad, K.; Alizadeh, M., Sinc approximation for numerical solution of integral equation arising in conductor-like screening model for real solvent, J Comput Appl Math, 251, 81-92 (2013) · Zbl 1291.65386 · doi:10.1016/j.cam.2013.03.043 |
| [36] | Maleknejad, K.; Shahabi, M., Application of hybrid functions operational matrices in the numerical solution of two-dimensional nonlinear integral equations, Appl Numer Math, 136, 46-65 (2019) · Zbl 1433.65355 · doi:10.1016/j.apnum.2018.09.014 |
| [37] | Maleknejad, K.; Rashidinia, J.; Eftekhari, T., Numerical solution of three-dimensional Volterra-Fredholm integral equations of the first and second kinds based on Bernstein’s approximation, Appl Math Comput, 339, 272-285 (2018) · Zbl 1429.65317 |
| [38] | Meerschaert, MM; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J Comput Appl Math, 172, 1, 65-77 (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033 |
| [39] | Mirzaee, F.; Hadadiyan, E., Three-dimensional triangular functions and their applications for solving nonlinear mixed Volterra-Fredholm integral equations, Alex Eng J, 55, 2943-2952 (2016) · doi:10.1016/j.aej.2016.05.001 |
| [40] | Najafalizadeh, S.; Ezzati, R., Numerical methods for solving two-dimensional nonlinear integral equations of fractional order by using two-dimensional block pulse operational matrix, Appl Math Comput, 280, 46-56 (2016) · Zbl 1410.65502 |
| [41] | Najafalizadeh, S.; Ezzati, R., A block pulse operational matrix method for solving two-dimensional nonlinear integro-differential equations of fractional order, J Comput Appl Math, 326, 159-170 (2017) · Zbl 1370.65079 · doi:10.1016/j.cam.2017.05.039 |
| [42] | Nemati, S.; Lima, PM; Ordokhani, Y., Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials, J Comput Appl Math, 242, 53-69 (2013) · Zbl 1255.65248 · doi:10.1016/j.cam.2012.10.021 |
| [43] | Nouri, K.; Torkzadeh, L.; Mohammadian, S., Hybrid Legendre functions to solve differential equations with fractional derivatives, Math Sci, 12, 129-136 (2018) · Zbl 1417.34020 · doi:10.1007/s40096-018-0251-7 |
| [44] | Paul S, Panja MM, Mandal BN (2019) Approximate solution of first kind singular integral equation with generalized kernel using Legendre multiwavelets. Comput Appl Math. 10.1007/s40314-019-0770-3 · Zbl 1438.65339 |
| [45] | Podlubony, I., Fractional differential equations (1999), San Diego: Academic Press, San Diego · Zbl 0924.34008 |
| [46] | Rajković PM, Marinković SD, Petković MD (2019) A class of orthogonal polynomials related to the generalized Laguerre weight with two parameters. Comput Appl Math. 10.1007/s40314-019-0783-y · Zbl 1438.33017 |
| [47] | Razzaghi, M.; Marzban, HR, A hybrid analysis direct method in the calculus of variations, Int J Comput Math, 75, 259-269 (2000) · Zbl 0969.49014 · doi:10.1080/00207160008804982 |
| [48] | Singh, V.; Pandey, R.; Singh, O., New stable numerical solutions of singular integral equations of Abel type by using normalized Bernstein polynomials, Appl Math Sci, 3, 241-255 (2009) · Zbl 1175.65152 |
| [49] | Wazwaz AM (2011) Linear and nonlinear integral equations: methods and applications. Higher Education Press, Beijing; Springer, Heidelberg · Zbl 1227.45002 |
| [50] | Yuste, SB; Acedo, L., An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J Numer Anal, 42, 5, 1862-1874 (2005) · Zbl 1119.65379 · doi:10.1137/030602666 |
| [51] | Zhuang, P.; Liu, F.; Anh, V.; Turner, I., Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J Numer Anal, 47, 1760-1781 (2009) · Zbl 1204.26013 · doi:10.1137/080730597 |
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.
