Existence and numerical analysis using Haar wavelet for fourth-order multi-term fractional differential equations. (English) Zbl 1513.65397
Summary: In this paper, a numerical technique is developed for the solution of multi-term fractional differential equations (FDEs) upto fourth order by using Haar collocation method (HCM). In Caputo sense, the fractional derivative is defined. The integral involved in the equations is calculated by the method of Lepik. The HCM converts the given multi-term FDEs into a system of linear equations. The convergence of the proposed method HCM is checked on some problems. Mean square root and maximum absolute errors are calculated for different numbers of collocation points (CPs), which are recorded in tables. The exact and approximate solution comparison is also given in figures. The time taken by CPU for numerical results is also given in the tables.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65T60 | Numerical methods for wavelets |
| 65F05 | Direct numerical methods for linear systems and matrix inversion |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Keywords:
fractional calculus; Haar wavelet; fixed-point theory; Gauss elimination method; numerical approximationReferences:
| [1] | Ahmad, O.; Sheikh, NA; Nisar, KS; Shah, FA, Biorthogonal wavelets on the spectrum, Math Methods Appl Sci, 44, 4479-4490 (2021) · Zbl 1472.42052 · doi:10.1002/mma.7046 |
| [2] | Ahmad, I.; Amin, R.; Abdeljawad, T.; Shah, K., A numerical method for fractional pantograph delay integro-differential equations on Haar wavelet, Int J Appl Comput Math, 7, 28, 1-13 (2021) · Zbl 1485.65073 |
| [3] | Alikhanov, AA, A new difference scheme for the fractional diffusion equation, J Comput Phys, 280, 424-438 (2015) · Zbl 1349.65261 · doi:10.1016/j.jcp.2014.09.031 |
| [4] | Alrabaiah H, Ahmad I, Amin R, Shah K (2022) A numerical method for fractional variable order pantograph differential equations based on Haar wavelet. Eng Comput 38(3): 2655-2668 |
| [5] | Amin, R.; Nazir, S.; Magarino, IG, Efficient sustainable algorithm for numerical solution of nonlinear delay Fredholm-Volterra integral equations via haar wavelet for dense sensor networks in emerging telecommunications, Trans Emerg Telecommun Technol, 20 (2020) |
| [6] | Amin, R.; Mahariq, I.; Shah, K.; Awais, M.; Elsayed, F., Numerical solution of the second order linear and nonlinear integro-differential equations using Haar wavelet method, Arab J Basic Appl Sci, 28, 11-19 (2021) |
| [7] | Amin, R.; Alshahrani, B.; Abdel-Aty, AH; Shah, K.; Wejdan, D., Haar wavelet method for solution of distributed order time-fractional differential equations, Alex Eng J, 60, 3295-3303 (2021) · doi:10.1016/j.aej.2021.01.039 |
| [8] | Amin, R.; Shah, K.; Asif, M.; Khan, I.; Ullah, F., An efficient algorithm for numerical solution of fractional integro-differential equations via Haar wavelet, J Comput Appl Math, 381, 1 (2021) · Zbl 1451.65230 · doi:10.1016/j.cam.2020.113028 |
| [9] | Awawdeh, F.; Rawashdeh, EA; Jaradat, HM, Analytic solution of fractional integro-differential equations, Ann Univ Craiova, 38, 1-10 (2011) · Zbl 1240.45015 |
| [10] | Aziz, I.; Amin, R., Numerical solution of a class of delay differential and delay partial differential equations via haar wavelet, Appl Math Model, 40, 10286-10299 (2016) · Zbl 1443.65089 · doi:10.1016/j.apm.2016.07.018 |
| [11] | Benchohra, M.; Bouriah, S.; Nieto, JJ, Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative, Demonstr Math, 52, 1, 437-450 (2019) · Zbl 1431.34006 · doi:10.1515/dema-2019-0032 |
| [12] | Debnath, L., Recent applications of fractional calculus to science and engineering, Int J Math Math Sci, 2003, 54, 3413-3442 (2003) · Zbl 1036.26004 · doi:10.1155/S0161171203301486 |
| [13] | Eslahchi, MR; Dehghan, M.; Parvizi, M., Application of the collocation method for solving nonlinear fractional integro-differential equations, J Comput Appl Math, 257, 105-128 (2014) · Zbl 1296.65106 · doi:10.1016/j.cam.2013.07.044 |
| [14] | Fonseca, G.; Fonseca, I.; Gangbo, W., Degree theory in analysis and applications (1995), Oxford: Oxford University Press, Oxford · Zbl 0852.47030 |
| [15] | Irfan, M.; Shah, FA; Nisar, KS, Fibonacci wavelet method for solving Pennes bioheat transfer equation, Int J Wavelets Multiresolut Inf Process, 19, 6 (2021) · Zbl 1486.65200 · doi:10.1142/S0219691321500235 |
| [16] | Kilbas, AA; Marichev, OI; Samko, SG, Fractional integrals and derivatives (theory and applications) (1993), Geneva: Gordon and Breach, Geneva · Zbl 0818.26003 |
| [17] | Lepik, U., Solving fractional integral equations by the Haar wavelet method, Appl Math Comput, 214, 468-478 (2009) · Zbl 1170.65106 |
| [18] | Li, D.; Zhang, C., Long time numerical behaviors of fractional pantograph equations, Math Comput Simul, 172, 244-257 (2020) · Zbl 1482.34189 · doi:10.1016/j.matcom.2019.12.004 |
| [19] | Lin, Y.; Xu, C., Finite difference/spectral approximations for time-fractional diffusion equation, J Comput Phys, 225, 1533-1552 (2007) · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001 |
| [20] | Nisar, KS; Shah, FA, A numerical scheme based on Gegenbauer wavelets for solving a class of relaxation-oscillation equations of fractional order, Math Sci (2022) · Zbl 1535.65236 · doi:10.1007/s40096-022-00465-1 |
| [21] | Podlubny, I., Fractional differential equations (1999), New York: Academic Press, New York · Zbl 0924.34008 |
| [22] | Rawashdeh, E., Numerical solution of semi-differential equations by collocation method, Appl Math Comput, 174, 869-876 (2006) · Zbl 1090.65097 |
| [23] | Shah, K.; Hussain, W., Investigating a class of nonlinear fractional differential equations and its Hyers-Ulam stability by means of topological degree theory, Numer Funct Anal Optim, 40, 12, 1355-1372 (2019) · Zbl 1425.34025 · doi:10.1080/01630563.2019.1604545 |
| [24] | Shah, FA; Irfan, M.; Nisar, KS; Matoog, RT; Mahmoud, EE, Fibonacci wavelet method for solving time-fractional telegraph equations with Dirichlet boundary conditions, Results Phys, 24 (2021) · doi:10.1016/j.rinp.2021.104123 |
| [25] | Shah, FA; Irfan, M.; Nisar, KS, Gegenbauer wavelet quasi-linearization method for solving fractional population growth model in a closed system, Math Methods Appl Sci, 45, 7, 3605-3623 (2022) · Zbl 1530.92193 · doi:10.1002/mma.8006 |
| [26] | Sheikh, NA; Jamil, M.; Ching, DLC; Khan, I.; Usman, M.; Nisar, KS, A generalized model for quantitative analysis of sediments loss: a Caputo time fractional model, J King Saud Univ Sci, 33 (2021) · doi:10.1016/j.jksus.2020.09.006 |
| [27] | Shiralashetti, SC; Deshi, AB, An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equation, Nonlinear Dyn, 83, 293-303 (2016) · doi:10.1007/s11071-015-2326-4 |
| [28] | Sun, H.; Zhang, Y.; Baleanu, D.; Chen, W.; Chen, Y., A new collection of real world applications of fractional calculus in science and engineering, Commun Nonlinear Sci Numer Simul, 64, 213-231 (2018) · Zbl 1509.26005 · doi:10.1016/j.cnsns.2018.04.019 |
| [29] | Wang, G.; Pei, K.; Agarwal, R., Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line, J Comput Appl Math, 343, 230-239 (2018) · Zbl 1524.34057 · doi:10.1016/j.cam.2018.04.062 |
| [30] | Wang, G.; Pei, K.; Chen, Y., Stability analysis of nonlinear Hadamard fractional differential system, J Frankl Inst, 356, 6538-6546 (2019) · Zbl 1418.34111 · doi:10.1016/j.jfranklin.2018.12.033 |
| [31] | Yang, XH; Xu, D.; Zhang, HX, Crank-Nicolson/quasi-wavelets method for solving fourth order partial integro-differential equation with a weakly singular kernel, J Comput Phys, 234, 317-329 (2013) · Zbl 1284.35454 · doi:10.1016/j.jcp.2012.09.037 |
| [32] | Zhang, H.; Han, X., Quasi-wavelet method for time-dependent fractional partial differential equation, Int J Comput Math, 90, 2491-2507 (2013) · Zbl 1284.65194 · doi:10.1080/00207160.2013.786050 |
| [33] | Zhang, HX; Han, X.; Yang, XH, Quintic B-spline collocation method for fourth order partial integro-differential equations with a weakly singular kernel, Appl Math Comput, 219, 6565-6575 (2013) · Zbl 1286.65188 |
| [34] | Zhao, J.; Xiao, J.; Ford, NJ, Collocation methods for fractional integro-differential equations with weakly singular kernels, Numer Algorithm, 65, 723-743 (2014) · Zbl 1298.65197 · doi:10.1007/s11075-013-9710-2 |
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.
