Keywords:-
Keywords:
Caputo fractional derivative, Legendre wavelets, Operational matrix. AMS Subject Classification (2010): 26A33, 34A08, 65T60
Article Content:-
Abstract
This article presents an efficient numerical algorithm based on Legendre wavelets operational matrix for solving some fractional models. Fractional integration operational matrix of Legendre wavelets is derived and it is employed to reduce fractional differential equations into a system of algebraic equations. Mixing problems, Newton law of cooling problems and sugar inversion problems are included to elucidate the applicability and the simplicity of the Legendre wavelet-based numerical algorithm.
References:-
References
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K.Diethelm. The analysis of fractional differential equations: An application-oriented exposition using differential operators of caputo type. Springer science & Business Media, 2010.
U.Farooq, H.Khan, D.Baleanu, and M.Arif. Numerical solutions of fractional delay differential equations using chebyshev wavelet method. Computational and applied mathematics, 38(4):1-13, 2019.
T.Ji and J.Hou. Numerical solution of the bagley-torvik equation using laguerre polynomials. SeMA Journal,77(1):97-106.2020.
N.Kadkhoda. A numerical approach for solveing variable order differential equations using Bernstein polynomials. Alexandria Engineering Journal, 59(5):3041-3047, 2020.
A.Kilicman and Z.A.A.A1 Zhour. Kronecker operational matrices for fractional calculus and some applications. Applied mathematics and computation, 187(1):250-265, 2007.
E.Kreyszig. Advanced Engineering Mathematics 9th Edition with wiley plus Set. John Wiley & Sons, 2007.
K.S. Miller ad B.Ross. An introduction to the fractional calculus and fractional differential equations. Wiley, 1993.
A.Mohammadi, N.Aghazadeh, and S.Rezapour. Haar wavelet collocation method for solving singular and nonlinear fractional time-dependent emden-fowler equations with initial and boundary conditions. Mathematical Science, 13(3):255-265, 2019.
K.Oldham and J.Spanier. The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, 1974.
I.Podlubny. Fractional equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, 1998.
P.Rahimkhani and R.Moeti. Numerical solution of the fractional order duffing-vander pol oscillator equation by using bernoulli wavelets collocation method. International Journal of Applied and computational Mathematics, 4(2):1-18, 2018.
P.Rahimkhani, Y.Ordokhani, and E.Babolian. Mu ̈ntz-legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations. Numerical algorithms, 77(4):1283-1305, 2018.
M.Razzagi and S.Yousefi. Legendre wavelets direct method for variational problems. Mathematics and computers in simulation, 53(3):185-192, 2000.
M.Razzagi and S,Yousefi. The legendre wavelets operational matrix of integration. Interational Journal of Systems Science, 32(4):495-502, 2001.
S.Sabermahani, Y.Ordokhani, and S.Yousefi. Numerical approach based on fractional-order lagrange polynomials for solving a class of fractional differential equations. Computational and Applied Mathematics, 37(3):3846-3868, 2018.
P.Sahu and B.Mallick. Approximate solution of fractional order lane-emden type differential equation by orthonormal bernoullis polynomials. International Journal of Applied and Computational Mathematics, 5(3):1-9, 2019.
Y. Talaei and M. Asgari. An operational matrix based on Chelyshkov polynomials for solving multi-order fractional differential equations. Neural Computing and Applications, 30(5):1369-1376, 2018.
P.T.Toan, T.N.Vo, and M.Razzaghi. Taylor wavelet method for fractional delay differential equations. Engineering with Computers, 37(1):231-240, 2021.
M.ur Rehman and R.A.Khan. The legendre wavelet method for solving fractional differential equations. Communications in Nonlinear Science and Numerical Simulation, 16(11):4163-4173, 2011
Y.Wang and Q.Fan. The second kind chebyshev wavelet method for solving fractional differential equations. Applied Mathematics and Computation, 218(17):8592-8601, 2012.
Y.Wang, L.Zhu, and Z.Wang. Solving the nonlinear variable order fractional differential equations by using euler wavelets. Computer Modeling in Engineering & Science, 118(2):339-350, 2019.
K.Diethelm. The analysis of fractional differential equations: An application-oriented exposition using differential operators of caputo type. Springer science & Business Media, 2010.
U.Farooq, H.Khan, D.Baleanu, and M.Arif. Numerical solutions of fractional delay differential equations using chebyshev wavelet method. Computational and applied mathematics, 38(4):1-13, 2019.
T.Ji and J.Hou. Numerical solution of the bagley-torvik equation using laguerre polynomials. SeMA Journal,77(1):97-106.2020.
N.Kadkhoda. A numerical approach for solveing variable order differential equations using Bernstein polynomials. Alexandria Engineering Journal, 59(5):3041-3047, 2020.
A.Kilicman and Z.A.A.A1 Zhour. Kronecker operational matrices for fractional calculus and some applications. Applied mathematics and computation, 187(1):250-265, 2007.
E.Kreyszig. Advanced Engineering Mathematics 9th Edition with wiley plus Set. John Wiley & Sons, 2007.
K.S. Miller ad B.Ross. An introduction to the fractional calculus and fractional differential equations. Wiley, 1993.
A.Mohammadi, N.Aghazadeh, and S.Rezapour. Haar wavelet collocation method for solving singular and nonlinear fractional time-dependent emden-fowler equations with initial and boundary conditions. Mathematical Science, 13(3):255-265, 2019.
K.Oldham and J.Spanier. The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, 1974.
I.Podlubny. Fractional equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, 1998.
P.Rahimkhani and R.Moeti. Numerical solution of the fractional order duffing-vander pol oscillator equation by using bernoulli wavelets collocation method. International Journal of Applied and computational Mathematics, 4(2):1-18, 2018.
P.Rahimkhani, Y.Ordokhani, and E.Babolian. Mu ̈ntz-legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations. Numerical algorithms, 77(4):1283-1305, 2018.
M.Razzagi and S.Yousefi. Legendre wavelets direct method for variational problems. Mathematics and computers in simulation, 53(3):185-192, 2000.
M.Razzagi and S,Yousefi. The legendre wavelets operational matrix of integration. Interational Journal of Systems Science, 32(4):495-502, 2001.
S.Sabermahani, Y.Ordokhani, and S.Yousefi. Numerical approach based on fractional-order lagrange polynomials for solving a class of fractional differential equations. Computational and Applied Mathematics, 37(3):3846-3868, 2018.
P.Sahu and B.Mallick. Approximate solution of fractional order lane-emden type differential equation by orthonormal bernoullis polynomials. International Journal of Applied and Computational Mathematics, 5(3):1-9, 2019.
Y. Talaei and M. Asgari. An operational matrix based on Chelyshkov polynomials for solving multi-order fractional differential equations. Neural Computing and Applications, 30(5):1369-1376, 2018.
P.T.Toan, T.N.Vo, and M.Razzaghi. Taylor wavelet method for fractional delay differential equations. Engineering with Computers, 37(1):231-240, 2021.
M.ur Rehman and R.A.Khan. The legendre wavelet method for solving fractional differential equations. Communications in Nonlinear Science and Numerical Simulation, 16(11):4163-4173, 2011
Y.Wang and Q.Fan. The second kind chebyshev wavelet method for solving fractional differential equations. Applied Mathematics and Computation, 218(17):8592-8601, 2012.
Y.Wang, L.Zhu, and Z.Wang. Solving the nonlinear variable order fractional differential equations by using euler wavelets. Computer Modeling in Engineering & Science, 118(2):339-350, 2019.
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Balaji, K., & Aruldoss, R. (2023). A Wavelet-Based Numerical Algorithm for Some Fractional Models. International Journal Of Mathematics And Computer Research, 11(7), 3543-3549. https://doi.org/10.47191/ijmcr/v11i7.07