### A Wavelet Collocation Method for some Fractional Models

#### Abstract

This article presents an effective numerical approach based on the operational matrix of fractional order integration of Haar wavelets for dealing with the fractional models of the mixing and the Newton law of cooling problems. A general procedure of obtaining the fractional integration operational matrix of Haar wavelets which converts the fractional models into a system of algebraic equations is derived so that the computation is very simple and it is much effective than the conventional numerical methods. The reliability and the applicability of the current numerical technique for fractional models are examined by comparing the achieved results with the precise solutions.

#### Full Text:

PDF#### References

Arikoglu and I. Ozkol. Solution of fractional integro-differential equations by using fractional differential transform method. Chaos, Solitons & Fractals, 40 (2):521–529, 2009.

R. Aruldoss and K. Balaji. Numerical inversion of laplace transform via wavelet operational matrix and its applications to fractional differential equations. International Journal of Applied and Computational Mathematics, 8(1):1–17, 2022.

R. Aruldoss, R. A. Devi, and P. M. Krishna. An expeditious wavelet-based numerical scheme for solving fractional differential equations. Computational and Applied Mathematics, 40(1):1–14, 2021.

Y. Chen, M. Yi, and C. Yu. Error analysis for numerical solution of fractional differential equation by haar wavelets method. Journal of Computational Science, 3(5):367–373, 2012.

S. Das. Analytical solution of a fractional diffusion equation by variational iteration method. Computers & Mathematics with Applications, 57(3):483–487, 2009.

K. Diethelm. The analysis of differential equations of fractional order: An application-oriented exposition using differential operators of caputo type. Lecture Notes in Mathematics, 2004, 2004.

S. El-Wakil, A. Elhanbaly, and M. Abdou. Adomian decomposition method for solving fractional nonlinear differential equations. Applied Mathematics and Computation, 182(1):313–324, 2006.

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 solving variable order differential equations using bernstein polynomials. Alexandria Engineering Journal, 59(5): 3041–3047, 2020.

A.Kilicman and Z. A. A. Al 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.

M. M. Meerschaert and C. Tadjeran. Finite difference approximations for twosided space-fractional partial differential equations. Applied numerical mathematics, 56(1):80–90, 2006.

K. S. Miller and 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 Sciences, 13(3):255–265, 2019.

Z. M. Odibat and N. T. Shawagfeh. Generalized taylors formula. Applied Mathematics and Computation, 186(1):286–293, 2007.

K. Oldham and J. Spanier. The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, 1974.

I.Podlubny. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Math. Sci. Eng, 198:340, 1999.

P. Rahimkhani and R. Moeti. Numerical solution of the fractional order duffingvan der 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. M¨untz-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.

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.

F. A. Shah, R. Abass, and L. Debnath. Numerical solution of fractional differential equations using haar wavelet operational matrix method. International Journal of Applied and Computational Mathematics, 3(3):2423–2445, 2017.

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.

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.

DOI: http://dx.doi.org/10.23755/rm.v43i0.859

### Refbacks

- There are currently no refbacks.

Copyright (c) 2022 Aruldoss R

This work is licensed under a Creative Commons Attribution 4.0 International License.

Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.