Operational Matrix of Fractional Order Integration and Its Application to Solve Fractional Differential Equations (FDEs) Using Haar Wavelet Collocation Method (HWCM)
Abstract
Abstract Views: 172
Wavelets play an essential part in numerical analysis. In this study, a novel numerical technique to solve fractional differential equations (FDEs) corresponding to initial conditions is presented using Haar wavelet approximations. Haar wavelet is first presented with an operational matrix of fractional order integration. Then, illustrative examples are presented to signify the validity and applicability of the proposed method.
Downloads
References
Bagley RL, Torvik PJ. Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J. 1985;23(6):918-925. https://doi.org/10.2514/3.9007
Baillie RT. Long memory processes and fractional integration in econometrics. J Econom. 1996;73(1):5-59. https://doi.org/10.1016/0304-4076(95)01732-1
Mainardi F. Fractional calculus: ‘Some basic problems in continuum and statistical mechanics’. In: Carpinteri A, Mainardi F. eds. Fractals and Fractional Calculus in Continuum Mechanics. Springer Verlag; 1997:291-348.
Rossikhin YA, Shitikova MV. Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl Mech Rev. 1997;50(1):15–67. https://doi.org/10.1115/1.3101682
Gaul L, Klein P, Kemple S. Damping description involving fractional operators. Mech Syst Signal Process. 1991;5(2):81-88. https://doi.org/10.1016/0888-3270(91)90016-X
Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic Press;1998.
Momani S, Al-Khaled K, Numerical solutions for systems of fractional differential equations by the decomposition method. Appl Math Comput. 2005;162(3):1351-1365.
Odibat Z, Momani S, Application of variational iteration method to nonlinear differential equations of fractional order. Int J Nonlinear Sci Numer Simul. 2006;7(1):27-34. https://doi.org/10.1515/IJNSNS.2006.7.1.27
Hashim I, Abdulaziz O, Momani S. Homotopy analysis method for fractional IVPs. Commun Nonlinear Sci Numer Simul. 2009;14(3):674-684. https://doi.org/10.1016/j.cnsns.2007.09.014
Talib I, Belgacem FBM, Asif NA, Khalil H. On Mixed Derivatives Type High Dimensions Multi-Term Fractional Partial Differential Equations Approximate Solutions. Paper presented at; AIP Conference Proceedings; 2017. https://doi.org/10.1063/1.4972616
Talib I, Tunc C, Noor ZA. New operational matrices of orthogonal legendre polynomials and their operational. J Taibah Univ Sci. 2019;13(1):377-389. https://doi.org/10.1080/16583655.2019.1580662
Talib I. Nonlinear fractional partial coupled systems approximate solutions through operational matrices approach. Nonlin Stud. 2019;26(4):955-971.
Talib I, Alam N, Baleanu D, Zaidi D, Marriyam A. A new integral operational matrix with applications to multi-order fractional differential equations. Aims Math. 2021;6(8):8742-8771.
Alam N, Talib I, Bazighifan O, Chalishajar DN, Almarri B. An analytical technique implemented in the fractional clannish random walker’s parabolic equation with nonlinear physical phenomena. Math. 2021;9(8):801. https://doi.org/10.3390/math9080801
Talib I, Jarad F, Mirza FM, Nawaz A, Riaz MB. A generalized operational matrix of mixed partial derivative terms with applications to multi-order fractional partial differential equations. Alex Eng J. 2022;61(1):135-145. https://doi.org/10.1016/j.aej.2021.04.067
Chui CK. Wavelets: A mathematical tool for signal analysis. SIAM; 1997.
Shamsi M, Razzaghi M. Solution of Hallen’s integral equation using multiwavelets, Comput Phys Commun. 2005;168(3):187-197. https://doi.org/10.1016/j.cpc.2005.01.016
Lakestani M, Razzaghi M, Dehghan M. Semi orthogonal spline wavelets approximation for Fredholm integro-differential equations. Math Probl Eng. 2006;2006: e096184. https://doi.org/10.1155/MPE/2006/96184
Beylkin G, Coifman R, Rokhlin V. Fast wavelet transforms and numerical algorithms I. Commun Pure Appl Math. 1991;44(2):141–183. https://doi.org/10.1002/cpa.3160440202
Lepik U. Numerical solution of differential equations using Haar wavelets. Math Comput Simul. 2005;68(2):127-143. https://doi.org/10.1016/j.matcom.2004.10.005
Lepik U. Numerical solution of evolution equations by the haar wavelet method. Appl Math Comput. 2007;185(1):695–704. https://doi.org/10.1016/j.amc.2006.07.077
Lepik U. Application of the Haar wavelet transform to solving integral and differential equations. Proc Estonian Acad Sci Phys Math. 2007;56(1):28-46.
Hariharan G, Kannan K, Sharma KR. Haar wavelet in estimating depth profile of soil temperature. Appl Mat. Comput. 2009;210(1):119–125. https://doi.org/10.1016/j.amc.2008.12.036
Islam S, Aziz I, Sarler B. The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets. Math Comput Model. 2010;50(9-10):1577-1590. https://doi.org/10.1016/j.mcm.2010.06.023
S. C. Shiralashetti SC, Deshi AB, Desai PBM. Haar wavelet collocation method for the numerical solution of singular initial value problems. AIN Shams Eng J. 2016;7(2):663-670. https://doi.org/10.1016/j.asej.2015.06.006
Shiralashetti SC, Angadi LM, Deshi AB, Kantli MH. Haar wavelet method for the numerical solution of Klein-Gordan equations. Asian-EurJ Math. 2016;9(1): e1650012. https://doi.org/10.1142/S1793557116500121
Shiralashetti SC, Kantli MH, Deshi AB. Haar wavelet based numerical solution of nonlinear differential equations arising in fluid dynamics. Int J Comput Mater Sci Eng. 2016;5(2): e1650010. https://doi.org/10.1142/S204768411650010X
Shiralashetti SC, Deshi AB. An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equations. Nonlinear Dyn. 2016; 83:293-303. https://doi.org/10.1007/s11071-015-2326-4
Copyright (c) 2022 A. B. Deshi, G. A. Gudodagi
This work is licensed under a Creative Commons Attribution 4.0 International License.
