Memory Effects in Diffusion Like Equation Via Haar Wavelets
Pure and Applied Mathematics Journal
Volume 5, Issue 4, August 2016, Pages: 130-140
Received: Jul. 17, 2016; Accepted: Jul. 26, 2016; Published: Aug. 10, 2016
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Authors
I. K. Youssef, Department of Mathematics, Ain Shams University, Cairo, Egypt
A. R. A. Ali, Department of Mathematics, Baghdad University, Baghdad, Iraq
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Abstract
The memory and hereditary effects of fractional derivatives as well as integral terms are considered in a diffusion like problem. The Haar wavelet operational matrix technique is employed to solve fractional order diffusion equation with time dependent integral term and time dependent boundary condition. The fractional derivative is described in the Caputo sense. The effect of using inverse fractional operator which combines the memory behaviors of the fractional derivatives to all other terms in the equation is disscused. Different Haar bases functions are used (8, 16, 32, 64) and comparison of the wavelet operational matrix is considered. Error analysis is considered. A general numerical example with four subproblems is considered, graphical representation of the different solutions as well as their errors are given.
Keywords
Haar Wavelet, Operational Matrix, Fractional Derivative, Fractional Order Diffusion Equation
To cite this article
I. K. Youssef, A. R. A. Ali, Memory Effects in Diffusion Like Equation Via Haar Wavelets, Pure and Applied Mathematics Journal. Vol. 5, No. 4, 2016, pp. 130-140. doi: 10.11648/j.pamj.20160504.17
Copyright
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Mathematics and its Applications, Springer Netherlands, 1982.
[2]
I. K. Youssef, A. M. Shukur, Modified variation iteration method for fraction space-time partial differential heat and wave equations, International Journal 2 (2) (2013) 1000–1013.
[3]
Ü. Lepik, Solving fractional integral equations by the haar wavelet method, Applied Mathematics and Computation 214 (2) (2009) 468–478.
[4]
I. Podlubny, Fractional Differential Equations, Camb. Academic Press, San Diego, CA, 1999.
[5]
K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: Wiley-Interscience Publ., 1993.
[6]
I. K. Youssef, R. A. Ibrahim, Boundary Value Problems, Fredholm Integral equations, SOR and KSOR Methods, Life Science Journal 10 (2) (2013) 304-312.
[7]
I. K. Youssef, A. M. Shukur, Precondition for discretized fractional boundary value problem, Pure and Applied Mathematics Journal 3 (1) (2014) 1-6.
[8]
I. K. Youssef, A. M. Shukur, The line method combined with spectral chebyshev for space-time fractional diffusion equation, Applied and Computational Mathematics 3 (6) (2014) 330- 336.
[9]
G. D. Smith, Numerical Solution of Partial Differential Equations Finite Difference Methods, Oxford University Press, 1978.
[10]
C. K. Chui, An introduction to wavelets, Vol. 1, Academic press, 2014.
[11]
Ü. Lepik, Buckling of elastic beams by the haar wavelet method, Estonian Journal of Engineering 17 (3) (2011) 271-284.
[12]
C. F. Chen, C. H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc., Control Theory Appl. 144 (1) (1997) 87–94. doi: 10.1049/ip-cta:19970702.
[13]
C. F. Chen, C. H. Hsiao, Wavelet approach to optimising dynamic systems, IEE Proc., Control Theory Appl. 146 (2) (1999) 213–219. doi: 10.1049/ip-cta:19990516.
[14]
Ü. Lepik, Application of the haar wavelet transform to solving integral and differential equations, Proc. Estonian Acad. Sci. Phys. Math. 56 (1) (2007) 28-46.
[15]
Ü. Lepik, Numerical solution of differential equations using haar wavelets, Mathematics and computers in simulation 68 (2) (2005) 127-143.
[16]
Ü. Lepik, Numerical solution of evolution equations by the haar wavelet method, Applied Mathematics and Computation 185 (1) (2007) 695-704.
[17]
C. H. Hsiao, Haar wavelet direct method for solving variational problems, Mathematics and Computers in Simulation 64 (5) (2004) 569-585.
[18]
C. H. Hsiao, W. J. Wang, Haar wavelet approach to nonlinear stiff systems, Mathematics and computers in simulation 57 (6) (2001) 347-353
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