Memory Effects Due to Fractional Time Derivative and Integral Space in Diffusion Like Equation Via Haar Wavelets
Applied and Computational Mathematics
Volume 5, Issue 4, August 2016, Pages: 177-185
Received: Aug. 6, 2016; Accepted: Aug. 15, 2016; Published: Sep. 2, 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
Memory and hereditary effects due to fractional time derivative are combined with the global behaviours due to space integral term. Haar wavelet operational matrix is adjusted to solve diffusion like equations with time fractional derivative, space derivatives and integral terms. The fractional derivative is understood in the Caputo sense. The memory behaviours is included in all the points of the domain due to the existence of space integral term and the inverse fractional operator treatment and this is ilustrated in error graphs introduced. A general example with four subproblems ranging from the simple classical heat equation to the fractional time diffusion equation with global integral term is proposed and the calculated results are displayed graphically.
Keywords
Haar Wavelet, Operational Matrix, Fractional Derivative, Diffusion Like Equation
To cite this article
I. K. Youssef, A. R. A. Ali, Memory Effects Due to Fractional Time Derivative and Integral Space in Diffusion Like Equation Via Haar Wavelets, Applied and Computational Mathematics. Vol. 5, No. 4, 2016, pp. 177-185. doi: 10.11648/j.acm.20160504.12
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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]
E. M. E. Elbarbary, S. M. Elsayed, I. K. Youssef, A. M. Khodier, Restrictive Chebyshev rational approximation and applications to heat-conduction problems, Applied Mathematics and Computation 136 (2-3) (2003) 395-403.
[2]
K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Mathematics and its Applications, Springer Netherlands, 1982.
[3]
A. M. A. El-Sayed, Fractional-order diffusion-wave equation, International Journal of Theoretical Physics, Springer Netherlands, 35 (2) (1996) 311-322.
[4]
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.
[5]
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.
[6]
I. K. Youssef, A. R. A. Ali, Memory Effects in Diffusion Like Equation Via Haar Wavelets, Pure and Applied Mathematics Journal, 5 (4), (2016) 130-140.
[7]
I. Podlubny, Fractional Differential Equations, Camb. Academic Press, San Diego, CA, 1999.
[8]
K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: Wiley-Interscience Publ., 1993.
[9]
Ü. Lepik, Solving fractional integral equations by the haar wavelet method, Applied Mathematics and Computation 214 (2) (2009) 468–478.
[10]
I. K. Youssef, A. M. Shukur, Precondition for discretized fractional boundary value problem, Pure and Applied Mathematics Journal 3 (1) (2014) 1-6.
[11]
G. D. Smith, Numerical Solution of Partial Differential Equations Finite Difference Methods, Oxford University Press, 1978.
[12]
C. K. Chui, An introduction to wavelets, Vol. 1, Academic press, 2014.
[13]
Ü. Lepik, Buckling of elastic beams by the haar wavelet method, Estonian Journal of Engineering 17 (3) (2011) 271–284.
[14]
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.
[15]
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.
[16]
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.
[17]
Ü. Lepik, Numerical solution of differential equations using haar wavelets, Mathematics and computers in simulation 68 (2) (2005) 127–143.
[18]
Ü. Lepik, Numerical solution of evolution equations by the haar wavelet method, Applied Mathematics and Computation 185 (1) (2007) 695–704.
[19]
C. H. Hsiao, Haar wavelet direct method for solving variational problems, Mathematics and Computers in Simulation 64 (5) (2004) 569–585.
[20]
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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