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Research Papers

A Comparative Study of Integer and Noninteger Order Wavelets for Fractional Nonlinear Fredholm Integro-Differential Equations

[+] Author and Article Information
F. Mohammadi

Department of Mathematics,
University of Hormozgan,
P. O. Box 3995,
Bandar Abbas 3995, Iran
e-mail: f.mohammadi62@hotmail.com

J. A. Tenreiro Machado

Department of Electrical Engineering,
Institute of Engineering,
Polytechnic of Porto,
Rua Dr. António Bernardino de Almeida,
Porto 4249-015, Portugal
e-mail: jtm@isep.ipp.pt

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 8, 2017; final manuscript received May 15, 2018; published online June 18, 2018. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(8), 081002 (Jun 18, 2018) (11 pages) Paper No: CND-17-1555; doi: 10.1115/1.4040343 History: Received December 08, 2017; Revised May 15, 2018

This paper compares the performance of Legendre wavelets (LWs) with integer and noninteger orders for solving fractional nonlinear Fredholm integro-differential equations (FNFIDEs). The generalized fractional-order Legendre wavelets (FLWs) are formulated and the operational matrix of fractional derivative in the Caputo sense is obtained. Based on the FLWs, the operational matrix and the Tau method an efficient algorithm is developed for FNFIDEs. The FLWs basis leads to more efficient and accurate solutions of the FNFIDE than the integer-order Legendre wavelets. Numerical examples confirm the superior accuracy of the proposed method.

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Topics: Wavelets
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References

Oldham, K. B. , and Spanier, J. , 1974, The Fractional Calculus, Academic Press, New York.
Podlubny, I. , 1999, Fractional Differential Equations, Academic Press, San Diego, CA. [PubMed] [PubMed]
Kilbas, A. A. , Srivastava, H. M. , and Trujillo, J. J. , 2006, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, CA.
Bagley, R. L. , and Torvik, P. J. , 1983, “ A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity,” J. Rheol., 27(3), pp. 201–210. [CrossRef]
Baillie, R. T. , 1996, “ Long Memory Processes and Fractional Integration in Econometrics,” J. Econometrics, 73(1), pp. 5–59. [CrossRef]
He, J. , 1998, “ Nonlinear Oscillation With Fractional Derivative and Its Applications,” International Conference Vibrating Engineering, Leuven, Belgium, Sept. 16–18, pp. 288–291.
He, J. , 1999, “ Some Applications of Nonlinear Fractional Differential Equations and Their Approximations,” Bull. Sci. Technol., 15(2), pp. 86–90.
Mainardi, F. , 1997, “ Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics,” Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi , eds., Springer-Verlag, Wien, Austria, pp. 291–348. [CrossRef]
Panda, R. , and Dash, M. , 2006, “ Fractional Generalized Splines and Signal Processing,” Signal Process, 86(9), pp. 2340–2350. [CrossRef]
Bohannan, G. , 2008, “ Analog Fractional Order Controller in Temperature and Motor Control Applications,” J. Vib. Control, 14(9–10), pp. 1487–1498. [CrossRef]
Kumar, S. , Kumar, D. , Abbasbandy, S. , and Rashidi, M. M. , 2014, “ Analytical Solution of Fractional Navier-Stokes Equation by Using Modified Laplace Decomposition Method,” Ain Shams Eng. J., 5(2), pp. 569–574. [CrossRef]
Li, Y. Y. , Zhao, Y. , Xie, G. N. , Baleanu, D. , Yang, X. J. , and Zhao, K. , 2014, “ Local Fractional Poisson and Laplace Equations With Applications to Electrostatics in Fractal Domain,” Adv. Math. Phys., 2014, p. 590574.
Arikoglu, A. , and Ozkol, I. , 2009, “ Solution of Fractional Integro-Differential Equations by Using Fractional Differential Transform Method,” Chaos Solitons Fractals, 40(2), pp. 521–529. [CrossRef]
Meerschaert, M. , and Tadjeran, C. , 2006, “ Finite Difference Approximations for Two-Sided Space-Fractional Partial Differential Equations,” Appl. Numer. Math., 56(1), pp. 80–90. [CrossRef]
Suarez, L. , and Shokooh, A. , 1997, “ An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives,” ASME J. Appl. Mech., 64(3), pp. 629–635. [CrossRef]
Doha, E. H. , Bhrawy, A. H. , and Ezz-Eldien, S. S. , 2012, “ A New Jacobi Operational Matrix: An Application for Solving Fractional Differential Equations,” Appl. Math. Modell., 36(10), pp. 4931–4943. [CrossRef]
Li, Y. , and Sun, N. , 2011, “ Numerical Solution of Fractional Differential Equations Using the Generalized Block Pulse Operational Matrix,” Comput. Math. Appl., 62(3), pp. 1046–1054. [CrossRef]
Tripathi, M. P. , Baranwal, V. K. , Pandey, R. K. , and Singh, O. P. , 2013, “ A New Numerical Algorithm to Solve Fractional Differential Equations Based on Operational Matrix of Generalized Hat Functions,” Commun. Nonlinear Sci. Numer. Simul., 18(6), pp. 1327–1340. [CrossRef]
Momani, S. , and Al-Khaled, K. , 2005, “ Numerical Solutions for Systems of Fractional Differential Equations by the Decomposition Method,” Appl. Math. Comput., 162(3), pp. 1351–1365.
Quan, X. J. , Han, H. L. , and Wang, J. , 2014, “ The Adomian Decomposition Method for Sloving Nonlinear Volterra Integral Equations of Fractional Order,” J. Jiangxi Normal Univ. (Natural Sci. Ed.), 5, p. 18.
Odibat, Z. , and Momani, S. , 2006, “ Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order,” Int. J. Nonlinear Sci. Numer. Simul., 7(1), pp. 27–34. [CrossRef]
Abdulaziz, O. , Hashim, I. , and Momani, S. , 2008, “ Solving Systems of Fractional Differential Equations by Homotopy-Perturbation Method,” Phys. Lett. A, 372(4), pp. 451–459. [CrossRef]
Hashim, I. , Abdulaziz, O. , and Momani, S. , 2009, “ Homotopy Analysis Method for Fractional IVPs,” Commun. Nonlinear Sci. Numer. Simul., 14(3), pp. 674–684. [CrossRef]
Baleanu, D. , Darzi, R. , and Agheli, B. , 2018, “ A Reliable Mixed Method for Singular Integro-Differential Equations of Non-Integer Order,” Math. Modell. Natural Phenom., 13(1). p. 4.
Yin, Y. , Yanping, C. , and Yunqing, H. , 2014, “ Convergence Analysis of the Jacobi Spectral-Collocation Method for Fractional Integro-Differential Equations,” Acta Math. Sci., 34(3), pp. 673–690. [CrossRef]
Mokhtary, P. , 2015, “ Reconstruction of Exponentially Rate of Convergence to Legendre Collocation Solution of a Class of Fractional Integro-Differential Equations,” J. Comput. Appl. Math., 279, pp. 145–158. [CrossRef]
Bhrawy, A. H. , and Zaky, M. A. , 2015, “ A Shifted Fractional-Order Jacobi Orthogonal Functions: An Application for System of Fractional Differential Equations,” Appl. Math. Modell., 40(2), pp. 832–845.
Bhrawy, B. A. H. , Alhamed, Y. A. , and Baleanu, D. , 2014, “ New Specral Techniques for Systems of Fractional Differential Equations Using Fractional-Order Generalized Laguerre Orthogonal Functions,” Fractional Calculus Appl. Anal., 17(4), pp. 1138–1157.
Kazem, S. , Abbasbandy, S. , and Kumar, S. , 2013, “ Fractional-Order Legendre Functions for Solving Fractional-Order Differential Equations,” Appl. Math. Modell., 37(7), pp. 5498–5510. [CrossRef]
Doha, E. H. , Bhrawy, A. H. , Baleanu, D. , Ezz-Eldien, S. S. , and Hafez, R. M. , 2015, “ An Efficient Numerical Scheme Based on the Shifted Orthonormal Jacobi Polynomials for Solving Fractional Optimal Control Problems,” Adv. Differ. Equations, 2015(1), pp. 1–17. [CrossRef]
Saeedi, H. , and Mohseni Moghadam, M. , 2011, “ Numerical Solution of Nonlinear Volterra Integro-Differential Equations of Arbitrary Order by CAS Wavelets,” Commun. Nonlinear. Sci. Numer. Simul, 16(3), pp. 1216–1226. [CrossRef]
Saeedi, H. , Moghadam, M. M. , Mollahasani, M. , and Chuev, G. N. , 2011, “ A CAS Wavelet Method for Solving Nonlinear Fredholm Integro-Differential Equations of Fractional Order,” Commun. Nonlinear. Sci. Numer. Simul., 16(3), pp. 1154–1163. [CrossRef]
Zhu, L. , and Fan, Q. , 2012, “ Solving Fractional Nonlinear Fredholm Integro-Differential Equations by the Second Kind Chebyshev Wavelet,” Commun. Nonlinear. Sci. Numer. Simul., 17(6), pp. 2333–2341. [CrossRef]
Zhu, L. , and Fan, Q. , 2013, “ Numerical Solution of Nonlinear Fractional-Order Volterra Integro-Differential Equations by SCW,” Commun. Nonlinear. Sci. Numer. Simul., 18(5), pp. 1203–1213. [CrossRef]
Mohammadi, F. , 2014, “ Numerical Solution of Bagley-Torvik Equation Using Chebyshev Wavelet Operational Matrix of Fractional Derivative,” Int. J. Adv. Appl. Math. Mech., 2(1), pp. 83–91.
Shiralashetti, S. C. , and Deshi, A. B. , 2016, “ An Efficient Haar Wavelet Collocation Method for the Numerical Solution of Multi-Term Fractional Differential Equations,” Nonlinear Dyn., 83(1–2), pp. 293–303. [CrossRef]
Wang, Y. , and Fan, Q. , 2012, “ The Second Kind Chebyshev Wavelet Method for Solving Fractional Differential Equations,” Appl. Math. Comput., 218(17), pp. 8592–8601.
Heydari, M. H. , Hooshmandasl, M. R. , and Mohammadi, F. , 2014, “ Two-Dimensional Legendre Wavelets for Solving Time-Fractional Telegraph Equation,” Adv. Appl. Math. Mech., 6(02), pp. 247–260. [CrossRef]
Mohammadi, F. , and Adhami, P. , 2016, “ Numerical Study of Stochastic Volterra-Fredholm Integral Equations by Using Second Kind Chebyshev Wavelets,” Random Operators Stochastic Equations, 24(2), pp. 129–141. [CrossRef]
Razzaghi, M. , and Yousefi, S. , 2001, “ The Legendre Wavelets Operational Matrix of Integration,” Int. J. Syst. Sci., 32 (4), pp. 495–502. [CrossRef]
Mohammadi, F. , and Hosseini, M. M. , 2011, “ A Comparative Study of Numerical Methods for Solving Quadratic Riccati Differential Equations,” J. Franklin Inst., 348(2), pp. 156–164. [CrossRef]
Mohammadi, F. , 2011, “ A New Legendre Wavelet Operational Matrix of Derivative and Its Applications in Solving the Singular Ordinary Differential Equations,” J. Franklin Inst., 348 (8), pp. 1787–1796. [CrossRef]
Mohammadi, F. , Hosseini, M. M. , and Mohyud-Din, S. T. , 2011, “ Legendre Wavelet Galerkin Method for Solving Ordinary Differential Equations With Non-Analytic Solution,” Int. J. Syst. Sci., 42(4), pp. 579–585. [CrossRef]
Mohammadi, F. , 2016, “ A Computational Wavelet Method for Numerical Solution of Stochastic Volterra-Fredholm Integral Equations,” Wavelet Linear Algebra, 3(1), pp. 13–25. http://wala.vru.ac.ir/article_19924.html
Mohammadi, F. , and Ciancio, A. , 2017, “ Wavelet-Based Numerical Method for Solving Fractional Integro-Differential Equation With a Weakly Singular Kernel,” Wavelet Linear Algebra, 4(1), pp. 53–73.
Xu, X. , and Xu, D. , 2017, “ Legendre Wavelets Method for Approximate Solution of Fractional-Order Differential Equations Under Multi-Point Boundary Conditions,” Int. J. Comput. Math., 95(5), pp. 998–1014.
Liu, N. , and Lin, E. B. , 2010, “ Legendre Wavelet Method for Numerical Solutions of Partial Differential Equations,” Numer. Methods Partial Differ. Equations, 26(1), pp. 81–94. [CrossRef]
Canuto, C. , Hussaini, M. , Quarteroni, A. , and Zang, T. , 1988, Spectral Methods in Fluid Dynamics, Springer, Berlin. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

(a) The exact and numerical solutions with M = 6, k = 1, and α=0.5 and (b) the absolute error of approximate solutions for several values of α and M=6,k=1 (Example 1)

Grahic Jump Location
Fig. 2

(a) Numerical solution for different values of ν=α with M = 6, k = 1 and (b) the absolute error of approximate solutions for ν = 1 and different values of the fractional order α, with M=6,k=1 (Example 2)

Grahic Jump Location
Fig. 3

The absolute value of the residual functions for different values of the fractional orders α and ν, with M=6,k=1: (a) ν=0.5, (b) ν=0.75, and (c) ν=0.95 (Example 2)

Grahic Jump Location
Fig. 4

(a) Numerical solution for different values of ν=α with M = 10, k = 1 and (b) the absolute error of approximate solutions for ν = 1 and different values of fractional oreder α with M=10,k=1 (Example 3)

Grahic Jump Location
Fig. 5

The absolute value of the residual function for various values of α, with M = 10 and k = 1: (a) ν=0.5, (b) ν=0.75, and (c) ν=0.95 (Example 3)

Grahic Jump Location
Fig. 6

(a) The approximate solutions for different values of ν=α and M = 14, k = 1 and (b) the absolute error for various ν=α and M = 14 and k = 1 (Example 4)

Grahic Jump Location
Fig. 7

The absolute value of the residual function for different values of α and M = 14 and k = 1: (a) ν=0.90 and (b) ν=0.95 (Example 4)

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