The Müntz–Legendre Wavelet Collocation Method for Solving Weakly Singular Integro-Differential Equations with Fractional Derivatives
Abstract
:1. Introduction
2. Müntz–Legendre Wavelets
Operational Matrix of Fractional Integration
3. Wavelet Collocation Method
Error Analysis
4. Numerical Simulations and Results
5. Conclusions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
Abbreviations | |
WSIDE | Weakly singular integro-differential equations with fractional derivatives |
ML | Müntz–Legendre |
RL | Riemann–Liouville |
FI | Fractional integration |
Nomenclatures | |
Space of Müntz–Legendre polynomials | |
Space of continuous functions on | |
Müntz–Legendre polynomials | |
Space of Müntz–Legendre wavelets | |
s | Refinement level |
r | Multiplicity |
Projection operator | |
Riemann–Liouville fractional integration | |
Müntz–Legendre wavelets | |
Residual function |
References
- Zhao, J.; Xiao, J.; Ford, N.J. Collocation methods for fractional integro-differential equations with weakly singular kernels. Numer. Algorithms 2014, 65, 723–743. [Google Scholar] [CrossRef]
- Angstmann, C.N.; Henry, B.I.; McGann, A.V. A fractional order recovery SIR model from a stochastic process. Bull. Math. Biol. 2016, 78, 468–499. [Google Scholar] [CrossRef] [PubMed]
- Eslahchi, M.R.; Dehghan, M.; Parvizi, M. Application of the collocation method for solving nonlinear fractional integro-differential equations. J. Comput. Appl. Math. 2014, 257, 105–128. [Google Scholar] [CrossRef]
- Aminikhah, H. A new analytical method for solving systems of linear integro-differential equations. J. King Saud Univ. Sci. 2011, 23, 349–353. [Google Scholar] [CrossRef]
- Arikoglu, A.; Ozkol, I. Solution of fractional integro-differential equations by using fractional differential transform method. Chaos Solitons Fractals 2009, 40, 521–529. [Google Scholar] [CrossRef]
- Momani, S.; Noor, M.A. Numerical methods for fourth order fractional integro-differential equations. Appl. Math. Comput. 2006, 182, 754–760. [Google Scholar] [CrossRef]
- Momani, S.; Qaralleh, A. An Efficient Method for Solving Systems of Fractional Integro-Differential Equations. Comput. Math. Appl. 2006, 52, 459–470. [Google Scholar] [CrossRef]
- Rawashdeh, E.A. Numerical solution of fractional integro-differential equations by collocation method. Appl. Math. Comput. 2006, 176, 1–6. [Google Scholar] [CrossRef]
- Chow, T.S. Fractional dynamics of interfaces between soft-nanoparticles and rough substrates. Phys. Lett. A 2005, 342, 148–155. [Google Scholar] [CrossRef]
- Mandelbrot, B. Some noises with 1/f spectrum, a bridge between direct current and white noise. IEEE Trans. Inform. Theory 1967, 13, 289–298. [Google Scholar] [CrossRef]
- Magin, R.L. Fractional Calculus in Bioengineering, Illustrated ed.; Begell House: Danbury, CT, USA, 2006. [Google Scholar]
- He, J.H. Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 1999, 15, 86–90. [Google Scholar]
- Rossikhin, Y.A.; Shitikova, M.V. Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 1997, 50, 15–67. [Google Scholar] [CrossRef]
- He, J.H. Nonlinear oscillation with fractional derivative and its applications. In Proceedings of the International Conference on Vibrating Engineering’98, Dalian, China, 25–28 May 1998; pp. 288–291. [Google Scholar]
- Metzler, R.; Klafter, J. The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 2004, 37, 161–208. [Google Scholar] [CrossRef]
- Mainardi, F. Fractional calculus: Some basic problems in continuum and statistical mechanics. In Fractals and Fractional Calculus in Continuum Mechanics; Carpinteri, A., Mainardi, F., Eds.; Springer: New York, NY, USA, 1997. [Google Scholar]
- Baillie, R.T. Long memory processes and fractional integration in econometrics. J. Econom. 1996, 73, 5–59. [Google Scholar] [CrossRef]
- Alquran, M.; Jaradat, H.M.; Syam, M.I. Analytical solution of the time-fractional Phi-4 equation by using modified residual power series method. Nonlinear Dynam. 2017, 90, 2525–2529. [Google Scholar] [CrossRef]
- El-Ajou, A.; Arqub, O.A.; Al Zhour, Z.; Momani, S. New results on fractional power series: Theories and applications. Entropy 2013, 15, 5305–5323. [Google Scholar] [CrossRef]
- Qazza, A.; Saadeh, R.; Salah, E. Solving fractional partial differential equations via a new scheme. AIMS Math. 2022, 8, 5318–5337. [Google Scholar] [CrossRef]
- Zhang, Y. A finite difference method for fractional partial differential equation. Appl. Math. Comput. 2009, 215, 524–529. [Google Scholar] [CrossRef]
- Bonyadi, S.; Mahmoudi, Y.; Lakestani, M.; Jahangiri rad, M. Numerical solution of space-time fractional PDEs with variable coefficients using shifted Jacobi collocation method. Comput. Methods Differ. Equ. 2023, 11, 81–94. [Google Scholar]
- Shahriari, M.; Saray, B.N.; Mohammadalipour, B.; Saeidian, S. Pseudospectral method for solving the fractional one-dimensional Dirac operator using Chebyshev cardinal functions. Phys. Scr. 2023, 98, 055205. [Google Scholar] [CrossRef]
- Yang, X.; Wu, L.; Zhang, H. A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity. Appl. Math. Comput. 2023, 457, 128192. [Google Scholar] [CrossRef]
- Zhang, H.; Yang, X.; Tang, Q.; Xu, D. A robust error analysis of the OSC method for a multi-term fourth-order sub-diffusion equation. Comput. Math. Appl. 2022, 109, 180–190. [Google Scholar] [CrossRef]
- Asadzadeh, M.; Saray, B.N. On a multiwavelet spectral element method for integral equation of a generalized Cauchy problem. BIT 2022, 62, 383–1416. [Google Scholar] [CrossRef]
- Li, C.; Li, Z.; Wang, Z. Mathematical analysis and the local discontinuous Galerkin method for Caputo–Hadamard fractional partial differential equation. J. Sci. Comput. 2020, 85, 41. [Google Scholar] [CrossRef]
- Mao, Z.; Shen, J. Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients. J. Comput. Phys. 2016, 307, 243–261. [Google Scholar] [CrossRef]
- Ford, N.J.; Xiao, J.; Yan, Y. A finite element method for time fractional partial differential equations. Fract. Calc. Appl. Anal. 2011, 14, 454–474. [Google Scholar] [CrossRef]
- Shah, N.A.; El-Zahar, E.R.; Akgül, A.; Khan, A.; Kafle, J. Analysis of Fractional-Order Regularized Long-Wave Models via a Novel Transform. J. Funct. Spaces 2022, 2022, 2754507. [Google Scholar] [CrossRef]
- Alpert, B.; Beylkin, G.; Coifman, R.R.; Rokhlin, V. Wavelet-like bases for the fast solution of second-kind integral equations. SIAM J. Sci. Stat. Comput. 1993, 14, 159–184. [Google Scholar] [CrossRef]
- Heller, V.; Strang, G.; Topiwala, P.N.; Heil, C. The application of multiwavelet filterbanks to image processing. IEEE Trans. Image Process. 1999, 8, 548–563. [Google Scholar]
- Saray, B.N. Abel’s integral operator: Sparse representation based on multiwavelets. BIT Numer. Math. 2021, 61, 587–606. [Google Scholar] [CrossRef]
- Saray, B.N. An effcient algorithm for solving Volterra integro-differential equations based on Alpert’s multi-wavelets Galerkin method. J. Comput. Appl. Math. 2019, 348, 453–465. [Google Scholar] [CrossRef]
- Saray, B.N. Sparse multiscale representation of Galerkin method for solving linear-mixed Volterra-Fredholm integral equations. Math. Method Appl. Sci. 2020, 43, 2601–2614. [Google Scholar] [CrossRef]
- Rahimkhani, P.; Ordokhani, Y.; Babolian, E. Müntz-Legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations. Numer. Algorithms 2018, 77, 1283–1305. [Google Scholar] [CrossRef]
- Jebreen, H.B.; Tchier, F. A New Scheme for Solving Multiorder Fractional Differential Equations Based on Müntz–Legendre Wavelets. Complexity 2021, 2021, 9915551. [Google Scholar] [CrossRef]
- Rahimkhani, P.; Ordokhani, Y. Numerical solution a class of 2D fractional optimal control problems by using 2D Müntz-Legendre wavelets. Optim. Contr. Appl. Met. 2018, 39, 1916–1934. [Google Scholar] [CrossRef]
- Almira, J.M. Müntz type theorems. I Surv. Approx. Theory 2007, 3, 152–194. [Google Scholar]
- Müntz, C.H. Über den Approximationssatz von Weierstrass. In Mathematische Abhandlungen Hermann Amandus Schwarz; Springer: Berlin/Heidelberg, Germany, 1914; pp. 303–312. [Google Scholar]
- Shen, J.; Wang, Y. Müntz-Galerkin methods and applicationa to mixed dirichlet-neumann boundary value problems. Siam J. Sci. Comput. 2016, 38, 2357–2381. [Google Scholar] [CrossRef]
- Borwein, P.; Erdélyi, T.; Zhang, J. Müntz systems and orthogonal Müntz–Legendre polynomials. Trans. Am. Math. Soc. 1994, 342, 523–542. [Google Scholar]
- Kilbas, A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations, 24; Elsevier B.V.: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Gu, X.M.; Huang, T.Z.; Zhao, Y.L.; Lyu, P.; Carpentieri, B. A fast implicit difference scheme for solving the generalized time–space fractional diffusion equations with variable coefficients. Numer. Methods Partial Differ. Equ. 2021, 37, 1136–1162. [Google Scholar] [CrossRef]
- Gu, X.M.; Sun, H.W.; Zhao, Y.L.; Zheng, X. An implicit difference scheme for time-fractional diffusion equations with a time-invariant type variable order. Appl. Math. Lett. 2021, 120, 107270. [Google Scholar] [CrossRef]
CPU Time | |||||||
---|---|---|---|---|---|---|---|
Chebyshev nodes | 5 | ||||||
9 | |||||||
Legendre nodes | 5 | ||||||
9 | |||||||
Uniform meshes | 5 | ||||||
9 |
Proposed Method | TCM [1] | |||
---|---|---|---|---|
Error |
CPU Time | |||||||
---|---|---|---|---|---|---|---|
Chebyshev nodes | 5 | ||||||
9 | |||||||
Legendre nodes | 5 | ||||||
9 | |||||||
Uniform meshes | 5 | ||||||
9 |
CPU Time | |||||||
---|---|---|---|---|---|---|---|
Chebyshev nodes | 12 | ||||||
20 | |||||||
Legendre nodes | 12 | ||||||
20 | |||||||
Uniform meshes | 12 | ||||||
20 |
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Bin Jebreen, H. The Müntz–Legendre Wavelet Collocation Method for Solving Weakly Singular Integro-Differential Equations with Fractional Derivatives. Fractal Fract. 2023, 7, 763. https://doi.org/10.3390/fractalfract7100763
Bin Jebreen H. The Müntz–Legendre Wavelet Collocation Method for Solving Weakly Singular Integro-Differential Equations with Fractional Derivatives. Fractal and Fractional. 2023; 7(10):763. https://doi.org/10.3390/fractalfract7100763
Chicago/Turabian StyleBin Jebreen, Haifa. 2023. "The Müntz–Legendre Wavelet Collocation Method for Solving Weakly Singular Integro-Differential Equations with Fractional Derivatives" Fractal and Fractional 7, no. 10: 763. https://doi.org/10.3390/fractalfract7100763