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Ranjbari, Sharareh, Baghmisheh, Mahdi, Jahangiri Rad, Mohammad, Nemati Saray, Behzad. (1404). On the wavelet Galerkin method for solving the fractional Fredholm integro-differential equations. سامانه مدیریت نشریات علمی, 13(3), 885-903. doi: 10.22034/cmde.2024.62193.2725
Sharareh Ranjbari; Mahdi Baghmisheh; Mohammad Jahangiri Rad; Behzad Nemati Saray. "On the wavelet Galerkin method for solving the fractional Fredholm integro-differential equations". سامانه مدیریت نشریات علمی, 13, 3, 1404, 885-903. doi: 10.22034/cmde.2024.62193.2725
Ranjbari, Sharareh, Baghmisheh, Mahdi, Jahangiri Rad, Mohammad, Nemati Saray, Behzad. (1404). 'On the wavelet Galerkin method for solving the fractional Fredholm integro-differential equations', سامانه مدیریت نشریات علمی, 13(3), pp. 885-903. doi: 10.22034/cmde.2024.62193.2725
Ranjbari, Sharareh, Baghmisheh, Mahdi, Jahangiri Rad, Mohammad, Nemati Saray, Behzad. On the wavelet Galerkin method for solving the fractional Fredholm integro-differential equations. سامانه مدیریت نشریات علمی, 1404; 13(3): 885-903. doi: 10.22034/cmde.2024.62193.2725

On the wavelet Galerkin method for solving the fractional Fredholm integro-differential equations

Computational Methods for Differential Equations
مقاله 12، دوره 13، شماره 3، مهر 2025، صفحه 885-903    اصل مقاله (607.03 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2024.62193.2725
نویسندگان
Sharareh Ranjbari1؛ Mahdi Baghmisheh* 1؛ Mohammad Jahangiri Rad1؛ Behzad Nemati Saray2
1Department of Mathematics, Ta.C., Islamic Azad University, Tabriz, Iran.
2Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran.
چکیده
An effective scheme is presented to estimate the numerical solution of fractional integro-differential equations (FIDEs). In the present method, to obtain the solution of the FIDEs, they must first be reduced to the corresponding Volterra Fredholm integral equations (VFIEs) with a weakly singular kernel. Then, by applying the matrix that represents the fractional integral (FI) based on biorthogonal Hermite cubic spline scaling bases (BHCSSb), and using the wavelet Galerkin method, the reduced problem can be solved. The combination of singularity and the challenge related to nonlinearity poses a formidable obstacle in solving the desired equations, but our method overcomes them well. An investigation of the method's convergence is provided, and it verifies that the convergence rate is $O(2^{-J})$ where $J\in \mathbb{N}_0$ is the refinement level. The verification of convergence has also been demonstrated through the presentation of several numerical examples. Compared to other methods, the results obtained show better accuracy.
کلیدواژه‌ها
Wavelet Galerkin method؛ Fractional integro-differential equation؛ Biorthogonal wavelet؛ Hermite cubic splines؛ Convergence analysis
مراجع
  • [1] B. Alpert, G. Beylkin, R. R. Coifman, and V. Rokhlin, Wavelet-like bases for the fast solution of second-kind integral equations, SIAM J. Sci. Statist. Comput., 14(1) (1993), 159–184.
  • [2] H. Aminikhah, A new analytical method for solving systems of linear integro-differential equations, J. King Saud Univer. Sci., 23(4) (2011), 349–353.
  • [3] C. N. Angstmann, B. I. Henry, and A. V. McGann, A fractional order recovery SIR model from a stochastic process, Bull. Math. Biol., 78(3) (2016), 468–499.
  • [4] A. Arikoglu and I. Ozkol, Solution of fractional integro-differential equations by using fractional differential transform method, Chaos, Solitons & Fractals, 40 (2009), 521–529.
  • [5] M. Asadzadeh and B. N. Saray, On a multiwavelet spectral element method for integral equation of a generalized Cauchy problem, BIT Numerical Mathematics, 62 (2022), 383-–1416.
  • [6] M. S. Asl, M. Javidi, and Y. Yan, A novel high-order algorithm for the numerical estimation of fractional differential equations, J. Comput. Appl. Math., 342 (2018), 180–201.
  • [7] R. L. Bagley and P. J. Torvik, Fractional Calculus in the Transient Analysis of Viscoelastically Damped Structures, AIAA. J., 23(6) (1985), 918–925.
  • [8] R. T. Baillie, Long Memory Processes and Fractional Integration in Econometrics, J. Econometrics, 73(1) (1996), 5–59.
  • [9] H. Bin Jebreen and I. Dassios, A Biorthogonal Hermite Cubic Spline Galerkin Method for Solving Fractional Riccati Equation, Mathematics, 10(9) (2022), 1461.
  • [10] V. Daftardar-Gejji, Y. Sukale, and S. Bhalekar, A new predictorcorrector method for fractional differential equations, Appl. Math. Comput., 244 (2014), 158–182.
  • [11] W. Dahmen, B. Han, R.Q. Jia, and A. Kunoth, Biorthogonal multiwavelets on the interval: Cubic Hermite splines, Constr. Approx., 16 (2000), 221–259.
  • [12] K. Diethelm, The analysis of fractional differential equations, An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer-Verlag Berlin Heidelberg, 2010.
  • [13] K. Diethelm, N. Ford, and A. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36(1) (2004), 31–52.
  • [14] M. R. Eslahchi, M. Dehghan, and M. Parvizi, Application of the collocation method for solving nonlinear fractional integro-differential equations, J. Comput. Appl. Math., 257 (2014), 105–128.
  • [15] J. H. He, Nonlinear Oscillation with Fractional Derivative and its Applications, Interna-tional Conference on Vibrating Engineering, (1998), 288–291.
  • [16] J. H. He, Some Applications of Nonlinear Fractional Differential Equations and Their Approximations, Bull. Sci. Technol., 2 (1999), 86–90.
  • [17] A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, 24. Elsevier B. V., Amsterdam, 2006.
  • [18] X. Li, Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method, Commun. Nonlinear Sci. Numer. Simulat., 17(10) (2012), 3934–3946.
  • [19] F. Mainardi, Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics. Fractals and Fractional Calculus in Continuum Mechanics, New York, Springer Verlag, 1997.
  • [20] B. Mandelbrot, Some Noises with 1/f Spectrum, a Bridge between direct Current and White Noise, IEEE Trans Inform Theory, 2 (1967), 289–298.
  • [21] M. Meerschaert and C. Tadjeran, Finite Difference Approximations for Two-Sided Space-Fractional Partial Differential Equations, Appl. Numer. Math., 56(1) (2006), 80–90.
  • [22] E. M. Mendes, G. H. Salgado, and L. A. Aguirre, Numerical solution of caputo fractional differential equations with infinity memory effect at initial condition, Commun. Nonlinear Sci. Numer. Simulat., 69 (2019), 237–247.
  • [23] R. Metzler and J. Klafter, The Restaurant at the End of the Random Walk: Recent Developments in the Description of Anomalous Transport by Fractional Dynamics, Journal of Physics A: Mathematical and General, 37 (2004), 161–208.
  • [24] S. Momani and K. Al-Khaled, Numerical Solutions for Systems of Fractional Differential Equations by the Decomposition Method, Appl. Math. Comput., 162(3) (2005), 1351–1365.
  • [25] S. Momani and M. A. Noor, Numerical methods for fourth order fractional integro-differential equations, Appl. Math. Comput., 182 (2006), 754–760.
  • [26] S. Momani and A. Qaralleh, An Efficient Method for Solving Systems of Fractional Integro-Differential Equations, Comput. Math. Appl., 52 (2006), 459–470.
  • [27] E. Pourfattah, M. Jahangiri Rad, and B. Nemati Saray, An efficient algorithm based on the Pseudospectral method for solving Abel’s integral equation using Hermite cubic spline scaling bases, Appl. Numer. Math., 185 (2023), 434-445.
  • [28] S. Paseban-Hag, E. Osgooei, and E. Ashpazzadeh, Alpert wavelet system for solving fractional nonlinear Fredholm integro-differential equations, Comput. Methods Differ. Equ., 9(3) (2021), 762–773.
  • [29] P. Rahimkhani, Y. Ordokhani, and E. Babolian, A New Operational Matrix based on Bernoulli Wavelets for Solving Fractional Delay Differential Equations, Numer. Algorithms., 74 (2017), 223–245.
  • [30] E. A. Rawashdeh, Numerical solution of fractional integro-differential equations by collocation method, Appl. Math. Comput., 176 (2006), 1–6.
  • [31] Y. A. Rossikhin and M. V. Shitikova, Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids, Appl. Mech. Rev., 50(1) (1997), 15–67.
  • [32] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual method for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), 856–869.
  • [33] H. Saeedi, M. Mohseni Moghadam, N. Mollahasani, and G. N. Chuev, A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order , Commun. Nonl. Sci. Numer. Simul., 16 (2011), 1154– 1163.
  • [34] P. K. Sahu and S. S. Ray, A numerical approach for solving nonlinear fractional Volterra Fredholm integrodifferential equations with mixed boundary conditions, Int. J. Wavelets Multi., 14(5) (2016), 1650036.
  • [35] G. H. O. Salgado and L. A. Aguirre, A hybrid algorithm for Caputo fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat., 33 (2016), 133–140.
  • [36] B. N. Saray, Abel’s integral operator: sparse representation based on multiwavelets, BIT Numerical Mathematics, 61 (2021), 587–606.
  • [37] B. N. Saray, An effcient algorithm for solving Volterra integro-differential equations based on Alpert’s multiwavelets Galerkin method, J. Comput. Appl. Math., 348 (2019), 453-465.
  • [38] Y. Yan, K. Pal, and N. Ford, Higher order numerical methods for solving fractional differential equations, BIT Numerical Mathematics, 54 (2014) 555–584.
  • [39] L. Zhu and Q. Fan, Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 2333–2341.
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