Convergence of collocation Bernoulli wavelet method in solving nonlinear Fredholm integro-differential equations of fractional order
Subject Areas : Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering
Abdolali Rooholahi
1
,
Saeed Akhavan
2
1 - a Department of Mathematics, Lorestan university, Khorramabad, Iran
2 - Department of Mathematics, Khomeinishahr Branch,
Islamic Azad University, Khomeinishahr/Isfahan, Iran
Keywords: Fractional calculus, Bernoulli wavelets, Fredholm integro-differential equations, collocation method.,
Abstract :
We provide a computer method for solving fractional order nonlinear Fredholm integro-differential equations in this study. This method transforms the core issue into a set of algebraic equations using Bernoulli wavelets. The operational Bernoulli wavelet with fractional integration is obtained and used. It works particularly well for technical applications. The convergence of the suggested strategy is the most crucial aspect to note here. The collocation approach for this issue has a unique approximation since these requirements can be shown using mathematical principles and matrices theory. Finally, some pertinent examples for which the exact solution is known are used in numerical simulation to confirm the effectiveness and relevance. Alternatively, these examples will demonstrate the viability and correctness of the suggested approach. We provide a computer method for solving fractional order nonlinear Fredholm integro-differential equations in this study. This method transforms the core issue into a set of algebraic equations using Bernoulli wavelets. The operational Bernoulli wavelet with fractional integration is obtained and used. It works particularly well for technical applications. The convergence of the suggested strategy is the most crucial aspect to note here. The collocation approach for this issue has a unique approximation since these requirements can be shown using mathematical principles and matrices theory. Finally, some pertinent examples for which the exact solution is known are used in numerical simulation to confirm the effectiveness and relevance. Alternatively, these examples will demonstrate the viability and correctness of the suggested approach.
[1] Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations (Vol. 204). elsevier.
[2] Podlubny, I. (1999). Fractional differential equations, mathematics in science and engineering.
[3] Panda, R., & Dash, M. (2006). Fractional generalized splines and signal processing. Signal Processing, 86(9), 2340-2350.
[4] Das, S. (2011). Functional fractional calculus (Vol. 1). Berlin: Springer.
[5] Bohannan, G. W. (2008). Analog fractional order controller in temperature and motor control applications. Journal of Vibration and Control, 14(9-10), 1487-1498.
[6] Bagley, R. L., & Torvik, P. J. (1983). A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology, 27(3), 201-210.
[7] Farhatnia, F., & Golshah, A. (2008). Investigation on Buckling of Orthotropic Circular and Annular Plates of Continuously Variable Thickness by Optimized Ritz Method. Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering, 1(2), 31-40.
[8] Alimoradzadeh, M., Salehi, M., & Mohammadi Esfarjani, S. (2017). Vibration Analysis of FG Micro-Beam Based on the Third Order Shear Deformation and Modified Couple Stress Theories. Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering, 10(3), 51-66.
[9] Fitt, A. D., Goodwin, A. R. H., Ronaldson, K. A., & Wakeham, W. A. (2009). A fractional differential equation for a MEMS viscometer used in the oil industry. Journal of Computational and Applied Mathematics, 229(2), 373-381.
[10] Daftardar-Gejji, V., & Jafari, H. (2007). Solving a multi-order fractional differential equation using Adomian decomposition. Applied Mathematics and Computation, 189(1), 541-548.
[11] Daftardar-Gejji, V., & Jafari, H. (2005). Adomian decomposition: a tool for solving a system of fractional differential equations. Journal of Mathematical Analysis and Applications, 301(2), 508-518.
[12] Yang, C., & Hou, J. (2013). An approximate solution of nonlinear fractional differential equation by Laplace transform and Adomian polynomials. Journal of information and computational science, 10(1), 213-222.
[13] Wang, Q. (2007). Homotopy perturbation method for fractional KdV equation. Applied Mathematics and Computation, 190(2), 1795-1802.
[14] Hemeda, A. A. (2014). Modified homotopy perturbation method for solving fractional differential equations. Journal of Applied Mathematics, 2014.
[15] Abdulaziz, O., Hashim, I., & Momani, S. (2008). Solving systems of fractional differential equations by homotopy-perturbation method. Physics Letters A, 372(4), 451-459.
[16] Jafari, H., & Seifi, S. (2009). Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation. Communications in Nonlinear Science and Numerical Simulation, 14(5), 2006-2012.
[17] Jafari, H., Das, S., & Tajadodi, H. (2011). Solving a multi-order fractional differential equation using homotopy analysis method. Journal of King Saud University-Science, 23(2), 151-155.
[18] Hemeda, A. A. (2013, January). New iterative method: an application for solving fractional physical differential equations. In Abstract and applied analysis (Vol. 2013). Hindawi.
[19] Scherer, R., Kalla, S. L., Tang, Y., & Huang, J. (2011). The Grünwald–Letnikov method for fractional differential equations. Computers & Mathematics with Applications, 62(3), 902-917.
[20] Heydari, M. H., Hooshmandasl, M. R., Maalek Ghaini, F. M., & Li, M. (2013). Chebyshev wavelets method for solution of nonlinear fractional integrodifferential equations in a large interval. Advances in Mathematical Physics, 2013.
[21] Doha, E. H., Bhrawy, A. H., & Ezz-Eldien, S. S. (2011). A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Computers & Mathematics with Applications, 62(5), 2364-2373.
[22] Ur Rehman, M., & Khan, R. A. (2011). The Legendre wavelet method for solving fractional differential equations. Communications in Nonlinear Science and Numerical Simulation, 16(11), 4163-4173.
[23] Yuanlu, L. I. (2010). Solving a nonlinear fractional differential equation using Chebyshev wavelets. Communications in Nonlinear Science and Numerical Simulation, 15(9), 2284-2292.
[24] Mohyud-Din, S. T., Khan, H., Arif, M., & Rafiq, M. (2017). Chebyshev wavelet method to nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions. Advances in Mechanical Engineering, 9(3), 1687814017694802.
[25] Akhavan, S. (2021). Convergence of Legendre and Chebyshev multiwavelets in Petrov-Galerkin method for solving Fredholm integro-differential equations of high orders. Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering, 13(3), 33-42.
[26] Li, Y., & Zhao, W. (2010). Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Applied mathematics and computation, 216(8), 2276-2285.
[27] Saeedi, H., Moghadam, M. M., Mollahasani, N., & Chuev, G. (2011). A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order. Communications in nonlinear science and numerical simulation, 16(3), 1154-1163.
[28] Yi, M., & Huang, J. (2015). CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel. International journal of computer mathematics, 92(8), 1715-1728.
[29] Zhu, L., & Fan, Q. (2012). Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet. Communications in nonlinear science and numerical simulation, 17(6), 2333-2341.
[30] Zhu, L., & Fan, Q. (2012). Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet. Communications in nonlinear science and numerical simulation, 17(6), 2333-2341.
[31] Meng, Z., Wang, L., Li, H., & Zhang, W. (2015). Legendre wavelets method for solving fractional integro-differential equations. International Journal of Computer Mathematics, 92(6), 1275-1291.
[32] Saeedi, H. (2013). Application of Haar wavelets in solving nonlinear fractional Fredholm integro-differential equations. Journal of Mahani Mathematical Research, 2(1), 15-28.
[33] Keshavarz, E., Ordokhani, Y., & Razzaghi, M. (2019). Numerical solution of nonlinear mixed Fredholm-Volterra integro-differential equations of fractional order by Bernoulli wavelets. Computational Methods for Differential Equations, 7(2), 163-176.
[34] Costabile, F., Dell'Accio, F., & Gualtieri, M. I. (2006). A new approach to Bernoulli polynomials. Rendiconti di Matematica e delle sue Applicazioni, 26, 1-12.
[35] Diethelm, K. (1997). An algorithm for the numerical solution of differential equations of fractional order. Electronic transactions on numerical analysis, 5(1), 1-6.
[36] Zhu, L., & Fan, Q. (2012). Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet. Communications in nonlinear science and numerical simulation, 17(6), 2333-2341.
[37] Chui, C. K. (1997). Wavelets: a mathematical tool for signal analysis. Society for Industrial and Applied Mathematics.
[38] Daftardar‐Gejji, V., & Bhalekar, S. (2007, December). An iterative method for solving fractional differential equations. In PAMM: Proceedings in Applied Mathematics and Mechanics (Vol. 7, No. 1, pp. 2050017-2050018). Berlin: WILEY‐VCH Verlag.
[39] Zhu, L., & Fan, Q. (2012). Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet. Communications in nonlinear science and numerical simulation, 17(6), 2333-2341.