Combination of Lucas wavelets with Legendre-Gauss quadrature for fractional Fredholm-Volterra integro-differential equations

Abstract In this paper, the numerical technique with the help of the Lucas wavelets (LWs) and the Legendre–Gauss quadrature rule is presented to study the solution of fractional Fredholm–Volterra integro-differential equations. The modified operational matrices of integration and pseudo-operational of fractional derivative for the proposed wavelet functions are calculated. These matrices in comparison to operational matrices existing in other methods are more accurate. The Lucas wavelets and their operational matrices provide the precise numerical scheme to get the approximate solution. Also, we exhibit the upper bound of error based on the method. We illustrate the behavior of the new scheme in several numerical examples with the help of tables and figures. The results confirm the accuracy and applicability of the numerical approach.

[1]  Shiva Sharma,et al.  Comparative study of three numerical schemes for fractional integro-differential equations , 2017, J. Comput. Appl. Math..

[2]  Yadollah Ordokhani,et al.  Fractional-order Legendre-Laguerre functions and their applications in fractional partial differential equations , 2018, Appl. Math. Comput..

[3]  R. Pandey,et al.  Galerkin and Collocation Methods for Weakly Singular Fractional Integro-differential Equations , 2018, Iranian Journal of Science and Technology, Transactions A: Science.

[4]  Mohammed Al-Smadi,et al.  Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method , 2013, Appl. Math. Comput..

[5]  M. Sezer,et al.  Lucas Polynomial Approach for System of High-Order Linear Differential Equations and Residual Error Estimation , 2015 .

[6]  Mehmet Sezer,et al.  Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients , 2008, J. Frankl. Inst..

[8]  Qibin Fan,et al.  Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW , 2013, Commun. Nonlinear Sci. Numer. Simul..

[9]  Carlo Cattani,et al.  Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations , 2014, Commun. Nonlinear Sci. Numer. Simul..

[10]  Jovan Popović,et al.  Fractional model for pharmacokinetics of high dose methotrexate in children with acute lymphoblastic leukaemia , 2015, Commun. Nonlinear Sci. Numer. Simul..

[11]  Ömer Oruç,et al.  A new numerical treatment based on Lucas polynomials for 1D and 2D sinh-Gordon equation , 2018, Commun. Nonlinear Sci. Numer. Simul..

[12]  K. Parand,et al.  Application of Bessel functions for solving differential and integro-differential equations of the fractional order ☆ , 2014 .

[13]  Muhammad Arif,et al.  Chebyshev wavelet method to nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions , 2017 .

[14]  M. Bahmanpour,et al.  Solving Fredholm integral equations of the first kind using Müntz wavelets , 2019, Applied Numerical Mathematics.

[15]  Yadollah Ordokhani,et al.  Fractional-order Bessel wavelet functions for solving variable order fractional optimal control problems with estimation error , 2020, Int. J. Syst. Sci..

[16]  F. B. Hildebrand,et al.  Introduction To Numerical Analysis , 1957 .

[17]  MEHDI DEHGHAN,et al.  Solution of a partial integro-differential equation arising from viscoelasticity , 2006, Int. J. Comput. Math..

[18]  R. Pandey,et al.  Adaptive Huber Scheme for Weakly Singular Fractional Integro-differential Equations , 2020, Differential Equations and Dynamical Systems.

[19]  Rajesh K. Pandey,et al.  Approximations of fractional integrals and Caputo derivatives with application in solving Abel’s integral equations , 2019, Journal of King Saud University - Science.

[20]  S. Jena,et al.  Differential Transformation Method (DTM) for Approximate Solution of Ordinary Differential Equation (ODE) , 2018, Advances in Modelling and Analysis B.

[21]  Y. Ordokhani,et al.  Numerical solution of linear Fredholm-Volterra integro-differential equations of fractional order ⇤ , 2016 .

[22]  Mehdi Dehghan,et al.  The numerical solution of the non-linear integro-differential equations based on the meshless method , 2012, J. Comput. Appl. Math..

[23]  Li Zhu,et al.  Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method , 2017, Advances in Difference Equations.

[24]  Shiva Sharma,et al.  Collocation method with convergence for generalized fractional integro-differential equations , 2018, J. Comput. Appl. Math..

[25]  M. I. Berenguer,et al.  A sequential approach for solving the Fredholm integro-differential equation , 2012 .

[26]  C. Christopoulos,et al.  Discrete transform technique for solving coupled integro-differential equations in digital computers , 1991 .

[27]  Abdollah Hadi-Vencheh,et al.  Direct method for solving integro differential equations using hybrid Fourier and block-pulse functions , 2005, Int. J. Comput. Math..

[28]  D. Benson,et al.  Fractional calculus in hydrologic modeling: A numerical perspective. , 2013, Advances in water resources.

[29]  Esmail Babolian,et al.  Convergence analysis of the Chebyshev-Legendre spectral method for a class of Fredholm fractional integro-differential equations , 2018, J. Comput. Appl. Math..

[30]  mer Oru.  A new algorithm based on Lucas polynomials for approximate solution of 1D and 2D nonlinear generalized BenjaminBonaMahonyBurgers equation , 2017 .

[31]  V. K. Patel,et al.  Two Dimensional Wavelets Collocation Scheme for Linear and Nonlinear Volterra Weakly Singular Partial Integro-Differential Equations , 2018, International Journal of Applied and Computational Mathematics.

[32]  António M. Lopes,et al.  Shifted Jacobi-Gauss-collocation with convergence analysis for fractional integro-differential equations , 2019, Commun. Nonlinear Sci. Numer. Simul..

[33]  Fengying Zhou,et al.  Numerical solution of fractional Volterra-Fredholm integro-differential equations with mixed boundary conditions via Chebyshev wavelet method , 2018, Int. J. Comput. Math..

[34]  Esmail Babolian,et al.  Application of He's homotopy perturbation method to nonlinear integro-differential equations , 2007, Appl. Math. Comput..

[35]  Suayip Yüzbasi,et al.  A collocation method based on Bernstein polynomials to solve nonlinear Fredholm-Volterra integro-differential equations , 2016, Appl. Math. Comput..

[36]  Yadollah Ordokhani,et al.  On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay , 2019, Numer. Linear Algebra Appl..

[37]  Mehdi Dehghan,et al.  SOLUTION OF AN INTEGRO-DIFFERENTIAL EQUATION ARISING IN OSCILLATING MAGNETIC FIELDS USING HE'S HOMOTOPY PERTURBATION METHOD , 2008 .

[38]  Khosrow Maleknejad,et al.  Numerical solution of high-order Volterra-Fredholm integro-differential equations by using Legendre collocation method , 2018, Appl. Math. Comput..

[39]  Limin Sun,et al.  Free vibrations of a taut cable with a general viscoelastic damper modeled by fractional derivatives , 2015 .