# Convergence of Legendre and Chebyshev multiwavelets in Petrov-Galerkin method for solving Fredholm integro-differential equations of high orders

Document Type : English

Author

Department of Mathematics, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr/Isfahan, Iran

Abstract

This work was intended as an attempt to motivate readers for a comparison study of constructions of Legendre multiwavelet and Chebyshev multiwavelet. It is also shown how to use them in Petrov-Galerkin approach for solving Fredholm integro-differential equation of high orders of the second kind. In fact, a numerical technique for the discretization method of Fredholm integro-differential equations is presented that yields linear system. The important point to note here is the convergence of presented methods. For the first time, two conditions are proved for convergence of Legendre and Chebyshev multiwavelets in Petrov-Galerkin method. The proof of these conditions with using linear algebra and matrix theory ensures that Petrov-Galerkin methods has a unique approximation. Finally, some relevent numerical examples, for which the exact solution is known, will indicate accuracy and applicability of the proposed method.

Keywords 20.1001.1.27834441.2021.13.3.4.1

#### References

 Abdul-Majid, W. A. Z. W. A. Z. (2011). Linear and Nonlinear Integral Equations: Methods and Applications.
 Davis, H. T. (1962). Introduction to Nonlinear Differential Integral Equations, US Atomic Energy Commission, Washington, DC, 1960. 592 p. Reprint: Dover, NewYork.
 Kanwal, R. P. (2013). Linear integral equations. Springer Science & Business Media.
 Maleknejad, K., Rabbani, M., Aghazadeh, N., & Karami, M. (2009). A wavelet Petrov–Galerkin method for solving integro-differential equations. International Journal of Computer Mathematics86(9), 1572-1590.
 Mustapha, K. (2008). A Petrov--Galerkin method for integro-differential equations with a memory term. ANZIAM Journal50, C610-C624.
 Alpert, B. K. (1993). A class of bases in L2 for the sparse representation of integral operators. SIAM journal on Mathematical Analysis24(1), 246-262.
 Chen, Z., & Xu, Y. (1998). The Petrov--Galerkin and Iterated Petrov--Galerkin Methods for Second-Kind Integral Equations. SIAM Journal on Numerical Analysis35(1), 406-434.
 Chen, Z., Micchelli, C. A., & Xu, Y. (1997). The Petrov–Galerkin method for second kind integral equations II: multiwavelet schemes. Advances in Computational Mathematics7(3), 199-233.
 Maleknejad, K., & Karami, M. (2003, August). The Petrov—Galerkin method for solving second kind Fredholm integral equations. In 34th Iranian Mathematics Conference (Vol. 30).
 Bialecki, B., Ganesh, M., & Mustapha, K. (2004). A Petrov–Galerkin method with quadrature for elliptic boundary value problems. IMA journal of numerical analysis24(1), 157-177.
 Jiang, Z. H. (1992). W. Schaufelberger, block Pulse Functions and Their Applications in Control Systems.
 Maleknejad, K., & Sohrabi, S. (2007). Numerical solution of Fredholm integral equations of the first kind by using Legendre wavelets. Applied Mathematics and Computation186(1), 836-843.
 Shang, X., & Han, D. (2007). Numerical solution of Fredholm integral equations of the first kind by using linear Legendre multi-wavelets. Applied Mathematics and Computation191(2), 440-444.
 Guf, J. S., & Jiang, W. S. (1996). The Haar wavelets operational matrix of integration. International Journal of Systems Science27(7), 623-628.
 Lepik, Ü. (2008). Solving integral and differential equations by the aid of non-uniform Haar wavelets. Applied Mathematics and Computation198(1), 326-332.
 Guner, O., & Bekir, A. (2017). Exp-function method for nonlinear fractional differential equations. Nonlinear Sci. Lett. A8(1), 41-49.
 Mahdy, A. M., & Mohamed, E. M. (2016). Numerical studies for solving system of linear fractional integro-differential equations by using least squares method and shifted Chebyshev polynomials. Journal of Abstract and Computational Mathematics1(1), 24-32.
 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 Engineering9(3), 1687814017694802.
 Nazari, D., & Shahmorad, S. (2010). Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions. Journal of Computational and Applied Mathematics234(3), 883-891.
 Ordokhani, Y., & Rahimi, N. (2014). Numerical solution of fractional Volterra integro-differential equations via the rationalized Haar functions. J. Sci. Kharazmi Univ14(3), 211-224.
 Pashayi, S., Hashemi, M. S., & Shahmorad, S. (2017). Analytical lie group approach for solving fractional integro-differential equations. Communications in Nonlinear Science and Numerical Simulation51, 66-77.
 Pedas, A., & Vikerpuur, M. (2021). Spline Collocation for Multi-Term Fractional Integro-Differential Equations with Weakly Singular Kernels. Fractal and Fractional5(3), 90.
 Sahu, P. K., & Ray, S. S. (2016). WITHDRAWN: A novel Legendre wavelet Petrov–Galerkin method for fractional Volterra integro-differential equations.
 Riahi Beni, M. (2021). Legendre wavelet method combined with the Gauss quadrature rule for numerical solution of fractional integro-differential equations. Iranian Journal of Numerical Analysis and Optimization.