A Novel Spectral Modified Pell Polynomials for Solving Singular Differential Equations
Keywords:Modified Pell polynomials, singular initial value problem, operation matrix, spectral method, product matrix of two modified Pell polynomials.
This paper studies the modified Pell polynomials. Some important properties of modified Pell polynomials are presented. An exact formula of modified Pell polynomials derivative in terms of modified Pell themselves is first derived with the proof and then a new relationship is constructed which relates the modified Pell polynomials expansion coefficients of a derivative in terms of their original expansion coefficients. An interesting new formula for the product operational matrix of modified Pell polynomials is also derived in this work. With modified Pell polynomials expansion scheme, the powers 1, x, …, xn are expressed in terms of such polynomials. The main goal of all presented formulas is to simplify the original equations and the determination of the coefficients of expansion based on modified Pell polynomials will be easy. Spectral techniques together with all the derived formulas of modified Pell polynomials are utilized to solve some singular initial value problems. Three test examples are solved in this work to illustrate the validity of the proposed method. The computational method is replaced by exact and explicit formulas. More accurate results are obtained than those presented by other existing methods in the literature.
Dkhilalli F., Borchani S. M., Rasheed M., Barelle R., Shihab S., Guidara K., & Megdiche M. (2018). Characterizations and morphology of sodium tungstate particles. Royal Society open science, (8) 1-16.
TSAY, S. C., & LEE, T. T. (1986). Solutions of integral equations via Taylor series. International Journal of Control, 44(3), 701-709.
Al-Rawy S. N. (2006). On the Solution of Certain Fractional Integral Equations, kirkuk university journal for scientific studies. 1(2) 125-136.
Kadkhoda, N. (2020). A numerical approach for solving variable order differential equations using Bernstein polynomials. Alexandria Engineering Journal, 59(5), 3041-3047.
Shihab S. N. & Abdalrehman A. A. (2012). Numerical solution of calculus of variations by using the second Chebyshev wavelets, Engineering and Technology Journal, 30(18) 3219-3229.
Yuanlu, L. I. (2010). Solving a nonlinear fractional differential equation using Chebyshev wavelets. Communications in Nonlinear Science and Numerical Simulation, 15(9), 2284-2292.
Dehghan, M., & Saadatmandi, A. (2008). Chebyshev finite difference method for Fredholm integro-differential equation. International Journal of Computer Mathematics, 85(1), 123-130.
Kojabad, E. A., & Rezapour, S. (2017). Approximate solutions of a sum-type fractional integro-differential equation by using Chebyshev and Legendre polynomials. Advances in Difference Equations, 2017(1), 1-18.
El-Gendi, S. E. (1969). Chebyshev solution of differential, integral and integro-differential equations. The Computer Journal, 12(3), 282-287.
Clenshaw, C. W., & Norton, H. J. (1963). The solution of nonlinear ordinary differential equations in Chebyshev series. The Computer Journal, 6(1), 88-92.
Sezer, M., & Kaynak, M. (1996). Chebyshev polynomial solutions of linear differential equations. International Journal of Mathematical Education in Science and Technology, 27(4), 607-618.
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.
Henderson, M., & Matthews, R. (1995). Permutation properties of Chebyshev polynomials of the second kind over a finite field. Finite Fields and Their Applications, 1(1), 115-125.
Mohsen, R., & Mohammad H. (1996). Application of Legendre series to the control problems governed by linear parabolic equations. Mathematics and computers in simulation, 42(1), 77-84.
Sarhan M. A., SHIHAB S., & RASHEED M. (2021). Some Results on a Two Variables Pell Polynomials. Al-Qadisiyah Journal of Pure Science. 26(1) 55-70.
Aziz S. H., SHIHAB S., & RASHEED M. (2021). On Some Properties of Pell Polynomials. Al-Qadisiyah Journal of Pure Science. 26(1) 39-54.
Maleknejad, K., & Mirzaee, F. (2005). Using rationalized Haar wavelet for solving linear integral equations. Applied Mathematics and Computation, 160(2), 579-587.
Shiralashetti, S. C., & Kumbinarasaiah, S. (2018). Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems. Alexandria engineering journal, 57(4), 2591-2600.
How to Cite
The journal has no restrictions for the author to hold the copyrights of his articles. The journal does not allow authors to republish the same article in other journals or conferences that is published in one of its volumes.