A Novel Spectral Modified Pell Polynomials for Solving Singular Differential Equations

Authors

  • Mohammed Abdelhadi Sarhan Mathematics Department, College of Sciences, Al-Mustansiriyah University.
  • Suha SHIHAB Applied Science Department, University of Technology, Baghdad, Iraq
  • Mohammed RASHEED Applied Science Department, University of Technology, Baghdad, Iraq

DOI:

https://doi.org/10.23851/mjs.v32i1.930

Keywords:

Modified Pell polynomials, singular initial value problem, operation matrix, spectral method, product matrix of two modified Pell polynomials.

Abstract

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.

Downloads

Download data is not yet available.

Author Biographies

  • Suha SHIHAB, Applied Science Department, University of Technology, Baghdad, Iraq

    She had her B.Sc., M.Sc. degrees in Applied Physics from University of Technology, Baghdad (Iraq) in 27/06/1992, 28/12/1995 respectively, and Ph.D. degree in the College of Science, Al-Mustansiriyah University Baghdad (Iraq) in 04/10/2005. He has become a Lecturer in 06/03/1999, Assistant professor at 15/03/2002 and Professor at 02/05/2012. She has research gate h-index 3, Scopus h-index 1 and more than 60 published articles inside, outside Iraq. She supervised on many students (Msc. and PHD.) degrees. He was having a many research group on numerical Analysis, optimal Control Systems, and Mathematical Physics. As well as; She have a group for the characterization of semiconducting thin films and nanostructures materials especially the Optical properties with mathematical models produced by different techniques of preparation such as: Spin Coating and Dip Coating (Sol Gel), Sputtering (DC and RF), PLD, chemical bath deposition (CBD) and ....etc. In addition; I have group for the characterization of Ceramic and Polymer materials mathematically produced by many techniques like solid-state and hydrothermal. Authored more than Five books published it in Lambert Academic Publishing in 2012 in Germany. She is reviewer for many international journals. She participate in more than 12 conferences all of them outside Iraq in France, Romania, Algeria and Tunisia.

  • Mohammed RASHEED, Applied Science Department, University of Technology, Baghdad, Iraq

    Mohammed S. Rasheed was born in Baghdad in 1971. He had her B.Sc., M.Sc. degrees in Applied Physics from the University of Technology, Baghdad (Iraq) on 03/09/1992, 17/05/1995 respectively, and a Ph.D. degree from the University of Angers, Angers (France) on 12/12/2017. He has become an Assistant professor at 11\11\2018. He has research gate h-index 3, Scopus h-index 3, and more than 30 published articles inside, outside Iraq, and most of them in Clarivate journals. He was having a many research groups on the Preparation and characterization of semiconducting thin films and nanostructures produced by the Spin Coating and Dip Coating (Sol-Gel) technique. As well, He is a member of research groups on the preparation and characterization of semiconducting thin films by chemical bath deposition (CBD) and preparation and characterization of Ceramic and Polymer materials produced by many techniques like solid-state and hydrothermal. Authored one book on the laser communication system and published it in Lambert Academic Publishing in 2012 in Germany. He is a reviewer for two international journals indexed in (Springer & Elsevier). He participates in more than 12 conferences all of them outside Iraq in France, Romania, Algeria, and Tunisia.

References

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.

CrossRef | PubMed

TSAY, S. C., & LEE, T. T. (1986). Solutions of integral equations via Taylor series. International Journal of Control, 44(3), 701-709.

CrossRef

Al-Rawy S. N. (2006). On the Solution of Certain Fractional Integral Equations, kirkuk university journal for scientific studies. 1(2) 125-136.

CrossRef

Kadkhoda, N. (2020). A numerical approach for solving variable order differential equations using Bernstein polynomials. Alexandria Engineering Journal, 59(5), 3041-3047.

CrossRef

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.

CrossRef

Dehghan, M., & Saadatmandi, A. (2008). Chebyshev finite difference method for Fredholm integro-differential equation. International Journal of Computer Mathematics, 85(1), 123-130.

CrossRef

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.

CrossRef

El-Gendi, S. E. (1969). Chebyshev solution of differential, integral and integro-differential equations. The Computer Journal, 12(3), 282-287.

CrossRef

Clenshaw, C. W., & Norton, H. J. (1963). The solution of nonlinear ordinary differential equations in Chebyshev series. The Computer Journal, 6(1), 88-92.

CrossRef

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.

CrossRef

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.

CrossRef

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.

CrossRef

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.

CrossRef

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.

CrossRef

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.

CrossRef

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.

CrossRef

Downloads

Key Dates

Published

24-02-2021

Issue

Vol. 32 No. 1 (2021)

Section

Original Article

License

(Starting May 5, 2024) Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution (CC-BY) 4.0 License that allows others to share the work with an acknowledgement of the work’s authorship and initial publication in this journal.

How to Cite

[1]
M. A. Sarhan, S. SHIHAB, and M. RASHEED, “A Novel Spectral Modified Pell Polynomials for Solving Singular Differential Equations”, Al-Mustansiriyah J. Sci., vol. 32, no. 1, pp. 18–24, Feb. 2021, doi: 10.23851/mjs.v32i1.930.
Crossref
Scopus
Google Scholar
Europe PMC

Similar Articles

1-10 of 227

You may also start an advanced similarity search for this article.