The Approximation of Weighted Hölder Functions by Fourier-Jacobi Polynomials to the Singular Sturm-Liouville Operator


  • Habeeb A. Aal-Rkhais Department of Mathematics, College of Computer Science and Mathematics, University of Thi-Qar, Thi-Qar, Iraq
  • Ahmed H. Kamil National University of Science and Technology, Thi-Qar, Iraq
  • Khalid Fanoukh Al Oweidi Department of Water Resources Management Engineering, College of Engineering, Al-Qasim Green University, Babylon, Iraq



Best approximation, Jacobi Polynomials, Moduli of smoothness, Sturm-Liouville equation, Weighted Ho ̈lder spaces


      In this work, a weighted H lder function that approximates a Jacobi polynomial which solves the second order singular Sturm-Liouville equation is discussed. This is generally equivalent to the Jacobean translations and the moduli of smoothness. This paper aims to focus on improving methods of approximation and finding the upper and lower estimates for the degree of approximation in weighted H lder spaces by modifying the modulus of continuity and smoothness. Moreover, some properties for the moduli of smoothness with direct and inverse results are considered.


Download data is not yet available.


Caruntu DI. Classical Jacobi polynomials closed-form solutions for transverse vibrations. J Sound Vib. 2007 Oct 9;306(3-5):467-94.

Grober OV, Grober TA, Bychkov AB. Convergence to boundary values in the topology of the inductive limit of generalized holder spaces. IOP Conf Ser: Mater Sci Eng. 2019 Dec 1; 698, 6: 066025. IOP Publishing.

Pirumyan N, Stakyan M. Mathematical model of pressure distribution in the main gas-transport system. IOP Conf Ser Mater Sci Eng. 2019 Dec 1; 698,6:066022. IOP Publishing.

Ditzian Z. Polynomial Approximation and ω_r^ϕ (.,t) Twenty Years Later. arXiv preprint arXiv:0709.2729. 2007 Sep 17.

Grober OV, Grober TA, Bychkov AB. An Analogue of Smirnov's Theorem in Spaces of the Functions which are Inductive Limits of some Hölder Spaces. Mater. Sci. Forum. 2018, 931: 184-187. Trans Tech Publications Ltd.

Platonov SS. Fourier-Jacobi harmonic analysis and approximation of functions. Izvestiya: Mathematics. 2014;78(1):106.

Abed BN, Majeed SJ, Aal-Rkhais HA. Qualitative Analysis and Traveling wave Solutions for the Nonlinear Convection Equations with Absorption. J Phys Conf Ser. 2020 Jul 1; 1591(1) 012052. IOP Publishing.

AL-Saidy S, AL-Jawari N, Zaboon AH. Best Multiplier Approximation of Unbounded Periodic Functions in L_(p,n ) (B),B=[0,2π] Using Discrete Linear Positive Operators. Baghdad Sci J. 2020; 17(3), 0882.

Aal-Rkhais HA, Ruba HQ. The Development of interfaces in a Parabolic p-Laplacian type diffusion equation with weak convection. J Phys Conf Ser. 2021; 963. 1. IOP Publishing.

Abdulla UG, Aal-Rkhais HA. Development of the Interfaces for the Nonlinear Reaction-Diffusion equation with Convection. IOP Conf Ser Mater Sci Eng. 2019 Jul 1; 571(1) 012012. IOP Publishing.

Al-Smadi M, Arqub OA, Hadid S. An attractive analytical technique for coupled system of fractional partial differential equations in shallow water waves with conformable derivative. Commun Theor Phys. 2020; 72(8), 085001.

Abed BN, Majeed SJ, and Aal-Rkhais HA. Qualitative Analysis and Traveling wave Solutions for the Nonlinear Convection Equations with Absorption. J. Phys. Conf. Ser. 2020; 1591 012052

Al-Smadi M, Arqub OA, Momani S. Numerical computations of coupled fractional resonant Schrödinger equations arising in quantum mechanics under conformable fractional derivative sense. Phys Scr. 2020; 95(7), 075218.

Al-Smadi M, Arqub OA, Zeidan D. Fuzzy fractional differential equations under the Mittag-Leffler kernel differential operator of the ABC approach: theorems and applications. Chaos Solitons Fractals. 2021; 146, 110891.

Al-Smadi M, Arqub OA. Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates. Appl Math Comput. 2019; 342, 280-294.‏

Al-Smadi M, Freihat A, Khalil H, Momani S, Ali Khan R. Numerical multistep approach for solving fractional partial differential equations. Int J Comput Methods. 2017; 4(03), 1750029.‏

Hameed EM, Aal-Rkhais HA, Majeed SJ. Some Estimates to Best Approximation of Functions by Fourier-Jacobi Differential Operators. IOP Conf Ser Mater Sci Eng. 2021 Feb 1, 1095(1) 012010. IOP Publishing.

Hussin CHC, Azmi A, Ismail AIM, Kilicman A, Hashim I. Approximate Analytical Solutions of Bright Optical Soliton for Nonlinear Schrödinger Equation of Power Law Nonlinearity. Baghdad Sci J. 2021; 18(1(Suppl.), 0836.

Khosravian-Arab H, Dehghan M, Eslahchi MR. Fractional Sturm–Liouville boundary value problems in unbounded domains: Theory and applications. J Comput Phys. 2015 Oct 15; 299: 526-60.

Jassim HK, Al-Rkhais HA. Fractional Sumudu Decomposition Method for Solving PDEs of Fractional Order. J. Appl. Comput. Mech. 7(1) (2021) 302–311.

Charalambides M, Waleffe F. Spectrum of the Jacobi tau approximation for the second derivative operator. SIAM J Numer Anal. 2008;46(1):280-94