# A new class of polynomial functions equipped with a parameter

- 894 Downloads

## Abstract

In this study, a new class of polynomial functions although equipped with a parameter is introduced. This class can be employed for computational solution of linear or non-linear functional equations, including ordinary differential equations or integral equations. The extra parameter permits us to obtain more accurate results. In the present paper, a number of numerical examples show the ability of this class of polynomial functions.

### Keywords

Lucas polynomials Bessel polynomials Boubaker polynomials Collocation method## Introduction

In numerous cases, we can find the solution of the linear or non-linear functional equations by Polynomial Expansion Scheme (PES) or matrix methods (in the sense of collocation scheme) based on famous polynomials such as Taylor, Chebyshev, Berstein, Legendre, Laguerre, Hermite, Bessel, Lucas, and Boubaker [1, 2, 3, 10, 11, 12, 13].

In many problems, utilizing polynomial functions can assist us in finding reasonable solutions. For example, in physics, engineering, and economics, we can find many computational methods applying polynomial functions. A list of polynomial functions and their applications can be found in [2]. In recent years, a great number of new families of polynomials are introduced, including *q*-analogue of Hermite polynomials [6], *d*-orthogonal polynomials by Cheikha and Romdhane [5], unified family of generalized Apostol-Bernoulli, and Euler and Genocchi polynomials by El-Desouky and Gomaa [7].

In numerical solution of an Ordinary Differential Equation (ODE), we usually put a linear combination of polynomial functions with unknown coefficients in the main equation and then find the unknowns. This can be done using collocation or, Galerkin or Tau methods. As we know, in this scheme, we can select a set of orthogonal or orthonormal polynomial functions for the solution.

In the current research, we aim to introduce a new class of polynomial functions equipped with a parameter that can be used for computational solution of linear or non-linear functional equations, including ordinary differential equations, partial differential equations, or integral equations. The extra parameter contributes to obtaining more accurate results. Numerical examples demonstrate the ability of this class of polynomial functions to obtain better solutions.

## Polynomial functions

In this section, a new class of polynomial functions is introduced. This definition is based on the Chebyshev polynomial of the second kind, \(U_n(x)\). These functions have already been employed. For example, Boubaker polynomials are defined as \(B_n(x)=U_n(\frac{x}{2})+3U_{n-2}(\frac{x}{2})\), for \(n\ge 2\). Furthermore, the modified Boubaker polynomials [13] are defined as \(\tilde{B}_n(x)=6 x U_{n-1}(x)-2U_n(x)\), for \(n\ge 1\).

**Definition 2.1**

*a*is an arbitrary constant and \(U_n(x)\) is the Chebyshev polynomial of the second kind. Let \(A_0(x)=1\) for any \(n \ge 1\). The new class of polynomial functions is defined by the following:

**Proposition 2.1**

Let \(T_n(x)\) be the Chebyshev polynomial of the first kind. Then, \(A_n(x)=(a+1)\frac{x}{n} T'_n(x)+T_n(x)\) for \(n\ge 1\).

**Proposition 2.2**

**Proposition 2.3**

**Proposition 2.4**

**Proposition 2.5**

**Corollary 2.1**

The system \(\{A_{4n}(x)\}_{n=0}^\infty\) is an orthogonal system of polynomial functions, with respect to the inner product defined in Proposition 2.5.

## The *m*th-order non-linear differential equation

In this section, the new class of polynomial functions is considered to find the numerical solutions of some non-linear differential equations. The solutions obtained show the efficiency of this class of functions. In this class, we have an unknown parameter, and by selecting it, we can reduce the error of PES. In all of the following examples, the least square method is applied so as to find the unknown parameter. All iterative algorithms, which we needed for our initial guess, were initialized at zero.

*m*th-order non-linear differential equation

*y*is an unknown function,

*g*and \(P_{k,r}\) are continuous functions, and \(a_{j,k}\), \(b_{j,k}\), and \(\gamma _j\) are real or complex constants, is considered in [13]. By PFS collocation method, we can solve this problem. In this method, we approximate the solution of (1) in the truncated series form:

*a*, of the new class of polynomial functions of \(A_k\). By adding one collocation point to the set of points, we can construct a non-linear system with \(N+2\) equations. Via the least square method, we can minimize the \(L_2\) norm of the residual in the augmented non-linear system to obtain

*a*. Afterwards, we can continue the matrix collocation method to find other unknowns like [13].

*Example 3.*1

Numerical results of the absolute error functions of Example (3.1), with \(a=2.59490427\)

\(x_i\) | \(e_4(x_i),\) [4] | \(e_5(x_i),\) [9] | \(e_5(x_i),\) [13] | \(e_5(x_i)\) |
---|---|---|---|---|

0 | 0 | 0 | 0 | 0 |

0.2 | 5.81371970e−3 | 1.52386520e4 | 3.10830422e−4 | 3.10830419e−4 |

0.4 | 8.16606722e−2 | 1.1431338878e2 | 8.84574370e−5 | 8.84574342e−5 |

0.6 | 3.73690421e−1 | 1.2118642056e−1 | 1.61217488e−4 | 1.61217491e−4 |

0.8 | 1.08914877e−0 | 6.1674343718e−1 | 1.07446634e−4 | 1.07446646e−4 |

1.0 | 2.48769984e−0 | 2.1293665035e−0 | 3.60206386e−3 | 3.60206385e−4 |

Numerical results of the absolute error functions of Example (3.1), with \(a=13.50332999\)

\(x_i\) | Boubaker functions | Present method |
---|---|---|

\(e_{10}(x_i)\) | \(e_{10}(x_i)\) | |

0 | 0 | 0 |

0.2 | 3.2921789994145e−7 | 3.2921717829648e−7 |

0.4 | 2.2017646683636e−7 | 2.2017604695002e−7 |

0.6 | 1.4863375341623e−7 | 1.4863314123925e−7 |

0.8 | 7.0191910173101e−8 | 7.0190931511505e−8 |

1.0 | 1.1546738372115e−5 | 1.1546784083105e−5 |

*Example 3.2*

Numerical results of the absolute error functions of Example (3.2), with \(a=4.01764012\)

\(x_i\) | Taylor method | Bessel functions | Boubaker functions | Present method |
---|---|---|---|---|

\(e_5(x_i),\) [8] | \(e_5(x_i)\) | \(e_5(x_i)\) | \(e_5(x_i)\) | |

0 | 0 | 0 | 0 | 0 |

0.2 | 8.64080e−8 | 6.8031096090e−7 | 6.8031096079e−7 | 6.8031096090e−7 |

0.4 | 5.379366e−6 | 2.2898841512e−7 | 2.2898841490e−7 | 2.2898841501e−7 |

0.6 | 5.9636094e−5 | 4.0321424399e−7 | 4.0321424366e−7 | 4.0321424377e−7 |

0.8 | 3.26297447e−4 | 4.4156961293e−7 | 4.4156961343e−7 | 4.4156961315e−7 |

1.0 | 1.212774501e−3 | 1.4263286683e−5 | 1.4263286683e−5 | 1.4263286682e−5 |

Numerical results of the absolute error functions of Example (3.1), with \(a=37.74361705\)

\(x_i\) | Boubaker functions | Present method |
---|---|---|

\(e_{10}(x_i)\) | \(e_{10}(x_i)\) | |

0 | 0 | 0 |

0.2 | 2.8976820942717e−14 | 2.8754776337792e−14 |

0.4 | 2.1649348980191e−14 | 2.1649348980191e−14 |

0.6 | 1.3322676295502e−14 | 1.3211653993039e−14 |

0.8 | 2.6090241078691e−15 | 2.0539125955565e−15 |

1.0 | 1.8128276657592e−12 | 1.8171020244040e−12 |

## Conclusion

In this study, a new class of polynomial functions equipped with a parameter was introduced. We found that the extra parameter permits us to obtain more accurate results. By solving Riccati and Abel non-linear differential equations, the ability and efficiency of this class of polynomial functions were supported. The application of this class of polynomials in solving complicated ordinary differential equations and partial differential equations and integral equations is expected.

## Notes

### Acknowledgements

The authors thanks the anonymous reviewers for helpful comments, which lead to definite improvement in the manuscript.

### References

- 1.Boubaker, K.: On modified Boubaker polynomials: some differential and analytical properties of the new polynomials issued from an attempt for solving bi-varied heat equation. Trends Appl. Sci. Res.
**2**(6), 540–544 (2007)CrossRefGoogle Scholar - 2.Boubaker, K.: The Boubaker polynomials, a new function class for solving bi-varied second order differential equations. Far East J. Appl. Math.
**31**(3), 299–320 (2008)MathSciNetMATHGoogle Scholar - 3.Boubaker, K., Van Gorder, R.A.: Application of the BPES to Lane-Emden equations governing polytropic and isothermal gas spheres. New Astronom.
**17**, 565–569 (2012)CrossRefGoogle Scholar - 4.Bulut, H., Evans, D.J.: On the solution of the Riccati equation by the decomposition method. Int. J. Comp. Math.
**79**, 103–109 (2002)MathSciNetCrossRefMATHGoogle Scholar - 5.Cheikha, Y.B., Romdhane, N.B.: \(d\)-Symmetric \(d\)-orthogonal polynomials of Brenke type. J. Math. Anal. Appl.
**416**(2), 735–747 (2014)MathSciNetCrossRefMATHGoogle Scholar - 6.Cigler, J., Zeng, J.: A curious \(q\)-analogue of Hermite polynomials. J. Combin. Theory Ser. A
**118**(1), 9–26 (2011)MathSciNetCrossRefMATHGoogle Scholar - 7.El-Desouky, B.S., Gomaa, R.S.: A new unified family of generalized Apostol-Euler, Bernoulli and Genocchi polynomials. Appl. Math. Comput.
**247**, 695–702 (2014)MathSciNetMATHGoogle Scholar - 8.Guler, C.: A new numerical algorithm for the Abel equation of the second kind. Int. J. Comp. Math.
**84**, 109–119 (2007)MathSciNetCrossRefMATHGoogle Scholar - 9.Gulsu, M., Sezer, M.: On the solution of the Riccati equation by the Taylor matrix method. Appl. Math. Comp.
**176**, 414–421 (2006)MathSciNetCrossRefMATHGoogle Scholar - 10.Işik, O.R., Güney, Z., Sezer, M.: Berstein series solutions of pantograph equations using polynomial interpolation. J. Diff. Equ. Appl.
**18**(3), 357–374 (2012)CrossRefMATHGoogle Scholar - 11.Karamete, A., Sezer, M.: A Taylor collocation method for the solution of linear integro-differential equations. Int. J. Comput. Math.
**79**(9), 987–1000 (2002)MathSciNetCrossRefMATHGoogle Scholar - 12.Labiadh, H., Boubaker, K.: A Sturm-Liouville shaped characteristic differential equation as a guide to establish a quasi-polynomial expression to the Boubaker polynomials. Differ. Equ. Control Process.
**2**, 117–133 (2007)MathSciNetGoogle Scholar - 13.Yüzbaşi, Ş., Şahin, N.: On the solutions of a class of nonlinear ordinary differential equations by the Bessel polynomials. J. Numer. Math.
**20**(1), 55–79 (2012)MathSciNetMATHGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.