On the Properties of Two-Dimensional Normalized Boubaker Polynomials

西 南 交 通 大 学 学 报

2020 年 6 月

JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY

Vol. 55 No. 3 June 2020

ISSN: 0258-2724 DOI：10.35741/issn.0258-2724.55.3.21 Research article

Mathematics

O

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二维归一化布巴克多项式的性质

Mohammed Abdelhadi Sarhan a, Suha Shihab b, *, Mohammed Rasheed b

a _{Mathematics Department, College of Sciences, Al-Mustansiriyah University}

Baghdad, Iraq, [email protected], [email protected] b

Applied Science Department, University of Technology

Baghdad, Iraq, [email protected], [email protected], [email protected], [email protected]

Received: February 2, 2020 ▪ Review: February 27, 2020 ▪ Accepted: April 21, 2020 This article is an open access article distributed under the terms and conditions of the Creative Commons

Attribution License (http://creativecommons.org/licenses/by/4.0)

Abstract

The aim of the present work deals with newly defined two-variable polynomials for normalized Boubaker . The operational matrices of derivatives with respect to the two variables are presented at first with explicit expression. Then, a normalized Boubaker polynomial approximation for the numerical solution of a class of partial differential equations is proposed, depending on a truncated, normalized Boubaker function series in the equation together with the operational matrices in the proposed partial differential equation. The original partial differential equation is reduced under consideration of a system of simply solvable algebraic equations. Due to the interesting derived properties of normalized Boubaker polynomials in two variables, the suggested method can achieve good results with few complexities. Using operational matrices of derivatives, one can save computation and more memory. Two-dimensional examples are listed to show the satisfactory level of the suggested method.

Keywords: Normalized Boubaker Polynomial, Operation Matrix of Derivative, Partial Differential Equation, Polynomials in Two Variables

摘要 本工作的目的是针对标准化的布贝克处理新定义的二变量多项式。首先用明确的表达式给出 关于两个变量的导数的运算矩阵。然后，针对一类偏微分方程的数值解，提出了归一化的布贝克 多项式逼近，具体取决于方程中的截断，归一化的布贝克函数系列以及所提出的偏微分方程中的 运算矩阵。考虑简单可解的代数方程组，可以简化原始的偏微分方程。由于在两个变量中归一化 的布贝克多项式具有有趣的派生性质，因此所建议的方法可以在不复杂的情况下获得良好的结果 。使用导数的运算矩阵，可以节省计算量和更多的内存。列出了二维示例以显示所建议方法的令

人满意的水平。

关键词: 标准化的布贝克多项式，导数运算矩阵，偏微分方程，两个变量的多项式

I. I

Due to the computational efficiency of orthogonal polynomials, they are successful in many fields of engineering and science. Orthogonal polynomials can make a connection with fast, approximate methods and play an interesting role in the approximate solution for partial differential equations, which are considered as a basis function. The orthogonal basis has the advantage of infinitely differentiable. In addition, their matrices of derivatives are dense in all dimensions. Much attention in recent years has been devoted to improving operational matrices of derivative and integration for orthogonal polynomials. Such types of operational matrices are derived using Boubaker wavelets [1], [2], Chebyshev wavelets [3], and shifted Chebyshev polynomials [4]. Researchers have developed various numerical methods based on operational matrices. For example, in [5], the authors established a new numerical scheme based on the operational matrix scheme for nonlinear integral equations of fractional integration in two dimensions, whereas the authors of [6] developed an operational matrix that yielded different results. In [7], the authors used spectral methods based on Legendre wavelets for nonlinear Klein/Sine-Gordon equations, and in [8] the authors used Legendre approximation for special hyperbolic partial differential equations (HPDEs) and provided a comparison with Taylor and Bernoulli algorithms. The authors of [9] solved a class of partial integro-differential equations in two variables with the aid of wavelets functions, whereas those of [10] and [11] proposed the solutions for problems of integro-partial differential equations and hyperbolic heat conduction using Haar wavelet operational matrix. Boubaker polynomials can be used in pure and applied physics [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39].

The purpose of this paper is to present a new general formulation of normalized (orthonormal) Boubaker polynomials in two variables and construct new important operational matrices of derivatives and then is utilized together with collocation method for approximate solution of a class of partial differential equation, which is

depending on truncated normalized Boubaker series of the functions. With the use of normalized Boubaker polynomials in two variables along with their derivative operation matrices with respect to two variables, the original partial differential equation will be transformed to algebric equations system. Our proposed technique is easy, and there is no computational complexity in solving the obtained algebraic equations.

The manuscript is organized as follows: Section II is about the new definition of normalized Boubaker polynomials in two dimensions, while Section III gives the condition for the function approximation.

Section IV deals with the operational derivatives with respect to the two variables for normalized Boubaker polynomials in two dimensions, while Section V is related to the application of the obtained operational matrices to transform the original partial differential equation to the corresponding system of algebraic equations. Some numerical examples are provided in Section VI, and a brief conclusion is provided in the last section.

II. T

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T

V

Boubaker polynomials with two variables in over the with respect to one dimensional Boubaker polynomials can be defined as

(1) where and are given by

(2)

(3)

(4) or

(5)

3

Thus, a complete polynomial is of the form

(6) For and the following polynomials are obtained

Boubaker polynomials in two variables can also be defined using the following recurrence relation

(7) and

(8)

III. F

A

The function defined over can be expressed as

(9) By truncating the series in Eq. 9 yields

(10) where

. Express Eq. 10 as the following vector matrix form

(11)

are coefficients matrix and Boubaker vector matrix, respectively. They are given by

(12)

(13)

IV. O

M

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Theorem 1: Let be the orthonormal Boubaker polynomials in two variables into

, then we have

(14)

where

where the matrices , and 0 are of dimensions

and , respectively.

In which the elements of R can be defined as

Theorem 2:

where

or

.

Proof: Suppose that the orthonormal expansion of function can be represented as (15) or (16) then (17)

then can be represented as

(18) where

, (19) Now, by taking in Eq. 2, one can obtain for , consequently

(20)

as a result .

With the aid of Eq. 6, the operational matrix can be obtained.

In the same way, one can obtain the operation matrix of derivative with respect to

(21)

Then can be represented as

(22) where or as a result .

V. I

E

The first partial differential equation is

(23)

with initial values , and represents the exact solution.

By using Eq. (11) and section 4, one can get Let (24) where , (25) (26) where , , (27)

Substituting Eqns. 25–27 into Eq. 23, yields

or .

Similarly, for the initial conditions

(28) and for

By equating the coefficients, one can obtain , , , , .

B. Example 2

Consider the second test problem

(29)

and (30)

Let the approximate solution be as follows (31) Then

, (32)

5

(33) Using similarity for the initial conditions in Eq. 30, one can obtain

(34) (35)

VI. C

A fractional space and time telegraph equation was studied in this paper (namely, a general fractional second order partial differential equation). Four cases of these equations were considered. The first, the linear and homogeneous fractional telegraph, was three preformed examples of different values of fractional order. Different examples were solved to show the power of this method as considered in the first three remarks. Example 4 was solved using the property in the first remark. The second remark was used to find the exact solution in Examples 5, 6, and 7. These examples were solved using the property of the third remark. This method is an easy way to find almost exact solutions, or obtain an idea about an exact solution, in all equations. In all cases, these examples have been given good approximation solutions.

R

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