Differentiable Function In Terms Research Articles

Some Results on a Two Variables Pell Polynomials

In this study, Pell polynomials in two variables, and their properties are investigated. Some formulas for two variables Pell polynomials are derived by matrices. By defining special formula for Pell polynomials in one variable, new important properties of Pell polynomials in two variables can be enabled to derive. A new exact formula expressing the partial derivatives of Pell polynomials explicitly of any degree in terms of Pell polynomials themselves is proved. A novel explicit formula, which constructs the two explicit formulas, which construct the two-dimension Pell polynomials expansion coefficients of a first partial derivative of a differentiable function in terms of their original expansion coefficients, is also included in the present article. The main advantage of the presented formulas is that the new properties of Pell polynomials in two variables greatly simplify the original problems and the result will lead to easy calculation of the coefficients of expansion. A direct spectral method along with the presented two variables Pell polynomials is proposed to solve the partial differential equation. Illustrated examples are included to demonstrate the validity of the technique.

On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain

Abstract Herein, three important theorems were stated and proved. The first relates the modified generalized Laguerre expansion coefficients of the derivatives of a function in terms of its original expansion coefficients; and an explicit expression for the derivatives of modified generalized Laguerre polynomials of any degree and for any order as a linear combination of modified generalized Laguerre polynomials themselves is also deduced. The second theorem gives new modified generalized Laguerre coefficients of the moments of one single modified generalized Laguerre polynomials of any degree. Finally, the third theorem expresses explicitly the modified generalized Laguerre coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its modified generalized Laguerre coefficients. Some spectral applications of these theorems for solving ordinary differential equations with varying coefficients and some specific applied differential problems, by reducing them to recurrence relations in their expansion coefficients of the solution are considered.

Spectral Solutions for Differential and Integral Equations with Varying Coefficients Using Classical Orthogonal Polynomials

Spectral methods for solving differential/integral equations are characterized by the representation of the solution by a truncated series of smooth functions. The unknowns to be determined are the expansion coefficients in such a representation. The goal of this article is to give an overview of numerical problems encountered when determining these coefficients and the rich variety of techniques proposed to solve these problems. Therefore, a series of explicit formulae expressing the derivatives, integrals and moments of a class of orthogonal polynomials of any degree and for any order in terms of the same polynomials are addressed. We restrict the current study to the orthogonal polynomials of the Hermite, generalized Laguerre, Bessel, and Jacobi (including Legendre, Chebyshev, and ultraspherical) families. Moreover, formulae expressing the coefficients of an expansion of these polynomials which have been differentiated or integrated an arbitrary number of times in terms of the coefficients of the original expansion are given. In addition, formulae for the polynomial coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its original expanded coefficients are also presented. A simple approach to build and solve recursively for the connection coefficients between different orthogonal polynomials is established. The essential results are summarized in tables which could serve as a useful reference to numerical analysts and practitioners. Finally, applications of these results in solving differential and integral equations with varying polynomial coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, are implemented.

Design optimization of a wind farm layout for maximum wind energy capture: A new constructive approach

Wind energy is becoming an attractive source of clean energy. However, this type of power source is subject to power reductions due to losses in wind energy conversion system and to frequent changes in wind velocity. For that reason, the important phase of a wind farm design is solving the wind farm layout optimization problem, which consists in optimally positioning the turbines within the wind farm so that the wake effects are minimized and therefore the expected power production is maximized. This problem has been receiving increasing attention from the scientific community. In this article, a mathematical optimization scheme is employed to optimize the locations of wind turbines with respect to maximizing the wind farm power production. To formulate the mathematical optimization problem, we used Jensen’s wake model. We calculate the wake loss and express the expected wind farm power as a differentiable function in terms of the locations of the wind turbines. Furthermore, we develop a new constructive approach to find the best solution to the wind turbines placement problem. Finally, our results are compared with those in some other ealier studies.

On the coefficients of differentiated expansions and derivatives of chebyshev polynomials of the third and fourth kinds

Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved. Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given. Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems, two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.

The general shifted Jacobi matrix method for solving generalized pantograph equations

In this paper, the general shifted Jacobi matrix method is applied to solve the generalized pantograph equations. Explicit formulae which express the Jacobi expansion coefficients for the shift of derivatives and moments of any differentiable function in terms of the original expansion coefficients of the function itself are given in the matrix form. The main importance of the approach is that using this scheme reduces solving the generalized pantograph equations to solve a system of linear algebraic equations, thus greatly simplifying the problem. Some experiments are given to demonstrate the validity and applicability of the method.

The General Two Dimensional Shifted Jacobi Matrix Method for Solving the Second Order Linear Partial Difference-differential Equations with Variable Coefficients

In this paper, a new and efficient approach for numerical approximation of second order linear partial differential-difference equations (PDDEs) with variable coefficients is introduced. Explicit formulae which express the two dimensional Jacobi expansion coefficients for the derivatives and moments of any differentiable function in terms of the original expansion coefficients of the function itself are given in the matrix form. The main importance of this scheme is that using this approach reduces solving the general linear PDDEs to solve a system of linear algebraic equations, wherein greatly simplify the problem. In addition, some experiments are given to demonstrate the validity and applicability of the method.

The general Jacobi matrix method for solving some nonlinear ordinary differential equations

In this paper, we obtain the approximate solutions for some nonlinear ordinary differential equations by using the general Jacobi matrix method. Explicit formulae which express the Jacobi expansion coefficients for the powers of derivatives and moments of any differentiable function in terms of the original expansion coefficients of the function itself are given in the matrix form. Three test problems are discussed to illustrate the efficiency of the proposed method.

On the Derivatives of Bernstein Polynomials: An Application for the Solution of High Even-Order Differential Equations

A new formula expressing explicitly the derivatives of Bernstein polynomials of any degree and for any order in terms of Bernstein polynomials themselves is proved, and a formula expressing the Bernstein coefficients of the general-order derivative of a differentiable function in terms of its Bernstein coefficients is deduced. An application of how to use Bernstein polynomials for solving high even-order differential equations by Bernstein Galerkin and Bernstein Petrov-Galerkin methods is described. These two methods are then tested on examples and compared with other methods. It is shown that the presented methods yield better results.

Recurrences and explicit formulae for the expansion and connection coefficients in series of ordinary Bessel polynomials

A formula expressing explicitly the derivatives of ordinary Bessel polynomials of any degree and for any order in terms of the ordinary Bessel polynomials themselves is proved. Another explicit formula, which expresses the ordinary Bessel expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of its original ordinary Bessel coefficients, is also given. A formula for the ordinary Bessel coefficients of the moments of one single ordinary Bessel polynomial of certain degree is proved. Formula for the ordinary Bessel coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its ordinary Bessel coefficients is also obtained. Application of these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between generalized Bessel and ordinary Bessel polynomials is described. Explicit formula for these coefficients between Jacobi and ordinary Bessel polynomials is given, of which the ultraspherical polynomials and its consequences are important special cases. Two analytical formulae for the connection coefficients between Laguerre–ordinary Bessel and Hermite–ordinary Bessel are also developed.

Recurrences and explicit formulae for the expansion and connection coefficients in series of Bessel polynomials

A formula expressing explicitly the derivatives of Bessel polynomials of any degree and for any order in terms of the Bessel polynomials themselves is proved. Another explicit formula, which expresses the Bessel expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of its original Bessel coefficients, is also given. A formula for the Bessel coefficients of the moments of one single Bessel polynomial of certain degree is proved. A formula for the Bessel coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Bessel coefficients is also obtained. Application of these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between Bessel–Bessel polynomials is described. An explicit formula for these coefficients between Jacobi and Bessel polynomials is given, of which the ultraspherical polynomial and its consequences are important special cases. Two analytical formulae for the connection coefficients between Laguerre–Bessel and Hermite–Bessel are also developed.

On the connection coefficients and recurrence relations arising from expansions in series of hermite polynomials

A formula expressing the Hermite coefficients of a general-order derivative of an infinitely differentiable function in terms of its original coefficients is proved, and a formula expressing explicitly the derivatives of Hermite polynomials of any degree and for any order as a linear combination of suitable Hermite polynomials is deduced. A formula for the Hermite coefficients of the moments of one single Hermite polynomial of certain degree is given. Formulae for the Hermite coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Hermite coefficients are also obtained. Two numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations in Hermite coefficients, are discussed. A simple approach in order to build and solve recursively for the connection coefficients between Jacobi–Hermite and Laguerre–Hermite polynomials is described. Explicit formula for these coefficients between Jacobi and Hermite polynomials is given, of which the ultraspherical polynomials and the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. Analytical formula for the connection coefficients between Laguerre and Hermite polynomials is also obtained.

On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials

Formulae expressing explicitly the Jacobi coefficients of a general-order derivative (integral) of an infinitely differentiable function in terms of its original expansion coefficients, and formulae for the derivatives (integrals) of Jacobi polynomials in terms of Jacobi polynomials themselves are stated. A formula for the Jacobi coefficients of the moments of one single Jacobi polynomial of certain degree is proved. Another formula for the Jacobi coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its original expanded coefficients is also given. A simple approach in order to construct and solve recursively for the connection coefficients between Jacobi–Jacobi polynomials is described. Explicit formulae for these coefficients between ultraspherical and Jacobi polynomials are deduced, of which the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. Two analytical formulae for the connection coefficients between Laguerre–Jacobi and Hermite–Jacobi are developed.

On the connection coefficients and recurrence relations arising from expansions in series of Laguerre polynomials

A formula expressing the Laguerre coefficients of a general-order derivative of an infinitely differentiable function in terms of its original coefficients is proved, and a formula expressing explicitly the derivatives of Laguerre polynomials of any degree and for any order as a linear combination of suitable Laguerre polynomials is deduced. A formula for the Laguerre coefficients of the moments of one single Laguerre polynomial of certain degree is given. Formulae for the Laguerre coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Laguerre coefficients are also obtained. A simple approach in order to build and solve recursively for the connection coefficients between Jacobi–Laguerre and Hermite–Laguerre polynomials is described. An explicit formula for these coefficients between Jacobi and Laguerre polynomials is given, of which the ultra-spherical polynomials of the first and second kinds and Legendre polynomials are important special cases. An analytical formula for the connection coefficients between Hermite and Laguerre polynomials is also obtained.

On the coefficients of differentiated expansions and derivatives of Jacobi polynomials

A formula expressing explicitly the derivatives of Jacobi polynomials of any degree and for any order in terms of the Jacobi polynomials themselves is proved. Another explicit formula, which expresses the Jacobi expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of its original Jacobi coefficients, is also given. The results for the special case of ultraspherical polynomials are considered. The results for Chebyshev polynomials of the first and second kinds and for Legendre polynomials are also noted. An application of how to use Jacobi polynomials for solving ordinary and partial differential equations is described.

The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function

A formula for the ultraspherical coefficients of the moments of one single ultraspherical polynomial of certain degree is given. Formulae for the ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its ultraspherical coefficients are also obtained. The corresponding formulae for the important special cases of Chebyshev polynomials of the first and second kinds and of Legendre polynomials are deduced. Two interesting numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations of lowest order in the ultraspherical expansion coefficients, in the sense of Lewanowicz, are discussed. Comparisons with the results obtained by optimal algorithm of Lewanowicz (1976) are also made.

On the legendre coefficients of the moments of the general order derivative of an infinitely differentiable function

Formulae for the Legendre coefficients of the moments of the general order derivative of an infinitely differentiable function in terms of its Legendre coefficients are derived. Two numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations of lowest order in the Legendre expansion coefficients are discussed. Comparisons with the results obtained by the optimal algorithm of Lewanowicz (1976) are made.

The first and second kind chebyshev coefficients of the moments for the general order derivative on an infinitely differentiable function

Expressions for the first and second kinds Chebyshev coefficients of the moments of the general order derivative of an infinitely differentiable function in terms of its Chebyshev coefficients are given. Two numerical applications of how to use these expressions for solving ordinary differential equations with polynomial coefficients are described. Comparisons with the results obtained by Lewanowicz optimum algorithm (1976) are noted.

The coefficients of differentiated expansions and derivatives of ultraspherical polynomials

A formula expressing the ultraspherical coefficients of the general order derivative of an infinitely differentiable function in terms of its original ultraspherical coefficients is stated in a more compact form and proved in a simpler way than the formula suggested by Karageorghis and Phillips in their recent report [5]. Formulas expressing explicitly the derivatives of ultraspherical polynomials of any degree and for any order in terms of the ultraspherical polynomials are given. The special cases of Chebyshev polynomials of the first and second kinds and of Legendre polynomials are considered. An application of how to use ultraspherical polynomials for solving ordinary and partial differential equations is described.

A note on the Chebyshev coefficients of the moments of the general order derivative of an infinitely differentiable function

An expression for the Chebyshev coefficients of the moments of the general order derivative of an infinitely differentiable function in terms of its Chebyshev coefficients is suggested.