On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations

AbstractAn analytical formula expressing the ultraspherical coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is stated in a more compact form and proved in a simpler way than the formula suggested by Phillips and Karageorghis (27 (1990) 823). A new formula expressing explicitly the integrals of ultraspherical polynomials of any degree that has been integrated an arbitrary number of times of ultraspherical polynomials is given. The tensor product of ultraspherical polynomials is used to approximate a function of more than one variable. Formulae expressing the coefficients of differentiated expansions of double and triple ultraspherical polynomials in terms of the original expansion are stated and proved. Some applications of how to use ultraspherical polynomials for solving ordinary and partial differential equations are described

The coefficients of differentiated expansions and derivatives of ultraspherical polynomials

AbstractA 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

Integrals of Bernstein polynomials: An application for the solution of high even-order differential equations

AbstractA new explicit formula for the integrals of Bernstein polynomials of any degree for any order in terms of Bernstein polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of high even-order boundary value problems (BVPs) with two point boundary conditions but by considering their integrated forms. The Bernstein–Petrov–Galerkin method (BPG) is applied to construct the numerical solution for such problems. The method is then tested on examples and compared with other methods. It is shown that the BPG yields better results

An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index

The shifted Legendre orthogonal polynomials are used for the numerical solution of a new formulation for the multi-dimensional fractional optimal control problem (M-DFOCP) with a quadratic performance index. The fractional derivatives are described in the Caputo sense. The Lagrange multiplier method for the constrained extremum and the operational matrix of fractional integrals are used together with the help of the properties of the shifted Legendre orthonormal polynomials. The method reduces the M-DFOCP to a simpler problem that consists of solving a system of algebraic equations. For confirming the efficiency and accuracy of the proposed scheme, some test problems are implemented with their approximate solutions

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

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

New ultraspherical wavelets collocation method for solving 2nth-order initial and boundary value problems

In this paper, a new spectral algorithm based on employing ultraspherical wavelets along with the spectral collocation method is developed. The proposed algorithm is utilized to solve linear and nonlinear even-order initial and boundary value problems. This algorithm is supported by studying the convergence analysis of the used ultraspherical wavelets expansion. The principle idea for obtaining the proposed spectral numerical solutions for the above-mentioned problems is actually based on using wavelets collocation method to reduce the linear or nonlinear differential equations with their initial or boundary conditions into systems of linear or nonlinear algebraic equations in the unknown expansion coefficients. Some specific important problems such as Lane–Emden and Burger’s type equations can be solved efficiently with the suggested algorithm. Some numerical examples are given for the sake of testing the efficiency and the applicability of the proposed algorithm