Numerical inversion method for the Laplace transform based on Boubaker polynomials operational matrix

We investigate through this research the numerical inversion technique for the Laplace transforms cooperated by the integration Boubaker polynomials operational matrix. The efficiency of the presented approach is demonstrated by solving some differential equations. Also, this technique is combined with the standard Laplace Homotopy Perturbation Method. The numerical results highlight that there is a very good agreement between the estimated solutions with exact solutions.

An Efficient Technique for Solving Bratu Type Equation via Wavelet Orthonormal Boubaker polynomials

The aim of this research is to show the applicability of new truncated orthonormal Boubaker wavelet polynomials (OBWP's) for solving one dimensional Bratu-type equation with numerically by the aid of iteration technique. Some numerical examples were added to show the ability of this kind of polynomials comparing with exact results using Matlab. Also illustrating graphs were added to verify the efficiency of the method.

STRUM-LIOUVILLE FORM AND OTHER IMPORTANT PROPERTIES OF MODIFIED ORTHOGONAL BOUBAKER POLYNOMIALS

In this paper, some classical properties of modified orthogonal Boubaker polynomials (MOBPs) are considered, which are: the three-term recurrence relation, Rodriguez formula, characteristic differential equation and the Strum-Liouville form. The only properties of the MOBPs known so far are orthogonality evidence, weight function, orthonormality evidence and zeros. The new properties established in this work will to the applicability of the MOBPs in different areas of science and engineering where the classical non-orthogonal Boubaker polynomials could be applied, and even in cases where these cannot be applied.

ON ORTHOGONALIZATION OF BOUBAKER POLYNOMIALS

In this work, we explore some unknown properties of the Boubaker polynomials. The orthogonalization of the Boubaker polynomials has not been discussed in the literature. Since most of the application areas of such polynomial sequences demand orthogonal polynomials, the orthogonality of the Boubaker polynomials will help extend its theareas of application. We investigate orthogonality of classical Boubaker polynomials using Sturm-Liouville form and then apply the Gram-Schmidt orthogonalization process to develop modified Boubaker polynomials which are also orthogonal. Some classical properties, like orthogonality and orthonormality relation and zeros, of the modified Boubaker polynomials, have been proved. The contributions from this study have an impact on the further application of modified Boubaker polynomials to not only the cases where classical polynomials could be used but also in cases where the classical ones could not be used due to orthogonality issue.

An Iterative Method for Solving Quadratic Optimal Control Problem Using Scaling Boubaker Polynomials

In this paper, an iteration method was used for solving a quadratic optimal control problem (QOCP) by the aid of state parameterization technique and scaling Boubaker polynomials. Some numerical examples were added to show the applicability of the method, also a comparison with other method was presented. The process steps were illustrated by some numerical examples with graphs done by Matlab.

An Efficient Technique for solving Lane-Emden Equation

The orthogonal Boubaker polynomials and their operational matrix of derivatives were deduced, introduced a new efficient approximate method for solving Lane-Emden equation with initial conditions via collocation method. Some numerical examples were given to demonstrate the applicability of this method. The results have been compared with exact solutions to show that they achieved a high accuracy.

Combinatorial Determinant Formulas for Boubaker Polynomials

In this paper, we evaluate several families of Toeplitz–Hessenberg determinants whose entries are the Boubaker polynomials. Equivalently, these determinant formulas may be also rewritten as combinatorial identities involving sum of products of Boubaker polynomials and multinomial coefficients. We also present new formulas for Boubaker polynomials via recurrent three-diagonal determinants.

Some Results for Orthonormal Boubaker Polynomials with Application

In this paper, we propose an efficient approximate indirect method for solving problems in calculus of variational. First, we introduce orthonormal Boubaker polynomials along with their important properties. These properties are employed to derive a general expansion of their operational matrices of derivation and integration along with a product operational matrix. The operational matrices are then used to approximate the solution. Finally, such approximations are substituted in the functional and necessary optimal condition, which then transform the variational problem under consideration into an algebraic system. Examples illustrate the validity and accuracy of the presented method.

A New Boubaker Wavelets Operational Matrix of Integration

Many fields of science and engineering have used wavelet functions. They are established from expansion of a single mother wavelet function. Boubaker wavelet functions are presented in this paper based on the important properties of Boubaker polynomials. The research goal of this article is to drive a Boubaker wavelets operation matrix of integration in general formulas. Then an approximate solution method for solving a singular initial value problem is presented using Boubaker wavelets along the obtained operational matrix of integration. The importance of this method is that it converts a singular initial value problem in order to solve algebraic examples as a system. The process is based on reducing by means of integration the original problem into integral equations using a Boubaker wavelets operation matrix of integration to predict the integral equation. Illustrative experiments are included. In addition, computational results obtained by a Boubaker wavelets operation matrix of integration are compared with the exact solutions and other existing methods.