Orthonormal Bernstein Polynomials Research Articles

Logarithmic Bernstein functions for fractional Rosenau-Hyman equation with the Caputo-Hadamard derivative

In this study, the Caputo-Hadamard derivative is fittingly used to define a fractional form of the Rosenau-Hyman equation. To solve this equation, the orthonormal logarithmic Bernstein functions (BFs) are created as a suitable basis for handling this type of derivative. The primary benefit of these functions lies in the ease of computing their Hadamard fractional integral and derivative. These logarithmic functions, combined with the orthonormal Bernstein polynomials (BPs), are simultaneously employed to develop a hybrid strategy for solving the aforementioned equation. More precisely, the orthonormal logarithmic BFs are utilized to approximate the solution in the temporal domain and the orthonormal BPs are employed in the spatial domain. In addition, a matrix is extracted for the Hadamard integral of the orthonormal logarithmic BFs due to the implementation of the presented method. The effectiveness of the established scheme in finding accurate numerical solutions is evaluated through the resolution of three examples.

Bernstein Polynomials for Solving Fractional Differential Equations with Two Parameters

This work presents a general framework for solving generalized fractional differential equations based on operational matrices of the generalized Bernstein polynomials. This method effectively obtains approximate analytical solutions of many fractional differential equations. The generalized fractional derivative of the Caputo type with two parameters and its properties are studied. Using orthonormal Bernstein polynomials has led to the development of fractional polynomials, which offer an approximate solution for ordinary fractional differential equations. The approach employs the generalized orthogonal Bernstein polynomials (FOBPs) and constructs their operational matrices for fractional integration and derivative in the generalized Caputo sense to achieve this objective. Operational matrices convert ordinary differential equations into a system of algebraic equations, which can be solved using Newton’s method. The convergence analysis and error estimate associated with the proposed problem have been investigated using the approximation of generalized orthogonal Bernstein polynomials (FOBPs). The effect of the new parameters of the fractional derivative is presented in several examples. Finally, several examples are included to clarify the proposed technique’s validity, efficiency, and applicability via generalized orthogonal Bernstein polynomials (FOBPs) approximation.

Numerical Methods Based on the Hybrid Shifted Orthonormal Polynomials and Block‐Pulse Functions for Solving a System of Fractional Differential Equations

This paper develops two numerical methods for solving a system of fractional differential equations based on hybrid shifted orthonormal Bernstein polynomials with generalized block‐pulse functions (HSOBBPFs) and hybrid shifted orthonormal Legendre polynomials with generalized block‐pulse functions (HSOLBPFs). Using these hybrid bases and the operational matrices method, the system of fractional differential equations is reduced to a system of algebraic equations. Error analysis is performed and some simulation examples are provided to demonstrate the efficacy of the proposed techniques. The numerical results of the proposed methods are compared to those of the existing numerical methods. These approaches are distinguished by their ability to work on the wide interval [0, a], as well as their high accuracy and rapid convergence, demonstrating the utility of the proposed approaches over other numerical methods.

Numerical solution of distributed-order fractional 2D optimal control problems using the Bernstein polynomials

In this work, a new class of two-dimensional optimal control problems is introduced with the help of distributed-order fractional derivative in the Caputo form. The orthonormal Bernstein polynomials are used to make a numerical method to solve these problems. Through this way, some operational matrices are obtained for the classical and fractional derivatives of these polynomials to make effective utilisation of them in constructing the proposed method. The main advantage of the established method is that it turns the solution of the problem under consideration into a system of algebraic equations by approximating the state and control variables by the expressed polynomials and applying the method of Lagrange multipliers. The accuracy and capability of the proposed approach are investigated by solving some numerical examples.

An efficient solution of system of generalized Abel integral equations using Bernstein polynomials wavelet bases

This work introduces a direct method based on orthonormal Bernstein polynomials wavelet bases, to present a stable algorithm for numerical inversion of a system of generalized Abel integral equations. The application of all the currently existing numerical inversion methods was strictly limited to only one portion of the generalized Abel integral equations. The proposed method is quite accurate, and several numerical illustrations demonstrate the convergence and utilization of the proposed method compared to some of the preexisting numerical solution techniques. The permanence of the numerical result under the effect of small perturbation in input data has been examined, which is depicted with the use of numerical illustrations.

Orthonormal Bernstein polynomials for solving nonlinear variable‐order time fractional fourth‐order diffusion‐wave equation with nonsingular fractional derivative

This article defines a novel version of nonlinear fourth‐order diffusion‐wave equation, which involves variable‐fractional differentiations with nonsingular kernel. This newly defined equation entails a highly accurate method for its numerical solution. The proposed scheme is based upon the orthonormal Bernstein polynomials and their operational matrices of variable‐order fractional differentiation together with the collocation method. In this paper, the formulation for obtaining these operational matrices are provided in details. The method converts the original problem to a system of algebraic equations that can be promptly solved using a software program. To clarify the validity and the high precision of the devised method, numerous illustrative examples with various boundary conditions are examined.

Orthonormal Bernstein polynomials for Volterra integral equations of the second kind

The purpose of this research is to provide an effective numerical method for solving linear Volterra integral equations of the second kind. The mathematical modeling of many phenomena in various branches of sciences lead into an integral equation. The proposed approach is based on the method of moments (Galerkin- Ritz) using orthonormal Bernstein polynomials. To solve a Volterra integral equation, the ap-proximation for a solution is considered as an expansion in terms of Bernstein orthonormal polynomials. Ultimately, the usefulness and extraordinary accuracy of the proposed approach will be verified by a few examples where the results are plotted in diagrams, Also the re-sults and relative errors are presented in some Tables. Â

Numerical study of singular fractional Lane–Emden type equations arising in astrophysics

The well-known Lane–Emden equation plays an important role in describing some phenomena in mathematical physics and astrophysics. Recently, a new type of this equation with fractional order derivative in the Caputo sense has been introduced. In this paper, two computational schemes based on collocation method with operational matrices of orthonormal Bernstein polynomials are presented to obtain numerical approximate solutions of singular Lane–Emden equations of fractional order. Four illustrative examples are implemented in order to verify the efficiency and demonstrate solution accuracy.

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

In this paper, a new two‐dimensional fractional polynomials based on the orthonormal Bernstein polynomials has been introduced to provide an approximate solution of nonlinear fractional partial Volterra integro‐differential equations. For this aim, the fractional‐order orthogonal Bernstein polynomials (FOBPs) are constructed, and its operational matrices of integration, fractional‐order integration, and derivative in the Caputo sense and product operational matrix are derived. These operational matrices are utilized to reduce the under study problem to a nonlinear system of algebraic equations. Using the approximation of FOBPs, the convergence analysis and error estimate associated to the proposed problem have been investigated. Finally, several examples are included to clarify the validity, efficiency, and applicability of the proposed technique via FOBPs approximation.

An Efficient Algorithm for Solving Integro-Differential Equation

In this paper, we study and modify an approximate method as well as a new collocation method, which is based on orthonormal Bernstein polynomials to find approximate solutions of mixed linear delay Fredholm integro-differential-difference equations under the mixed conditions. The main purpose of this paper is to study and develop some approximate methods to solve the mixed linear delay Fredholm integro-differential-difference equations. We employ a new algorithm to find approximate solution via perpendicular Bernstein polynomials on the interval [0,1], and we construct a new matrix of derivatives that will be used to find an approximate solution of matrix equation, that will reduce it to the systems of linear algebraic equations. We study the convergence approximate solutions to the exact solutions. Finally, two examples are given and their results are shown in figures to illustrate the efficiency and accuracy of this method. All the computations are implemented using Math14.

Numerical solution based on two-dimensional orthonormal Bernstein polynomials for solving some classes of two-dimensional nonlinear integral equations of fractional order

In this paper, we develop a numerical scheme based on two-dimensional orthonormal Bernstein polynomials (2D-OBPs) to solve two-dimensional nonlinear integral equations of fractional order. The fractional integral considered here is in the Riemann–Liouville sense. By using definition of Riemann–Liouville fractional integral, two-dimensional nonlinear fractional integral equations is transformed into two-dimensional nonlinear ordinary integral equations. Operational matrices method based on 2D-OBPs are applied to obtain an approximate solution with high accuracy for these equations. In addition, error analysis of the proposed method is discussed and an upper error bound is provided under weak assumptions. Some linear and nonlinear examples are given to demonstrate the accuracy, efficiency and speed of the suggested method.

A numerical approach for solving weakly singular partial integro‐differential equations via two‐dimensional‐orthonormal Bernstein polynomials with the convergence analysis

In this paper, we develop an efficient matrix method based on two‐dimensional orthonormal Bernstein polynomials (2D‐OBPs) to provide approximate solution of linear and nonlinear weakly singular partial integro‐differential equations (PIDEs). First, we approximate all functions involved in the considerable problem via 2D‐OBPs. Then, by using the operational matrices of integration, differentiation, and product, the solution of Volterra singular PIDEs is transformed to the solution of a linear or nonlinear system of algebraic equations which can be solved via some suitable numerical methods. With a small number of bases, we can find a reasonable approximate solution. Moreover, we establish some useful theorems for discussing convergence analysis and obtaining an error estimate associated with the proposed method. Finally, we solve some illustrative examples by employing the presented method to show the validity, efficiency, high accuracy, and applicability of the proposed technique.

Alternative methods for solving nonlinear two-point boundary value problems

Abstract In this sequel, the numerical solution of nonlinear two-point boundary value problems (NTBVPs) for ordinary differential equations (ODEs) is found by Bezier curve method (BCM) and orthonormal Bernstein polynomials (OBPs). OBPs will be constructed by Gram-Schmidt technique. Stated methods are more easier and applicable for linear and nonlinear problems. Some numerical examples are solved and they are stated the accurate findings.

A reliable algorithm based on the shifted orthonormal Bernstein polynomials for solving Volterra–Fredholm integral equations

ABSTRACTThis paper deals with the numerical solution of Volterra–Fredholm integral equations. In this work, we approximate the unknown functions based on the shifted orthonormal Bernstein polynomials, in conjunction with the least-squares approximation method. The method is using a simple computational manner to obtain a quite acceptable approximate solution. The merits of this method lie in the fact that, on the one hand, the problem will be reduced to a system of algebraic equations. On the other hand, the efficiency and accuracy of the method for solving these equations are high. The convergence analysis of proposed method have been discussed through some theorems. Moreover, we will obtain an estimation of error bound for this algorithm. Finally, some examples are given to show the capability of presented method in comparison with four well-known algorithms in the literature namely the Legendre collocation method, Taylor collocation method, Taylor polynomial method and Lagrange collocation method.

Numerical solution of high order linear complex differential equations via complex operational matrix method

In this work, a numerical method based on a complex operational matrix is utilized to solve high order linear complex differential equations under mixed initial conditions. For this aim, we introduce orthonormal Bernstein polynomials (OBPs), and we obtain their complex operational matrix of differentiation. The main advantage of the proposed method is that by using this method complex differential equations reduce to a linear system of algebraic equations which can be solved by using an appropriate iterative method. To, some results concerning the error analysis associated with the present method are discussed. Finally, we give some numerical examples to reveal accuracy and efficiency of the proposed method. Also, the numerical results obtained by this method are compared with numerical results achieved from other existing methods.

Parameters estimation of HIV infection model of CD4+T-cells by applying orthonormal Bernstein collocation method

The HIV infection model of CD4[Formula: see text][Formula: see text]T-cells corresponds to a class of nonlinear ordinary differential equation systems. In this study, we provide the approximate solution of this model by using orthonormal Bernstein polynomials (OBPs). By applying the proposed method, the nonlinear system of ordinary differential equations reduces to a nonlinear system of algebraic equations which can be solved by using a suitable numerical method such as Newton’s method. We prove some useful theorems concerning the convergence and error estimate associated to the present method. Finally, we apply the proposed method to get the numerical solution of this model with the arbitrary initial conditions and values. Furthermore, the numerical results obtained by the suggested method are compared with the results achieved by other previous methods. These results indicate that this method agrees with other previous methods.

Investigation of magneto-hemodynamic flow in a semi-porous channel using orthonormal Bernstein polynomials

In this paper, the problem of the magneto-hemodynamic laminar viscous flow of a conducting physiological fluid in a semi-porous channel under a transverse magnetic field is investigated numerically. Using a Berman’s similarity transformation, the two-dimensional momentum conservation partial differential equations can be written as a system of nonlinear ordinary differential equations incorporating Lorentizian magneto-hydrodynamic body force terms. A new computational method based on the operational matrix of derivative of orthonormal Bernstein polynomials for solving the resulting differential systems is introduced. Moreover, by using the residual correction process, two types of error estimates are provided and reported to show the strength of the proposed method. Graphical and tabular results are presented to investigate the influence of the Hartmann number (Ha) and the transpiration Reynolds number (Re on velocity profiles in the channel. The results are compared with those obtained by previous works to confirm the accuracy and efficiency of the proposed scheme.

Application of orthonormal Bernstein polynomials to construct a efficient scheme for solving fractional stochastic integro-differential equation

Fractional stochastic integro-differential equations arise in many problems in mechanics, finance, biology, medical, social sciences. In many situation, we can not solve these equations analytically, therefore we try to get an approximate solution for these equations. In this article, we apply operational matrices method based on orthonormal Bernstein polynomials (OBPs) to solve fractional stochastic integro-differential equations. For this aim, we introduce Bernstein polynomials and OBPs and explain that how an arbitrary function can be expanded by using these polynomials. We get operational matrix and stochastic operational matrix based on OBPs. By applying these matrices fractional stochastic integro-differential equations convert to a linear system of algebraic equations which can be solved by a appropriate numerical method. Also, convergence and error analysis of the proposed method is discussed and an upper error bound of the numerical method are given. Finally, Accuracy and efficiency of the present method are shown with two examples.

Solving generalized pantograph equations by shifted orthonormal Bernstein polynomials

In this paper, we introduce Shifted Orthonormal Bernstein Polynomials (SOBPs) and derive the operational matrices of integration and delays for these polynomials. Then, we apply them to convert the pantograph equations to a system of linear equations. An important property of this method is that the condition number of the coefficient matrix of the system is small which confirms that our method is stable. Error analysis and comparison with other methods are given to confirm the validity, efficiency and applicability of the proposed method.

New Operational Matrices of Seventh Degree Orthonormal Bernstein Polynomials

Based on analyzing the properties of Bernstein polynomials, the extended orthonormal Bernstein polynomials, defined on the interval [0, 1] for n=7 is achieved. Another method for computing operational matrices of derivative and integration D_b and R_(n+1)^B respectively is presented. Also the result of the proposed method is compared with true answers to show the convergence and advantages of the new method.