Bernstein Polynomials Method Research Articles

Bernstein polynomials method for solving multi-order fractional neutral pantograph equations with error and stability analysis

In this investigation, we present a new method for addressing fractional neutral pantograph problems, utilizing the Bernstein polynomials method. We obtain solutions for the fractional pantograph equations by employing operational matrices of differentiation, derived from fractional derivatives in the Caputo sense applied to Bernstein polynomials. Error analysis, along with Chebyshev algorithms and interpolation nodes, is employed for solution characterization. Both theoretical and practical stability analyses of the method are provided. Demonstrative examples indicate that our proposed techniques occasionally yield exact solutions. We compare the algorithms using several established analytical methods. Our results reveal that our algorithm, based on Bernstein series solution methods, outperforms others, exhibiting superior performance with higher accuracy orders compared to those obtained from Chebyshev spectral methods, Bernoulli wavelet method, and Spectral Tau method.

Multi-objective optimization of snap-through instability of helicoidal composite imperfect beams using Bernstein polynomials method

This paper deals with the multi-objective optimization problems of snap-through instability of geometrically imperfect simply supported helicoidal composite beam with nonlinear orientation schemes. The objective function is to maximize the load required to initiate snap-through and minimize the instability region. It is assumed that the curved bioinspired beam is embedded on nonlinear elastic foundation and is subjected to different types of lateral loads. The presented formulation is based on Euler-Bernoulli theory, nonlinear higher order strain, and imperfect curvature. Bernstein polynomials (BPs) method is employed to solve the nonlinear nonhomogeneous fourth order integro-differential governing equation. Developed model can considered the effect of imperfection, lamination scheme, and elastic foundation on the nonlinear deflection and snap-through behaviors of helicoidal composite structures. Analytical closed form formula of the nonlinear load-deflection relationship is developed. The analytical relation is validated by comparing it with exact solution and numerical results available in literature. Multi-objective particle swarm optimization (MOPSO) algorithm is employed to determine Pareto-optimal set for limiting load and region of instability. In these analyses, optimal lamination parameter, amplitude of imperfection and elastic foundation constants are found for helicoidal composite beam, and the results are presented as 2D Pareto-optimal design points.

Approximate Solution for linear Fredhom Integro-Differential Equation and Integral Equation by Using Bernstein Polynomials method

في هذا البحث ، يتم استخدام طريقة بيرنشتاين متعددة الحدود لإيجاد قيمة تقريبية حل معادلة فريدهولم التكاملية التفاضلية والمعادلة التكاملية لـ النوع الثاني. تعد هذه كثيرة الحدود أدوات رياضية مفيدة بشكل لا يصدق ، لأنه يتم تعريفها ببساطة ، يمكن حسابها بسرعة على أنظمة الكمبيوتر وتمثل مجموعة هائلة من الوظائف. يمكن تمييزها و تتكامل بسهولة.

Snap-through instability of helicoidal composite imperfect beams surrounded by nonlinear elastic foundation

Bernstein polynomials are exploited, for the first time, in investigation of the nonlinear bending and snap-through instability phenomena of geometrically imperfect bioinspired composite beams with helicoidal orientation scheme. The curved bioinspired beam subjected to different types of lateral loading and embedded on three parameters elastic foundations is considered in formulation and analysis. The nonlinear equilibrium equation of the system based on Euler-Bernoulli beam hypothesis, von Kármán nonlinear strain, and initial imperfect curvature is developed. The equilibrium equation portrayed by nonlinear fourth order integro-differential equation is solved by Bernstein polynomials method and closed form solution of the load-deflection relationship is obtained. The accuracy of the present model and solution procedure is examined by comparing our results with previous results. The developed model is exploited to present the influence of lamination scheme, amplitude of initial imperfection and elastic foundation constants on the nonlinear bending and snap-through/reverse snap-through behaviors in detail. The numerical results show that the Bernstein polynomials approach can capture successfully the nonlinear equilibrium path and overcoming limit (turning) points, which usually appear in the nonlinear response of curved structures. The developed model can be used in design and control the snap-through response and open opportunities for this type of structures in many real applications.

Generalization of the Modified Bernstein Polynomials Method for Solving Fractional Coupled Equal Width Wave Equations

The objective of this research paper is to find the approximate solution to the fractional partial differential equations, which are difficult to find the exact solution, and therefore, by relying on the modified Bernstein polynomial, the study suggested generalizing the method to be available for both space and time fractional partial differential equations. In order to investigate the ability of the method to reach the approximate solution, the method was applied to the coupled of space-time-fractional of the equal width wave equations(FCEWE) using Caputo’s definition of the fractional derivative, the results of the application showed the ability of the method to find the approximate solution.

Iterative fuzzy Bernstein polynomials method for nonlinear fuzzy Volterra integral equations

An iterative method based on fuzzy Bernstein polynomials is presented for solving nonlinear fuzzy Volterra integral equations. To prove the convergence of the method, an error estimate is given in terms of Lipschitz constants. The accuracy of this method is illustrated by some numerical experiments that confirm the convergence stated in the theoretical result.

Collocation method for solving nonlinear fractional optimal control problems by using Hermite scaling function with error estimates

SummaryThis article presents an efficient numerical method for solving fractional optimal control problems (FOCPs) by utilizing the Hermite scaling function operational matrix of fractional‐order integration. The proposed technique is applied to transform the state and control variables into nonlinear programming (NLP) parameters at collocation points. The NLP solver is then used to solve FOCP. Furthermore, the L2‐error estimates in the approximation of unknown variables and the approximation of block pulse operational matrix of fractional‐order integration are derived and illustrative examples are included to demonstrate the applicability of the proposed method. Moreover, the results are compared with the Haar wavelet collocation method, hybrid of block‐pulse and Taylor polynomials method, Bernstein polynomials method, and the Boubaker hybrid function method to show the superiority of the proposed method.

Modelling leaf surface reconstruction using Bernstein polynomials method

In this research paper, we adapt for the first time Bernstein polynomials for reconstructing a continuous differentiable surfaces of a real leaf from 3D scanned data points. As such model is essential for modelling the plant architecture, fluid movement, and biological structure such as photosynthesis. To evaluate the accuracy of the proposed method, we apply it on a real scattered data points collected from a genuine Frangipani and Anthurium leaf surface using a laser scanner. We found that the Bernstein polynomials develop precise model of the leaf surface.

Numerical solution of the first order nonlinear differen-tial equations with the mixed nonlinear conditions by using PLSM(comparison with Bernstein polynomials method)

We use the Polynomial Least Squares Method (PLSM), which allows us to compute analytical approximate polynomial solutions for nonlinear ordinary differential equations with the mixed nonlinear conditions. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using Bernstein polynomials method.

Solving multipoint problems with linear Volterra–Fredholm integro-differential equations of the neutral type using Bernstein polynomials method

In this paper, the Bernstein polynomials method is proposed for the numerical solution of a class of multipoint problems with linear Volterra–Fredholm integro-differential equations of the neutral type. The main characteristic behind 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 Bernstein polynomials method for solving these equations are high, which will be shown by preparing some theorems. Also, the existence and uniqueness of solution have been proved. Finally, the numerical experiments are presented to show the excellent behavior and high accuracy of proposed scheme in comparison with some other well-known methods.

Numerical algorithm for time-fractional Sawada-Kotera equation and Ito equation with Bernstein polynomials

The generalized KdV equation arises in many problems in mathematical physics. In this paper, an effective numerical method is proposed to solve two types of time-fractional generalized fifth-order KdV equations, the time-fractional Sawada-Kotera equation and Ito equation, the idea is to use Bernstein polynomials. Firstly, Bernstein basis polynomials are utilized to approximate unknown function and the error bound is given. Secondly, the representation of Bernstein basis polynomials are proposed to easily and quickly obtain the integer and fractional differential operator of unknown function, by which the studied equations can be displayed as the combination of operator matrices. Finally, comparison with Chebyshev wavelets method and the error data are presented to demonstrate the high accuracy and efficiency of Bernstein polynomials method for this kind of wave equations.

Solving directly third-order ODEs using operational matrices of Bernstein polynomials method with applications to fluid flow equations

In this paper, we adapt for the first time the operational matrices of Bernstein polynomials method for solving directly a class of third-order ordinary differential equations (ODEs). This method gives a numerical solution by converting the equation into a system of algebraic equations which is solved directly. Applications of the present method to the famous Blasius equation describing a boundary layer flow over a flat plate and third-order ODE for thin film flow are presented. Some numerical examples are also given to show the applicability of the method.

Bernstein polynomials method and it’s error analysis for solving nonlinear problems in the calculus of variations: Convergence analysis via residual function

In this paper, Bernstein polynomials method (BPM) and their operational matrices are adopted to obtain approximate analytical solutions of variational problems. The operational matrix of differentiation is introduced and utilized to reduce the calculus of variations problems to the solution of system of algebraic equations. The solutions are obtained in the form of rapidly convergent finite series with easily computable terms. Comparison between the present method and the homotopy perturbation method (HPM), the non-polynomial spline method and the B-spline collocation method are made to show the effectiveness and efficiency for obtaining approximate solutions of the calculus of variations problems. Moreover, convergence analysis based on residual function is investigated to verified the numerical results.

Bernstein polynomials method for numerical solutions of integro-differential form of the singular Emden-Fowler initial value problems

Department of Mathematics, University Kasdi-Merbah, Ouargla, 30000, Algeria. Department of Mathematics, Ferhat-Abbas University, S ´ etif, 19000, Algeria.

Approximation Solution to Solving Linear Volterra-Fredholm Integro-Differential Equations of the Second Kind by Using Bernstein Polynomials Method

The aim of this study is to present numerical method for solving the linear Volterra-Fredholm integro-differential equations of the second kind. This method is called the Bernestein polynomials method. This technique transforms the integro-differential equations to the system of algebraic equations. Some numerical results are presented to illustrate the efficiency and accuracy of this method.

Collocation Orthonormal Bernstein Polynomials Method for Solving Integral Equations

Collocation Orthonormal Bernstein Polynomials Method for Solving Integral Equations

Numerical solution for the variable order linear cable equation with Bernstein polynomials

In this paper, Bernstein polynomials method is proposed for the numerical solution of a class of variable order fractional linear cable equation. In this paper, we adopted Bernstein polynomials basis defined on the interval [0,R] to solve the equations defined on the section Ω=[0,X]×[0,T]. The main characteristic behind this approach in this paper is that we derive two kinds of operational matrixes of Bernstein polynomials. With the operational matrixes, the initial equation is transformed into the products of several dependent matrixes which can also be viewed as the system of linear equations after dispersing the variable. By solving the linear system of algebraic equations, the numerical solutions are acquired. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical examples are provided to show that the method is computationally efficient.

On Bernstein Polynomials Method to the System of Abel Integral Equations

This paper deals with a new implementation of the Bernstein polynomials method to the numerical solution of a special kind of singular system. For this aim, first the truncated Bernstein series polynomials of the solution functions are substituted in the given problem. Using some properties of these polynomials, the solution of the problem is reduced to solve a linear system of algebraic equations. In order to confirm the reliability and accuracy of the proposed method, some weakly Abel integral equations systems with comparisons are solved in detail as numerical examples.

Solution of Fractional Differential Equations by Using Bernstein Polynomials Method

Solution of Fractional Differential Equations by Using Bernstein Polynomials Method

Bernstein Polynomials Method For Solving Linear Volterra Integral Equation of The Second Kind

Bernstein Polynomials Method For Solving Linear Volterra Integral Equation of The Second Kind