Bernstein Collocation Method Research Articles

Numerical solution of coupled system of Emden-Fowler equations using artificial neural network technique

In this paper, a deep artificial neural network technique is proposed to solve the coupled system of Emden-Fowler equations. A vectorized form of algorithm is developed. Implementation and simulation of this technique is performed using Python code. This technique is implemented in various numerical examples, and simulations are conducted. We have shown graphically how accurately this method works. We have shown the comparison of numerical solution and exact solution using error tables. We have also conducted a comparative analysis of our solution with alternative methods, including the Bernstein collocation method and the Homotopy analysis method. The comparative results are presented in error tables. The efficiency and accuracy of this method are demonstrated by these graphs and tables.

Approximate solutions for a fractional thermostat model boundary value problem via Bernstein's collocation method with Legendre polynomials

Numerous classical models of thermostats have been analyzed in the literature, and while existence and uniqueness results have been established, the investigation of other properties, such as stability, can present significant challenges. Therefore, there is a need for fractional mathematical models that are nonlocal problems and allow for the straightforward study of the asymptotic behavior of their solutions. However, solving these problems is often exceedingly complex or even impossible, necessitating the use of numerical methods to obtain approximate solutions. The focus of this research is a generalized boundary value problem that is based on a classical thermostat model. Our initial step involves determining the associated integral equation for the problem at hand. The properties of existence and uniqueness are proved by exploiting the classical contraction principle of Banach together with another kind called ‐ ‐contraction. Afterwards, thanks to the Bernstein polynomials and fractional matrix of derivative, the approximate solutions of our nonlinear fractional thermostat model problem are obtained. This approach utilizes the Legendre operational matrix of fractional derivatives in combination with a truncated Legendre series for the numerical integration of fractional differential equations. By doing so, the problem can be reduced to solving a system of algebraic equations, resulting in significant simplification. This method has been successfully used to solve both linear and nonlinear fractional differential equations. Finally, an example on this type of problem have been studied and the approximate solution are plotted.

A numerical approach based on Bernstein collocation method: Application to differential Lyapunov and Sylvester matrix equations

In this paper, we apply the Bernstein collocation method to construct the solution set of the Sylvester matrix differential equation (Sy-MDE) which involves the Lyapunov matrix differential equation. The method depends on the collocation method and Bernstein polynomials. The main advantage of the proposed method is that by using this method Sy-MDE reduces to a linear system of algebraic equations which can be solved by using an appropriate iterative method. We analyze the error and give a theorem that bounds the error. We also give the residual correction procedure to estimate the error. By using the procedure, we obtain a new approximate solution, namely a corrected Bernstein collocation solution. To illustrate how the proposed method is applied, several examples are given. Numerical experiments show the effectiveness and accuracy of the method for solving such types of Sy-MDE.

Bernstein Collocation Method for Solving High-Order Singular Differential–Difference Equations

Bernstein Collocation Method for Solving High-Order Singular Differential–Difference Equations

An efficient numerical approach for solving three-point Lane–Emden–Fowler boundary value problem

For three-point Lane–Emden–Fowler boundary value problems (LEFBVPs), we propose two robust algorithms consisting of Bernstein and shifted Chebyshev polynomials coupled with the collocation technique. The first algorithm uses the Bernstein collocation method with uniform collocation points, while the second is based on the shifted Chebyshev collocation method with roots as its collocation points. In both algorithms, the equivalent integral equations of three-point LEFBVPs are constructed. Then the approximation and collocation approach are used to generate a system of nonlinear equations. Then we implement Newton’s method to solve this system. The current method is different from the traditional method as we do not require to approximate u′ and u′′ appearing in the problem. It reduces not only the computational time but also the truncation error. The existence of a unique solution is discussed. Error analysis of both algorithms is demonstrated. A well known property of the proposed techniques is that it provides efficient solution for a few collocation points. The numerical results verify the same.

An effective approach based on Smooth Composite Chebyshev Finite Difference Method and its applications to Bratu-type and higher order Lane–Emden problems

The Smooth Composite Chebyshev Finite Difference method is generalized for higher order initial and boundary value problems. Round-off and truncation error analyses and convergence analysis of the method are also extended to higher order. The proposed method is applied to obtain the highly precise numerical solutions of boundary or initial value problems of the Bratu and higher order Lane Emden types. To visualize the competency of the presented method, the obtained results are compared with nine different methods, namely, Bezier curve method, Adomian decomposition method, Operational matrix collocation method, Direct collocation method, Haar Wavelet Collocation, Bernstein Collocation Method, Improved decomposition method, Quartic B-Spline method and New Cubic B-spline method. The comparisons show that the presented method is highly accurate than the other numerical methods and also gets rid of the singularity of the given problems.

Bernstein Collocation Method for Solving MHD Jeffery–Hamel Blood Flow Problem with Error Estimations

In this paper, the Bernstein collocation method (BCM) is used for the first time to solve the nonlinear magnetohydrodynamics (MHD) Jeffery–Hamel arterial blood flow issue. The flow model described by nonlinear partial differential equations is first transformed to a third-order one-dimensional equation. By using the Bernstein collocation method, the problem is transformed into a nonlinear system of algebraic equations. The residual correction procedure is used to estimate the error; it is simple to use and can be used even when the exact solution is unknown. In addition, the corrected Bernstein solution can be found. As a consequence, the solution is estimated using a numerical approach based on Bernstein polynomials, and the findings are verified by the 4th-order Runge–Kutta results. Comparison with the homotopy perturbation method shows that the present method gives much higher accuracy. The accuracy and efficiency of the proposed method were supported by the analysis of variance (ANOVA) and 95% of confidence on interval error. Finally, the results revealed that the MHD Jeffery–Hamel flow is directly proportional to the product of the angle between the plates α and Reynolds number Re .

Numerical Analysis of Electrohydrodynamic Flow in a Circular Cylindrical Conduit by Using Neuro Evolutionary Technique

This paper analyzes the mathematical model of electrohydrodynamic (EHD) fluid flow in a circular cylindrical conduit with an ion drag configuration. The phenomenon was modelled as a nonlinear differential equation. Furthermore, an application of artificial neural networks (ANNs) with a generalized normal distribution optimization algorithm (GNDO) and sequential quadratic programming (SQP) were utilized to suggest approximate solutions for the velocity, displacements, and acceleration profiles of the fluid by varying the Hartmann electric number (Ha2) and the strength of nonlinearity (α). ANNs were used to model the fitness function for the governing equation in terms of mean square error (MSE), which was further optimized initially by GNDO to exploit the global search. Then SQP was implemented to complement its local convergence. Numerical solutions obtained by the design scheme were compared with RK-4, the least square method (LSM), and the orthonormal Bernstein collocation method (OBCM). Stability, convergence, and robustness of the proposed algorithm were endorsed by the statistics and analysis on results of absolute errors, mean absolute deviation (MAD), Theil’s inequality coefficient (TIC), and error in Nash Sutcliffe efficiency (ENSE).

Application of Bernstein collocation method for solving the generalized regularized long wave equations

Abstract The regularized and the modified regularized long wave (RLW and MRLW) equations are solved numerically by the Bernstein polynomials in both the space and time directions based on Kronecker product. In this paper, we applied a fully different Bernstein collocation method than the other methods which used Bernstein polynomials to solve the problems. The approximate solution is defined by the Bernstein polynomials in all directions. A general form for any m derivative of any Bernstein polynomials is constructed. A general matrix form for the vector of any m derivative of any Bernstein polynomials is also constructed. Convergence study for the proposed numerical scheme is investigated. To determine the conservation properties of the RLW and MRLW equations, three invariants of motion ( I 1 , I 2 and I 3 ) are computed. To test the accuracy, two error norms ( ‖ E ‖ 2 and ‖ E ‖ ∞ ) are evaluated. Numerical outcomes and comparisons with other techniques for the single and the interaction of two solitary waves for RLW and MRLW equations are presented.

Numerical solution of system of Emden-Fowler type equations by Bernstein collocation method

The system of Emden-Fowler equations occurs frequently in physical and natural sciences. These system of equations are very difficult to handle due to their singularity and the nonlinearity. We provide a numerically robust technique for approximating the solution of the system of Emden-Fowler equations. This algorithm is based on the Bernstein polynomial accompanied by the collocation method. First, we transform the system of Emden-Fowler equations into their integral form. Then we implement the Bernstein polynomial approximation and the collocation technique to acquire a nonlinear system of equations. We apply the Newton-Raphson method to analyze the resulting nonlinear system of equations. We also discuss the error analysis of this method. We provide the numerical results of the $$L_{\infty }$$ error, the $$L_{2}$$ -norm error, and the residual error of some numerical supporting problems to examine the accuracy of the current technique. The advantage of the present method is exhibited by comparing the numerical results obtained by the proposed technique and other known techniques.

An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs

In this paper, we introduce two new methods to solve systems of ordinary differential equations. The first method is constituted of the generalized Bernstein functions, which are obtained by Bernstein polynomials, and operational matrix of differentiation with collocation method. The second method depends on tau method, the generalized Bernstein functions and operational matrix of differentiation. These methods produce a series which is obtained by non-polynomial functions set. We give the standard Bernstein polynomials to explain the generalizations for both methods. By applying the residual correction procedure to the methods, one can estimate the absolute errors for both methods and may obtain more accurate results. We apply the methods to some test examples including linear system, non-homogeneous linear system, nonlinear stiff systems, non-homogeneous nonlinear system and chaotic Genesio system. The numerical shows that the methods are efficient and work well. Increasing m yields a decrease on the errors for all methods. One can estimate the errors by using the residual correction procedure.

Solving Fractional Coupled EW and Coupled MEW Equations Using Bernstein Collocation Method

This paper deals with approximation solution for coupled of space-time-fractional of both the equal width wave equation(FCEWE) and the modified equal width wave equation (FCMEWE) using Bernstein polynomials with collocation method and employing the Caputo definition for fractional derivatives. The method reduces the coupled system to a system of algebraic equations which is simple in handling and gives the best results.

Bernstein collocation method for neutral type functional differential equation

Functional differential equations of neutral type are a class of differential equations in which the derivative of the unknown functions depends on the history of the function and its derivative as well. Due to this nature the explicit solutions of these equations are not easy to compute and sometime even not possible. Therefore, one must use some numerical technique to find an approximate solution to these equations. In this paper, we used a spectral collocation method which is based on Bernstein polynomials to find the approximate solution. The disadvantage of using Bernstein polynomials is that they are not orthogonal and therefore one cannot use the properties of orthogonal polynomials for the efficient evaluation of differential equations. In order to avoid this issue and to fully use the properties of orthogonal polynomials, a change of basis transformation from Bernstein to Legendre polynomials is used. An error analysis in infinity norm is provided, followed by several numerical examples to justify the efficiency and accuracy of the proposed scheme.

Numerical results of Emden–Fowler boundary value problems with derivative dependence using the Bernstein collocation method

In this paper, we propose an efficient numerical technique based on the Bernstein polynomials for the numerical solution of the equivalent integral form of the derivative dependent Emden–Fowler boundary value problems which arises in various fields of applied mathematics, physical and chemical sciences. The Bernstein collocation method is used to convert the integral equation into a system of nonlinear equations. This system is then solved efficiently by suitable iterative method. The error analysis of the present method is discussed. The accuracy of the proposed method is examined by calculating the maximum absolute error and the $$L_{2}$$ error of four examples. The obtained numerical results are compared with the results obtained by the other known techniques.

Novel efficient collocation method for Sturm–Liouville problems with nonlocal integral boundary conditions

This paper presents a study of the performance of the collocation methods using sinc and Bernstein bases functions to determine the eigenvalues of Sturm–Liouville problems with nonlocal boundary conditions. The two methods have the linear systems solved by the Q-Z method. This study shows that Bernstein-collocation method performs better than the sinc-collocation method for the cases considered.

An efficient numerical technique for Lane–Emden–Fowler boundary value problems: Bernstein collocation method

In this paper, we propose an efficient numerical technique for numerical solutions of the equivalent integral form of Emden–Fowler type boundary value problems (BVPs), which model many phenomena in mathematical physics and astrophysics. The Bernstein collocation method is used to convert the integral equation into a system of nonlinear equations. The iterative method is applied to solve the system numerically. The error analysis of the proposed method is provided. Several examples are provided to demonstrate the accuracy, applicability, and efficiency of the present method.

A new approach for numerical solution of two-dimensional nonlinear Fredholm integral equations in the most general kind of kernel, based on Bernstein polynomials and its convergence analysis

Regarding the efficient previous method for approximation of one-dimensional functions integrals through Bernstein polynomials in Amirfakhrian (2011), this paper presents a development of this method for approximation of two-dimensional functions integrals for the first time. Then, by combining this approximation with Bernstein collocation method for numerical solution of two-dimensional nonlinear Fredholm integral equations, the kernels double integrals of integral equations will be approximated. Combination of two-dimensional functions numerical integration method with numerical solution of integral equations method (in both methods, Bernstein polynomials were used) will result in increase of convergence speed and accuracy of the method. The convergence analysis of the method is completely presented. Finally, numerical examples are presented to illustrate the efficiency and superiority of our method in comparing it with other methods.

Multistage Bernstein collocation method for solving strongly nonlinear damped systems

In this paper, we propose an approximate solution method, called multistage Bernstein collocation method (MBCM), to solve strongly nonlinear damped systems. The method is given with an error analysis. The systems investigated include step function external excitation and periodical function external excitation. We found that the MBCM gives more accurate approximate solutions than the standard collocation method. Since MBCM splits the domain of the problem into n parts, one can overcome oscillations arising from the standard collocation method by increasing n. The results obtained from MBCM compare favorably with that of the fourth-order Runge–Kutta method (RK4).

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.

Convergence of 2D-orthonormal Bernstein collocation method for solving 2D-mixed Volterra–Fredholm integral equations

In this paper, an efficient numerical method is used for solving 2D-mixed Volterra–Fredholm integral equations. This method is based on 2D-orthonormal Bernstein polynomials (2D-OBPs) together with collocation method. This approach is applied to convert the problem under study into a system of algebraic equations which can be solved by using a convenient numerical method. Several useful theorems are proved which are concerned with the convergence and error estimate associated to the suggested scheme. Finally, by comparing the values of absolute error achieved from this method with values of absolute error obtained from other previous methods, we show that this method is very accurate and efficient.