Bernoulli polynomial and the numerical solution of high-order boundary value problems

Abstract

In this work we present a fast and accurate numerical approach for the higher-order boundary value problems via Bernoulli collocation method. Properties of Bernoulli polynomial along with their operational matrices are presented which is used to reduce the problems to systems of either linear or nonlinear algebraic equations. Error analysis is included. Numerical examples illustrate the pertinent characteristic of the method and its applications to a wide variety of model problems. The results are compared to other methods.

Bernoulli-Type Spectral Numerical Scheme for Initial and Boundary Value Problems with Variable Order

“…In this section, we present a procedure for approximating a function [32]. In this regard, consider that…”

“…We recollect the Bernoulli operational matrices of fractional-order integration and differentiation from [32]. A new operational matrix based on boundary conditions is developed in the same subsection.…”

“…Recently, Bernoulli polynomials have been used to study various problems of fractional calculus for numerical solutions (see [28][29][30]). For further properties and applications of Bernoulli polynomials, readers should refer to [31][32][33]. The same operational matrices of Bernoulli polynomials have been extended to variable-order optimal control problems by Nemati et al [34].…”

Solving a new design of nonlinear second-order Lane–Emden pantograph delay differential model via Bernoulli collocation method

“…These polynomials are a type of non-orthogonal polynomials and have been used for solving different types of problems. For more details about some of the models that have been solved using Bernoulli polynomials, one may refer to [48][49][50] and the references therein. Bernoulli polynomial B n (η) is usually generated by the following function as:…”

Pell–Lucas collocation method for numerical solutions of two population models and residual correction

“…Recently, some model problems related to HIV infection, population growth, reactiondiffusion, gas dynamic and the non-linear porous fin problem have been solved by various numerical methods in [6][7][8][9][10]. Also, the differential equations and integral equations has been solved by numerical techniques based on various polynomial bases [11][12][13][14][15][16][17][18][19][20][21][22][23]. Besides these, lately, there are some recent developments that have obtained numerical results for non-linear PDEs [24][25][26][27][28][29][30].…”