A new operational matrix based on Bernoulli polynomials

In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized to transform the differential equation to a matrix equation which corresponds to a system of algebraic equations with unknown Bernoulli coefficients. This method can be used for many problems such as differential equations, integral equations and so on. Numerical examples show the method is computationally simple and also illustrate the efficiency and accuracy of the method

On An Operational Matrix Method Based On Generalized Bernoulli Polynomials Of Level M

An operational matrix method based on generalized Bernoulli polynomials of level m is introduced and analyzed in order to obtain numerical solutions of initial value problems. The most innovative component of our method comes, essentially, from the introduction of the generalized Bernoulli polynomials of level m, which generalize the classical Bernoulli polynomials. Computational results demonstrate that such operational matrix method can lead to very ill-conditioned matrix equations

A new approach to find an approximate solution of linear initial value problems with high degree of accuracy

This work investigates a new approach to find closed form solution to linear initial value problems (IVP). Classical Bernoulli polynomials have been used to derive a finite set of orthonormal polynomials and a finite operational matrix to simplify derivatives in IVP. These orthonormal polynomials together with the operational matrix of relevant order provides a robust approximation to the solution of a linear initial value problem by converting the IVP into a set of algebraic equations. Depending upon the nature of a problem, a polynomial of degree n or numerical approximation can be obtained. The technique has been demonstrated through four examples. In each example, obtained solution has been compared with available exact or numerical solution. High degree of accuracy has been observed in numerical values of solutions for considered problems

Shifted Legendre polynomial solutions of nonlinear stochastic Itô - Volterra integral equations

In this article, we propose the shifted Legendre polynomial-based solution for solving a stochastic integral equation. The properties of shifted Legendre polynomials are discussed. Also, the stochastic operational matrix required for our proposed methodology is derived. This operational matrix is capable of reducing the given stochastic integral equation into simultaneous equations with N+1 coefficients, where N is the number of terms in the truncated series of function approximation. These unknowns can be found by using any well-known numerical method. In addition to the capability of the operational matrices, an essential advantage of the proposed technique is that it does not require any integration to compute the constant coefficients. This approach may also be used to solve stochastic differential equations, both linear and nonlinear, as well as stochastic partial differential equations. We also prove the convergence of the solution obtained through the proposed method in terms of the expectation of the error function. The upper bound of the error in L² norm between exact and approximate solutions is also elaborately discussed. The applicability of this methodology is tested with a few numerical examples, and the quality of the solution is validated by comparing it with other methods with the help of tables and figures.Publisher's Versio