The principal aim of this paper is to serve the numerical solution of an integral-algebraic equation (IAE) by using the Bernoulli polynomials and the residual correction method. After implementation of our scheme, the main problem would be transformed into a system of algebraic equations such that its solutions are the unknown Bernoulli coefficients. This method gives an analytic solution when ...

In this paper‎, ‎a modern method is presented to solve a class of fractional optimal control problems (FOCPs) indirectly‎. ‎First‎, ‎the necessary optimality conditions for the FOCP are obtained in the form of two fractional differential equations (FDEs)‎. ‎Then‎, ‎the unknown functions are approximated by the hybrid functions‎, ‎including Bernoulli polynomials and Block-pulse functions based o...

This paper proposes an approach for the robust averaged control of random vibrations for the Bernoulli-Euler beam equation under uncertainty in the flexural stiffness and in the initial conditions. The problem is formulated in the framework of optimal control theory and provides a functional setting which is so general as to include different types of random variables and second order random fi...

Systems of first order differential equations have been arisen in science and engineering. Specially, the systems normalized linear appear geometry kinematics problems. Solution them is quite difficult analytically; therefore, numerical methods need for approximate solution. In this study, by means a matrix method related to truncated Bernoulli series we find solutions Frenet-Like system with v...

In this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained replacing x with xα, positive α. Fractional derivatives are in Caputo sense. With help incomplete beta functions, able to build exactly Riemann–Liouville fractional integral operator associated FOHBPB. This operator, together Newton–Cotes ...

Abstract This paper describes an experiment that attempts to compare a range of existing collocation extraction techniques as well as the implementation of a new technique based on tests for lexical substitutability. After a description of the experiment details, the techniques are discussed with particular emphasis on any adaptations that are required in order to evaluate it in the way propose...

We reveal the relationship between a Petrov–Galerkin method and a spectral collocation method at the Chebyshev points of the second kind (±1 and zeros of Uk) for the two-point boundary value problem. Derivative superconvergence points are identified as the Chebyshev points of the first kind (Zeros of Tk). Super-geometric convergent rate is established for a special class of solutions.

We present optimal error estimates for spectral Petrov–Galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval. We also develop Laguerre spectral Petrov–Galerkin methods and collocation methods for fractional equations on the half line. Numerical results confirm the error estimates.