In this paper, we present a numerical method for solving nonlinear Hammerstein fractional integral equations. The approximates the solution by Picard iteration together with integration designed weakly singular integrals. Error analysis of proposed is also investigated. Numerical examples approve its efficiency in terms accuracy and computational cost.

Introduction The investigation of Lane Emden HLEEL equation has a long standing history Hsee, e.g., @1 10D, to mention only part of a vast literatureL. Recently Schaudt @6D has shown that Picard type iteration scheme may be used to show the existence, uniqueness and regularity of global solutions of LEE , in particular, for n 3 1. In this paper it was shown that straightforward usage of Picard ...

The well known Ostrowski theorem [1] gives a sufficient condition (the spectral radius of the Jacobian of the iteration mapping in the fixed point to be less than 1) for the local convergence of Picard iteration. “However, no estimate for the size of an attraction ball is known” [2] (2009). The problem of estimating the local radius of convergence for different iterative methods was considered ...

This paper investigates a polygonal finite element (PFE) to solve two-dimensional (2D) incompressible steady fluid problem in cavity square. It is well-known standard benchmark (i.e., lid-driven flow)-to evaluate the numerical methods solving problems controlled by Navier–Stokes (N–S) equation system. The approximation solutions provided this research are based on our developed equal-order mixe...

Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteratio...

We derive, implement, and test a posteriori error estimates for numerical methods for a non-autonomous linear system that involve iterative solution of the discrete equations. We consider two iterations: the Picard iteration and the Jacobi iteration for solving the discrete matrix-vector equations. To carry out the analysis, we define an appropriate adjoint problem for the numerical approximati...

Preconditioners for the Conjugate Gradient method are studied to solve the Newton system with symmetric positive definite (SPD) Jacobian. Following the theoretical work in [1] we start from a given approximation of the inverse of the initial Jacobian, and we construct a sequence of preconditioners by means of a low rank update, for the linearized systems arising in the Picard-Newton solution of...

In this paper we lay the foundation for a numerical algorithm to simulate high-dimensional coupled FBSDEs under weak coupling or monotonicity conditions. In particular, we prove convergence of a time discretization and a Markovian iteration. The iteration differs from standard Picard iterations for FBSDEs in that the dimension of the underlying Markovian process does not increase with the numbe...

In this paper we lay the foundation for a numerical algorithm to simulate high-dimensional coupled FBSDEs under weak coupling or monotonicity conditions. In particular, we prove convergence of a time discretization and a Markovian iteration. The iteration differs from standard Picard iterations for FBSDEs in that the dimension of the underlying Markovian process does not increase with the numbe...

In this paper, two examples of quasi-firmly type nonexpansive mappings are given to prove that the concept is different from nonexpansive mapping. Furthermore, it is studied to the convergence of the sequence of successive approximations for this class of mappings only when the super limit of iteration coefficients is less than 1. In particular, the Picard iteration {T x0} of such a mapping con...