Linear ordinary differential equations (ODEs) with periodic coefficients appear in various interesting applications, such as determining the linear stability regions of systems of vertically driven multiple pendula. Sinha and Butcher [1, 2] have obtained very good approximations to the solutions of such equations by calculating approximate Picard iterates symbolically in the parameters on which...

In this paper, we introduce a modification of the Picard iteration method (PIM) using Padé approximation and so called Picard-Padé technique. Special attention is given to study the convergence analysis of the proposed method. Convergence analysis is reliable enough to estimate the maximum absolute error of the solution given by PIM. A basic enzyme kinetics is used to test the effectiveness of ...

S U M M A R Y The stationary Reynolds equation is solved over a rectangular region. The problem is linearized by Picard linearization. The ADI method is used to solve the resulting set of linear equations. A set of parameters is introduced to speed up convergence as well for the Picard linearization as for the ADI method. A comparison is made with Booy-Coleman's method. Results are given for be...

In this article, we propose a new multigrid-based algorithm for solving integral equations of the reference interactions site model (RISM). We also investigate the relationship between the parameters of the algorithm and the numerical accuracy of the hydration free energy calculations by RISM. For this purpose, we analyzed the performance of the method for several numerical tests with polar and...

Non-linear magnetic problems are commonly solved by iterative methods. This paper discusses the application of the Picard-method and the Newton-method in the context of solving time-harmonic problems. It is shown that the overall computation time can be decreased by initiating the non-linear solution process using Picard-iterations and by switching to Newton-iterations as soon as an estimator i...

Recently Bai and Yang in [On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math. 59 (2009) 2923–2936.] proposed the Picard-HSS iteration method to solve the system of nonlinear equations Ax = φ(x), where A ∈ Cn×n is a nonHermitian positive definite matrix and φ : D ⊂ C → C is continuously differentiable function defined on the open complex domain D in the n-dimensional ...

Periodic solutions of certain large scale systems of ODEs can be computed eeciently using a hybrid Newton-Picard scheme, especially in a continuation context. In this paper we describe and analyse a numerical method to accurately compute period doubling bifurcation points using the Newton-Picard approach. The method avoids the computation of the full monodromy matrix, which is present in the de...

This paper discusses the diffusion equation with a damping term as follows ut = div(|Dum|p−2Dum)− u1 |Du|1 , where p > 2,m > 1, and p > p1, q1+p1m > m(p−1) > 1. By the standard Picard iteration method, a sufficient condition is given to the existence of the singular self-similar solutions. Moreover, the paper gives a classification for these singular self-similar solutions. Key–Words: Diffusion...

Using the Modified Picard Method of Parker and Sochacki [1, 2], we derive a hybrid scheme using the analytical Picard method with approximations to the differential operators. This new method, called Discretized Picard’s Method, uses approximations in the space dimensions to compute derivatives but utilizes a continuous approximation in the time dimension. We illustrate the method using finite ...