This paper introduces a natural extension of Kolchin’s differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard-Vessiot theory recently developed by Matzat and van der Put. We use the methods and framework provided by the model theory of iterative differential fields. We offer a definition of strongly normal ex...

In this paper, we introduce and study a class of new Picard-Mann iterative methods with mixed errors for common fixed points of two different nonexpansive and contraction operators. We also give convergence and stability analysis of the new Picard-Mann iterative approximation and propose numerical examples to show that the new Picard-Mann iteration converges more effectively than the Picard ite...

In this paper we present an efficient branch-following procedure that can be used not only to compute branches of periodic solutions of periodically forced dynamical systems but also to determine the stability of the periodic solutions. The procedure combines Broyden’s method with a subspace iteration method to determine the dominant eigenvalues. The method has connections with the hybrid Newto...

We present a modified Picard-like method to solve absolute value equations by equivalently expressing the implicit fixed-point equation form of as two-by-two block nonlinear equation. This unifies some existing matrix splitting algorithms and improves efficiency algorithm introducing parameter ω. For choice ω in new method, we give way determine quasi-optimal values. Numerical examples are give...

This paper presents a collocation method with an iterative linear system solver to compute periodic solutions of a system of autonomous delay differential equations (DDEs). We show that the linearized collocation system is equivalent to a discretization of the linearized periodic boundary value problem (BVP). This linear BVP is solved using the Newton-Picard single shooting method ([Int. J. Bif...

This paper presents a new feedback-accelerated Picard iteration method for solving long-term orbit-propagation problemsandperturbedLambert’s problems.Thismethod is developedby combining the collocationmethodand the variational iterationmethod over large time steps. The resulting iterative formulas are explicitly derived so that they canbedirectly adopted to solve problems in orbitalmechanics. S...

The problem of downward continuation of the gravity field from the Earth’s surface to the reference ellipsoid arises from the fact that the solution to the boundary value problem for geoid determination without applying Stokes formula is sought in terms of the disturbing potential on the ellipsoid but the gravity observations are only available on the Earth’s surface. Downward continuation is a...

Periodic solutions of certain large-scale systems of ODEs can be computed ee-ciently using a hybrid Newton{Picard scheme, especially in a continuation context. In this paper we describe and analyse how this approach can be extended to the direct computation of period doubling bifurcation points. The Newton{Picard scheme is based on shooting and a splitting of the state space in a low-dimensiona...

We study differential equations satisfied by modular forms of two variables associated to Γ1×Γ2, where Γi (i = 1, 2) are genus zero subgroups of SL2(R) commensurable with SL2(Z), e.g., Γ0(N) or Γ0(N)∗ for some N . In some examples, these differential equations are realized as the Picard–Fuchs differential equations of families of K3 surfaces with large Picard numbers, e.g., 19, 18, 17, 16. Our ...