Efficient high-order methods based on golden ratio for nonlinear systems

We derive new iterative methods with order of convergence four or higher, for solving nonlinear systems, by composing iteratively golden ratio methods with a modified Newton’s method. We use different efficiency indices in order to compare the new methods with other ones and present several numerical tests which confirm the theoretical results.     

A class of Steffensen type methods with optimal order of convergence

In this paper, a family of Steffensen type methods of fourth-order convergence for solving nonlinear smooth equations is suggested. In the proposed methods, a linear combination of divided differences is used to get a better approximation to the derivative of the given function. Each derivative-free member of the family requires only three evaluations of the given function per iteration. Therefore, this class of methods has efficiency index equal to 1.587. Kung and Traub conjectured that the order of convergence of any multipoint method without memory cannot exceed the bound 2^d-1, where d is the number of functional evaluations per step. The new class of methods agrees with this conjecture for the case d=3. Numerical examples are made to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other ones.     

Graph convergence for the H(·, ·)-accretive operator in Banach spaces with an application

In this paper, a concept of graph convergence concerned with the H(·, ·)-accretive operator is introduced in Banach spaces and some equivalence theorems between of graph-convergence and resolvent operator convergence for the H(·, ·)-accretive operator sequence are proved. As an application, a perturbed algorithm for solving a class of variational inclusions involving the H(·, ·)-accretive operator is constructed. Under some suitable conditions, the existence of the solution for the variational inclusions and the convergence of iterative sequence generated by the perturbed algorithm are also given.     

LDG method for reaction-diffusion dynamical systems with time delay

In this paper we introduce a local discontinuous Galerkin method to solve nonlinear reaction-diffusion dynamical systems with time delay. Stability and convergence of the schemes are obtained. Finally, numerical examples on two biologic models are shown to demonstrate the accuracy and stability of the method.     

Convergence of an adaptive Ka?anov FEM for quasi-linear problems

We design an adaptive finite element method to approximate the solutions of quasi-linear elliptic problems. The algorithm is based on a Ka?anov iteration and a mesh adaptation step is performed after each linear solve. The method is thus inexact because we do not solve the discrete nonlinear problems exactly, but rather perform one iteration of a fixed point method (Ka?anov), using the approximation of the previous mesh as an initial guess. The convergence of the method is proved for any reasonable marking strategy and starting from any initial mesh. We conclude with some numerical experiments that illustrate the theory.     

A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients

We provide a rate for the strong convergence of Euler approximations for stochastic differential equations (SDEs) whose diffusion coefficient is not Lipschitz but only (1/2+α)-Hölder continuous for some α≥0.     

Entropy solution theory for fractional degenerate convection–diffusion equations

We study a class of degenerate convection-diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection-diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Lévy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.     

Three-step iterative methods with optimal eighth-order convergence

In this paper, based on Ostrowski’s method, a new family of eighth-order methods for solving nonlinear equations is derived. In terms of computational cost, each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative, so that their efficiency indices are 1.682, which is optimal according to Kung and Traub’s conjecture. Numerical comparisons are made to show the performance of the new family.     

On the convergence of spline collocation methods for solving fractional differential equations

In the first part of this paper we study the regularity properties of solutions of initial value problems of linear multi-term fractional differential equations. We then use these results in the convergence analysis of a polynomial spline collocation method for solving such problems numerically. Using an integral equation reformulation and special non-uniform grids, global convergence estimates are derived. From these estimates it follows that the method has a rapid convergence if we use suitable nonuniform grids and the nodes of the composite Gaussian quadrature formulas as collocation points. Theoretical results are verified by some numerical examples.     

Hybrid polynomial approximation to higher derivatives of rational curves

In this paper, we extend the results published in JCAM volume 214 pp. 163-174 in 2008. Based on the bound estimates of higher derivatives of both Bernstein basis functions and rational Bézier curves, we prove that for any given rational Bézier curve, if the convergence condition of the corresponding hybrid polynomial approximation is satisfied, then not only the l-th (l=1,2,3) derivatives of its hybrid polynomial approximation curve uniformly converge to the corresponding derivatives of the rational Bézier curve, but also this conclusion is tenable in the case of any order derivative. This result can expand the area of applications of hybrid polynomial approximation to rational curves in geometric design and geometric computation.