Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients

A class of weighted finite difference methods (WFDMs) for solving a class of initial-boundary value problems of space fractional partial differential equations with variable coefficients is presented. Their stability and convergence properties are considered. It is proven that the WFDMs are unconditionally-stable for , and conditionally-stable for , here r is the weighting parameter subjected to 0≤r≤1. Some convergence results are given. These methods, problems and results generalize the corresponding methods, problems and results given in [7], [8] and [10]. Some numerical examples are provided to show the effectiveness of the methods with different weighting parameters.     

On the local convergence of a deformed Newton's method under Argyros-type condition

For the iteration which was independently proposed by King [R.F. King, Tangent method for nonlinear equations, Numer. Math. 18 (1972) 298-304] and Werner [W. Werner, Über ein Verfarhren der Ordnung zur Nullstellenbestimmung, Numer. Math. 32 (1979) 333-342] for solving a nonlinear operator equation in Banach space, we established a local convergence theorem under the condition which was introduced recently by Argyros [I.K. Argyros, A unifying local-semilocal convergence analysis and application for two-point Newton-like methods in Banach space, J. Math. Anal. Appl. 298 (2004) 374-397].     

Derivative free two-point methods with and without memory for solving nonlinear equations

Two families of derivative free two-point iterative methods for solving nonlinear equations are constructed. These methods use a suitable parametric function and an arbitrary real parameter. It is proved that the first family has the convergence order four requiring only three function evaluations per iteration. In this way it is demonstrated that the proposed family without memory supports the Kung-Traub hypothesis (1974) on the upper bound 2ⁿ of the order of multipoint methods based on n + 1 function evaluations. Further acceleration of the convergence rate is attained by varying a free parameter from step to step using information available from the previous step. This approach leads to a family of two-step self-accelerating methods with memory whose order of convergence is at least and even in special cases. The increase of convergence order is attained without any additional calculations so that the family of methods with memory possesses a very high computational efficiency. Numerical examples are included to demonstrate exceptional convergence speed of the proposed methods using only few function evaluations.     

A linear backward Euler scheme for the saturation equation: Regularity results and consistency

We consider a linearization of a numerical scheme for the saturation equation (or porous medium equation) , through first order expansions of the fractional function f and the inverse of the function , after a regularization of the porous medium equation. We establish a regularity result for the Continuous Galerkin Method and a regularity result for the linearized scheme analogous to the corresponding nonlinear scheme. We then show that the linearized scheme is consistent with the nonlinear scheme analyzed in a previous work.     

Stability of perturbed nonlinear parabolic equations with Sturmian property

We study stability of an equilibrium f∗ of autonomous dynamical systems under asymptotically small perturbations of the equation. We show that such stability takes place if the domain of attraction of the equilibrium f∗ contains a one-parametric ordered family . In the stability analysis we need a special S-relation (a kind of “restricted partial ordering”) to be preserved relative to the family . This S-relation is inherited from the Sturmian zero set properties for linear parabolic equations. As main applications, we prove stability of the self-similar blow-up behaviour for the porous medium equation, the p-Laplacian equation and the dual porous medium equation in with nonlinear lower-order perturbations. For such one-dimensional parabolic equations the S-relation is Sturm's Theorem on the nonincrease of the number of intersections between the solutions and particular solutions with initial data in . This Sturmian property plays a key role and is true for the unperturbed PME, but is not true for perturbed equations.     

Approximations for strongly singular evolution equations

The problem of specification of self-adjoint operators corresponding to singular bilinear forms is very important for applications, such as quantum field theory and theory of partial differential equations with coefficient functions being distributions. In particular, the formal expression corresponds to a non-trivial self-adjoint operator ? in the space only if d?3. For spaces of larger dimensions (this corresponds to the strongly singular case), the construction of ? is much more complicated: first one should consider the space as a subspace of a wider Pontriagin space, then one implicitly specifies ?. It is shown in this paper that Schrodinger, parabolic and hyperbolic equations containing the operator ? can be approximated by explicitly defined systems of evolution equations of a larger order. The strong convergence of evolution operators taking the initial condition of the Cauchy problem to the solution of the Cauchy problem is proved.     

Pointwise estimates and Lp convergence rates to diffusion waves for p-system with damping

By introducing a new approximate Green function, we obtain the pointwise estimates on the solutions of Euler equations with linear frictional damping, from which we can deduce the optimal convergence rates to the nonlinear diffusion waves. The pointwise estimates and convergence rates given in this paper are new.     

Optimal decay rates to diffusion wave for nonlinear evolution equations with ellipticity

We derive the optimal convergence rates to diffusion wave for the Cauchy problem of a set of nonlinear evolution equations with ellipticity and dissipative effects

     

A convergence analysis of block accelerated over-relaxation iterative methods for weak block H-matrices to partition π

The aim of this paper is to establish the convergence of the block iteration methods such as the block successively accelerated over-relaxation method (BAOR) and the symmetric block successively accelerated over-relaxation method (BSAOR): Let be a weak block H-matrix to partition π, then for ,

     

A new Filter-Levenberg-Marquardt method with disturbance for solving nonlinear complementarity problems

Recently, filter methods are extensively studied to handle nonlinear programming problems. Because of good numerical results, filter techniques are attached importance to. The nonlinear complementarity problem can be reformulated as the least l₂-norm solution of an optimization problem. In this paper, basing on the filter technique and the new smoothing function, we present a new Filter-Levenberg-Marquardt method for solving the equation system with the disturbance . Under the assumption that the lever set of the problem is compact, we prove its global convergence.