Convergence rate estimates for the kernelized predictor corrector method for fractional order initial value problems

This manuscript presents a kernelized predictor corrector (KPC) method for fractional order initial value problems, which replaces linear interpolation with interpolation by a radial basis function (RBF) in a predictor-corrector scheme. Specifically, the class of Wendland RBFs is employed as the basis function for interpolation, and a convergence rate estimate is proved based on the smoothness of the particular kernel selected. Use of the Wendland RBFs over Mittag-Leffler kernel functions employed in a previous iteration of the kernelized method removes the problems encountered near the origin in [11]. This manuscript performs several numerical experiments, each with an exact known solution, and compares the results to another frequently used fractional Adams-Bashforth-Moulton method. Ultimately, it is demonstrated that the KPC method is more accurate but requires more computation time than the algorithm in [4].

Newton-type methods near critical solutions of piecewise smooth nonlinear equations

It is well-recognized that in the presence of singular (and in particular nonisolated) solutions of unconstrained or constrained smooth nonlinear equations, the existence of critical solutions has a crucial impact on the behavior of various Newton-type methods. On the one hand, it has been demonstrated that such solutions turn out to be attractors for sequences generated by these methods, for wide domains of starting points, and with a linear convergence rate estimate. On the other hand, the pattern of convergence to such solutions is quite special, and allows for a sharp characterization which serves, in particular, as a basis for some known acceleration techniques, and for the proof of an asymptotic acceptance of the unit stepsize. The latter is an essential property for the success of these techniques when combined with a linesearch strategy for globalization of convergence. This paper aims at extensions of these results to piecewise smooth equations, with applications to corresponding reformulations of nonlinear complementarity problems.

On the convergence of the barycentric method in solving internal Dirichlet and Neumann problems in $\mathbb{R}^2$ for the Helmholtz equation

The application of the barycentric method for the numerical solution of Dirichlet and Neumann problems for the Helmholtz equation in the bounded simply connected domain $\Omega\subset\mathbb{R}^2$ is considered. The main assumption in the solution is to set the $\Omega$ boundary in a piecewise linear representation. A distinctive feature of the barycentric method is the order of formation of a global system of vector basis functions for $\Omega$ via barycentric coordinates. The existence and uniqueness of the solution of Dirichlet and Neumann problems for the Helmholtz equation by the barycentric method are established and the convergence rate estimate is determined. The features of the algorithmic implementation of the method are clarified.

Fractional order convergence rate estimate of finite-difference method for the heat equation with concentrated capacity

The convergence of difference scheme for initial-boundary value problem for the heat equation with concentrated capacity and time-dependent coefficient of the space derivatives, is considered. Fractional order convergence rate estimate in a special discrete Sobolev norms, compatible with the smoothness of the coefficient and solution, is proved.

Additive difference scheme for two-dimensional fractional in time diffusion equation

An additive finite-difference scheme for numerical approximation of initial-boundary value problem for two-dimensional fractional in time diffusion equation is proposed. Its stability is investigated and a convergence rate estimate is obtained.

Factorized difference scheme for 2D fractional in time diffusion equation

A factorized finite-difference scheme for numerical approximation of initial-boundary value problem for two-dimensional fractional in time diffusion equation is proposed. Its stability is investigated and a convergence rate estimate is obtained.

On the Convergence of Biogeography-Based Optimization for Binary Problems

Biogeography-based optimization (BBO) is an evolutionary algorithm inspired by biogeography, which is the study of the migration of species between habitats. A finite Markov chain model of BBO for binary problems was derived in earlier work, and some significant theoretical results were obtained. This paper analyzes the convergence properties of BBO on binary problems based on the previously derived BBO Markov chain model. Analysis reveals that BBO with only migration and mutation never converges to the global optimum. However, BBO with elitism, which maintains the best candidate in the population from one generation to the next, converges to the global optimum. In spite of previously published differences between genetic algorithms (GAs) and BBO, this paper shows that the convergence properties of BBO are similar to those of the canonical GA. In addition, the convergence rate estimate of BBO with elitism is obtained in this paper and is confirmed by simulations for some simple representative problems.

Finite difference approximation of a parabolic problem with variable coefficients

We study the convergence of a finite difference scheme that approximates the third initial-boundary-value problem for a parabolic equation with variable coefficients on a unit square. We assume that the generalized solution of the problem belongs to the Sobolev space W s,s/2 2, s?3. An almost second-order convergence rate estimate (with additional logarithmic factor) in the discrete W 1,1/2 2 norm is obtained. The result is based on some nonstandard a priori estimates involving fractional order discrete Sobolev norms.