An expectation formula with applications

In this paper, we derive an expectation formula of a random variable having distribution W(x;q). As applications of the expectation formula, we give a transformation formula and an expansion of Sears? transformation formula.     

Zero distribution of solutions of complex linear differential equations determines growth of coefficients

It is shown that the exponent of convergence λ(f) of any solution f of with entire coefficients A₀(z), …, A_k?2(z), satisfies λ(f) ? λ ∈ [1, ∞) if and only if the coefficients A₀(z), …, A_k?2(z) are polynomials such that for j = 0, …, k ? 2. In the unit disc analogue of this result certain intersections of weighted Bergman spaces take the role of polynomials. The key idea in the proofs is W. J. Kim’s 1969 representation of coefficients in terms of ratios of linearly independent solutions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Convergence analysis and performance of the extended artificial physics optimization algorithm

This paper presents extended artificial physics optimization (EAPO), a population-based, stochastic, evolutionary algorithm (EA) for multidimensional search and optimization. EAPO extends the physicomimetics-based Artificial Physics Optimization (APO) algorithm by including each individual’s best fitness history. Including the history improves EAPO’s search capability compared to APO. EAPO and APO invoke a gravitational metaphor in which the force of gravity may be attractive or repulsive, the aggregate effect of which is to move individuals toward local and global optima. A proof of convergence is presented that reveals the conditions under which EAPO is guaranteed to converge. Discrete-time linear system theory is used to develop a second-order difference equation for an individual’s stochastic position vector as a function of time step. Stable solutions require eigenvalues inside the unit circle, leading to explicit convergence criteria relating the run parameters {m_i, w, G}. EAPO is tested against several benchmark functions with excellent results. The algorithm converges more quickly than APO and with better diversity.     

kth Order convergence for a semilinear elliptic boundary value problem in the divergence form

The convergence problem of approximate solutions for a semilinear elliptic boundary value problem in the divergence form is studied. By employing the method of quasilinearization, a sequence of approximate solutions converging with the kth (k ? 2) order convergence to a weak solution for a semilinear elliptic problem is obtained via the variational approach.     

An optimal Steffensen-type family for solving nonlinear equations

In this paper, a general family of Steffensen-type methods with optimal order of convergence for solving nonlinear equations is constructed by using Newton’s iteration for the direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub [H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21 (1974) 634-651] that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2^m−1. Its error equations and asymptotic convergence constants are obtained. Finally, it is compared with the related methods for solving nonlinear equations in the numerical examples.     

A new globalization technique for nonlinear conjugate gradient methods for nonconvex minimization

It is well-known that the HS method and the PRP method may not converge for nonconvex optimization even with exact line search. Some globalization techniques have been proposed, for instance, the PRP+ globalization technique and the Grippo-Lucidi globalization technique for the PRP method. In this paper, we propose a new efficient globalization technique for general nonlinear conjugate gradient methods for nonconvex minimization. This new technique utilizes the information of the previous search direction sufficiently. Under suitable conditions, we prove that the nonlinear conjugate gradient methods with this new technique are globally convergent for nonconvex minimization if the line search satisfies Wolfe conditions or Armijo condition. Extensive numerical experiments are reported to show the efficiency of the proposed technique.     

Asymptotic properties of fractional delay differential equations

In this paper we study the asymptotic properties of d-dimensional linear fractional differential equations with time delay. We present necessary and sufficient conditions for asymptotic stability of equations of this type using the inverse Laplace transform method and prove polynomial decay of stable solutions. Two examples illustrate the obtained analytical results.     

New third and fourth order nonlinear solvers for computing multiple roots

We present a new third order method for finding multiple roots of nonlinear equations based on the scheme for simple roots developed by Kou et al. [J. Kou, Y. Li, X. Wang, A family of fourth-order methods for solving non-linear equations, Appl. Math. Comput. 188 (2007) 1031-1036]. Further investigation gives rise to new third and fourth order families of methods which do not require second derivative. The fourth order family has optimal order, since it requires three evaluations per step, namely one evaluation of function and two evaluations of first derivative. The efficacy is tested on a number of relevant numerical problems. Computational results ascertain that the present methods are competitive with other similar robust methods.