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Abstract: We describe a method that computes the number of roots of a polynomial inside a region bounded by the curve , with an analysis of its computational cost. It is based on the number of roots being the same as the winding number of . While the usual methods for computing the winding number involve numerical integration, in this paper we use a geometrical construction. We show its correctness without referring to global information about (like its Lipschitz constant on ). The analysis of its cost is based on the distance from the roots to , expressed using a condition number suitably defined. The method can be used in a divide-and-conquer root-finding algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, we develop new numeric modified Adomian decomposition algorithms by using the Wazwaz–El-Sayed modified decomposition recursion scheme, and investigate their practicality and efficiency for several nonlinear examples. We show how we can conveniently generate higher-order numeric algorithms at will by this new approach, including, by using examples, 12th-order and 20th-order numeric algorithms. Furthermore, we show how we can achieve a much larger effective region of convergence using these new discrete solutions. We also demonstrate the superior robustness of these numeric modified decomposition algorithms including a 4th-order numeric modified decomposition algorithm over the classic 4th-order Runge–Kutta algorithm by example. The efficiency of our subroutines is guaranteed by the inclusion of the fast algorithms and subroutines as published by Duan for generation of the Adomian polynomials to high orders. [Copyright &y& Elsevier]

Abstract: This paper deals with an inverse problem of determining a source term in the one-dimensional fractional advection–dispersion equation (FADE) with a Dirichlet boundary condition on a finite domain, using final observations. On the basis of the shifted Grünwald formula, a finite difference scheme for the forward problem of the FADE is given, by means of which the source magnitude depending upon the space variable is reconstructed numerically by applying an optimal perturbation regularization algorithm. Numerical inversions with noisy data are carried out for the unknowns taking three functional forms: polynomials, trigonometric functions and index functions. The reconstruction results show that the inversion algorithm is efficient for the inverse problem of determining source terms in a FADE, and the algorithm is also stable for additional data having random noises. [Copyright &y& Elsevier]

Abstract: This paper explores the problems of generalized center conditions and integrability of resonant infinity for a complex polynomial differential system. The method is based on converting resonant infinity into an elementary singular point by a homeomorphic transformation. The calculation of generalized singular point quantities is an effective way of finding necessary conditions for integrable systems. A new recursive algorithm for computing generalized singular point quantities at the origin of the transformed system is derived. At the same time, a necessary and sufficient condition for resonant infinity to be a generalized complex center is presented. As an application of the new recursive algorithm, the generalized center conditions for resonant infinity for a class of cubic systems are discussed. To the best of our knowledge, this is the first time that the generalized center problem has been considered for resonant infinity. [Copyright &y& Elsevier]

Abstract: One of the main problems dealing with iterative methods for solving polynomial systemsis the initialization of the iteration. This paper provides an algorithm to initialize the search of solutions of polynomial systems. [Copyright &y& Elsevier]