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We present an efficient and practical numerical method for solving fractional integro-differential equations with a weakly singular kernel in complex two and three-dimensional domains. The reproducing kernel pseudo-spectral method is used to discretize the multidimensional regular and irregular spatial domains, while time discretization is done using a third-order backward differentiation formula. We have investigated the unconditional stability of the utilized temporal discretization. Finally, we have used the proposed method to solve the examples presented in the articles and compared the results. The obtained results show the accuracy and ability of the presented method. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to design a simple and accurate a-posteriori Lagrangian-based error estimator is developed for the class of backward differentiation formula (BDF) algorithms with variable time step size, and the adaptive time-stepping in BDF algorithms is demonstrated for efficient time-dependent simulations in fluid flow and heat transfer. Design/methodology/approach: The Lagrange interpolation polynomial is used to predict the time derivative, and then the accurate primary result is obtained by the Gauss integral, which is applied to evaluate the local error. Not only the generalized formula of the proposed error estimator is presented but also the specific expression for the widely applied BDF1/2/3 is illustrated. Two essential executable MATLAB functions to implement the proposed error estimator are appended for practical applications. Then, the adaptive time-stepping is demonstrated based on the newly proposed error estimator for BDF algorithms. Findings: The validation tests show that the newly proposed error estimator is accurate such that the effectivity index is always close to unity for both linear and nonlinear problems, and it avoids under/overestimation of the exact local error. The applications for fluid dynamics and coupled fluid flow and heat transfer problems depict the advantage of adaptive time-stepping based on the proposed error estimator for time-dependent simulations. Originality/value: In contrast to existing error estimators for BDF algorithms, the present work is more accurate for the local error estimation, and it can be readily extended to practical applications in engineering with a few changes to existing codes, contributing to efficient time-dependent simulations in fluid flow and heat transfer. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to show that the computation of time-periodic signals for coupled antenna-circuit problems can be substantially accelerated by means of the single shooting method. This allows an efficient analysis of nonlinearly loaded coupled loop antennas for near field communication (NFC) applications. Design/methodology/approach: For the modelling of electrically small coupled field-circuit problems, the partial element equivalent circuit (PEEC) method shows to be very efficient. For analysing the circuit-like description of the coupled problem, this paper developed a generalised modified nodal analysis (MNA) and applied it to specific NFC problems. Findings: It is shown that the periodic steady state (PSS) solution of the resulting differential-algebraic system can be computed very time efficiently by the single shooting method. A speedup of roughly 114 to conventional transient approaches can be achieved. Practical implications: The proposed approach appears to be an efficient alternative for the computation of time PSS solutions for nonlinear circuit problems coupled with discretised conductive structures, where the homogeneous solution is not of interest. Originality/value: The present paper explores the implementation and application of the shooting method for nonlinearly loaded coupled antenna-circuit problems based on the PEEC method and shows the efficiency of this approach. [ABSTRACT FROM AUTHOR]

Abstract: We derive two reaction–diffusion systems modeling the impact of a nuclear waste contamination on food chain population sets. We exhibit the numerical difficulties the problem features and provide second order accurate nonnegative approximate solutions in two space dimensions. [Copyright &y& Elsevier]