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This paper proposes a heterogeneous time integration algorithm to analyze the dynamic response of structures with localized hysteretic nonlinearities. The critical point is to treat the whole structure as a combination of a linear substructure governed by the second-order dynamic equation and a nonlinear substructure controlled by the first-order differential formulation. With this partitioning, tailored numerical integration algorithms with heterogeneous time steps are applied directly to calculate the displacements and hysteretic forces from the linear and the nonlinear substructures, respectively. Subsequently, the dynamic responses of the whole structure are solved by a predictor–corrector procedure, where the hysteretic forces and displacement responses are coupled and exchanged to update the partitioning solutions. Furthermore, the energy balance method is derived to verify the stability of the proposed heterogeneous time integration algorithm. Dynamic responses of structures with friction damper, hysteretic models, and bolted joint models are studied to demonstrate the accuracy and efficiency of the proposed algorithm.

A new general multistep predictor-corrector (PC) pair form is introduced for the numerical integration of second-order initial-value problems. Using this form, a new symmetric eight-step predictor-corrector method with minimal phase-lag and algebraic order ten is also constructed. The new method is based on the multistep symmetric method of Quinlan–Tremaine,¹ with eight steps and 8th algebraic order and is constructed to solve numerically the radial time-independent Schrödinger equation. It can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with minimal phase-lag is the most efficient of all the compared methods and for all the problems solved.

We propose a new primal-dual interior-point predictor–corrector algorithm in Ai and Zhang’s wide neighborhood for solving monotone linear complementarity problems (LCP). Based on the understanding of this neighborhood, we use two new directions in the predictor step and in the corrector step, respectively. Especially, the use of new corrector direction also reduces the duality gap in the corrector step, which has good effects on the algorithm’s convergence. We prove that the new algorithm has a polynomial complexity of $O(\sqrt{n}L)$, which is the best complexity result so far. In the paper, we also prove a key result for searching for the best step size along some direction. Considering local convergence, we revise the algorithm to be a variant, which enjoys both complexity of $O(\sqrt{n}L)$ and Q-quadratical convergence. Finally, numerical result shows the effectiveness and superiority of the two new algorithms for monotone LCPs.

Since the ability to control the investor’s income or loss within a certain range, barrier option has been among the most popular path-dependent options where its payoff depends on whether or not the underlying asset’s price reaches a given “barrier”. First, assuming the underlying asset as an uncertain variable for the case that the Caputo fractional-order derivative is adopted instead of the ordinary derivative, the real financial market is better modeled by the uncertain fractional-order differential equation with Caputo type. Then, a first-hitting time model which can measure the exercise ability is innovatively presented. Second, based on the first-hitting time theorem of the uncertain fractional-order differential equation, the reliability index (including validity and survival index) for the proposed model is obtained, and four types of European barrier option (including up-and-in call, down-and-in put, up-and-out put, and down-and-out call options) pricing formulas are obtained accordingly. Lastly, applying the predictor–corrector method, numerical algorithms are provided for calculating European barrier and the reliability index, numerical experiments and corresponding sensitivity analysis are also illustrated concerning various conditions.

Ocean acoustic wave propagation can be predicted by applying numerical methods to solve representative wave equations computationally. For this purpose, numerical methods have been introduced; a latest introduction was the Predictor-Corrector Method. An important question arises: Whether or not these numerical methods can produce satisfactory required accurate results? This may cause an accuracy concerned by the users. This paper introduces a new Predict-Correct Procedure to examine whether or not the result meets the accuracy requirement. If not, the procedure can improve the result until it becomes satisfactorily accurate. Discussions will be given on the mathematical and computational developments of the Predictor-Corrector Method as well as the Predict-Correct Procedure. Following that is a discussion on how the Predict-Correct Procedure works. An important part of this paper is devoted to show how this new procedure can achieve the goal of obtaining the required accurate prediction results.