In this work, we are concerned with the explicit multistep scheme for solving the coupled forward backward stochastic differential equations (FBSDEs). Based on the Lagrange interpolation and first-order derivative approximations, we will propose a fully discrete explicit high-order multistep scheme for solving coupled FBSDEs. Its high accuracy, efficiency and stability are verified by the numerical experiments.

The study of this paper is two-fold. On the one hand, we reduce the subdiffusion ($0<\alpha<1$) and diffusion-wave ($1<\alpha<2$) problems on unbounded strips to initial boundary value problems (IBVPs) by deriving high-order local artificial boundary conditions (ABCs). After that, the IBVPs with our high-order local ABCs are proved to be stable in the L2-norm. On the other hand, unconditionally stable schemes are constructed to numerically solve the IBVPs by using L1 approximation to discretize the temporal derivative and using finite difference methods to discretize the spatial derivative. We provide the complete error estimates for the subdiffusion case and sketch the proof for the diffusion-wave case. To further reduce the computational and storage cost for the evaluation of the fractional derivatives, the fast algorithm presented in [14] is employed for the case of $0<\alpha<1$ and a similar algorithm for the case of  $1<\alpha<2$ is first introduced in this article. Numerical examples are provided to verify the effectiveness and performance of our ABCs and numerical methods.

In a previous paper (Adv. Appl. Math. Mech., 10 (2018), pp. 1025-1056), we used the Buchwald representation to construct several families of separable cylindrical solutions to the Navier-Lamé equation; these solutions had the property of being $2\pi$-periodic in the circumferential coordinate. In this paper, we extend the analysis and obtain the complementary set of separable solutions whose circumferential parts are elementary $2\pi$-aperiodic functions. Collectively, we construct eighteen distinct families of separable solutions; in each case, the circumferential part of the solution is one of three elementary $2\pi$-aperiodic functions. These solutions are useful for solving a wide variety of dynamical problems that involve cylindrical geometries and for which $2\pi$-periodicity in the angular coordinate is incompatible with the given boundary conditions. As illustrative examples, we show how the obtained solutions can be used to solve certain forced-vibration problems involving open cylindrical shells and open solid cylinders where (by virtue of the boundary conditions) $2\pi$-periodicity in the angular coordinate is inappropriate. As an addendum to our prior work, we also include an illustrative example of a certain type of asymmetric problem that can be solved using the particular $2\pi$-periodic subsolutions that ensue when there is no explicit dependence on the circumferential coordinate.

In this article, we apply the discontinuous Galerkin method by patch reconstruction for solving convection-diffusion problems. The proposed method is highly efficient that it uses only one degree of freedom per element to achieve higher order approximation. It also enjoys the implementation flexibility on the general polygonal meshes. A priori error estimates of energy norm is devised. Numerical examples are presented to illustrate the theoretical results and the efficiency of the method.

In this paper we develop a novel numerical method for pricing American options under regime-switching jump-diffusion models which are governed by a system of partial integro-differential complementarity problems (PIDCPs). Based on a penalty approach, the PIDCPs results in a set of coupled nonlinear partial integro-differential equations (PIDEs). To numerically solve these nonlinear penalized PIDEs, we introduce a fitted finite volume method for the spatial discretization, coupled with the backward Euler and Crank-Nicolson time stepping schemes. We show that these schemes are consistent, stable and monotone, hence it ensures the convergence to the solution of continuous problem. To solve the discretized nonlinear system effectively, an iterative method is designed. Numerical experiments are presented to demonstrate the accuracy, efficiency and robustness of the new numerical method.

In this paper, a type of cascadic adaptive finite element method is proposed for eigenvalue problem based on the complementary approach. In this new scheme, instead of solving the eigenvalue problem in each adaptive finite element space directly, we only need to do some smoothing steps for a boundary value problems on each adaptive space and solve some eigenvalue problems on a low dimensional space. Hence the efficiency can be improved since we do not need to solve the eigenvalue problems on each adaptive space which is time-consuming. Further, the complementary error estimate for eigenvalue problem will be introduced. This estimate can not only provide an accurate error estimate for eigenvalue problem but also provide the way to refine mesh and control the number of smoothing steps for the cascadic adaptive algorithm. Some numerical examples are presented to validate the efficiency of the proposed algorithm in this paper.

Lightweight and flexible structures are often used in the engineering field which may produce the long time residual vibration of flexible appendages. In order to reduce the residual vibration of the flexible manipulator, a control method of combining the multi-impulse robust shaper with the linear quadratic regulator controller is put forward in this paper. The Lagrange equation and assumed mode method are used to obtain the dynamic equation of the manipulator. Then the dynamic equation is rewritten in state space, and the state equation of the closed-loop system is obtained after the optimal control is used. The multi-impulse robust shaper is designed according to the system frequency and damping ratio and is used to shape the required angle. The input torque is obtained by the linear quadratic regulator controller. The proposed combination controller is compared with the combination controller of ZVD shaper and the optimal control method, the combination controller of the EI shaper and the optimal control method, and optimal controller. The robustness of the proposed controller is then investigated. The simulation results show that the proposed combination controller has a better advantage in suppressing residual vibration than other controllers and has a good performance in the robustness of natural frequency.

In this paper, an efficient spectral method is applied to solve fourth order elliptic eigenvalue problems in circular domain. Firstly, we derive the essential pole conditions and the equivalent dimension reduction schemes of the original problem. Then according to the pole conditions, we define the corresponding weighted Sobolev spaces. Together with the minimax principle and approximation properties of orthogonal polynomials, the error estimates of approximate eigenvalues are proved. Thirdly, we construct an appropriate set of base functions contained in approximation spaces and establish the matrix formulations for the discrete variational form, whose mass matrix and stiff matrix are all sparse so that we can solve the numerical solutions efficiently. Finally, we provide some numerical experiments to validate the theoretical results and algorithms.

In this paper, we consider a semilinear elliptic equation with Dirac right-hand side. An equivalent a posteriori error estimator for the $L^{s}$ norm is obtained. We note that the a posteriori error estimator can be used to design adaptive finite element algorithms. In the end, some examples are provided to examine the quality of the derived estimator.

In this paper, we propose a non-convex and convex coupling variational model for image segmentation. We design the non-convex and convex regularization terms based on total generalized p-variation (TGpV) regularizer to preserve the boundary of segmented parts and detect the structure in the image. Our method has two stages. The first stage is to approximate the Mumford-Shah model. The second stage is to segment the smoothed $u$ into different phases by using a thresholding strategy. We develop a scheme based on the alternating direction method of multipliers (ADMM) algorithm,  generalized $p$-shrinkage operation and K-means clustering method to carry out our method. We perform numerical experiments on many kinds of images such as real Bacteria image, Tubular magnetic resonance angiography (MRA) image, magnetic resonance (MR) images, anti-mass images, artificial images, noisy or blurred images. Some comparisons are arranged to show the effectiveness and advantages of our method.