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Developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering. In this paper, we explore the regularization method for the coupled elliptic sine-Gordon equations along with Cauchy data.The system of equations originates from the static case of the coupled hyperbolic sine-Gordon equations modeling the coupled Josephson junctions in superconductivity, and so far it addresses the Josephson πjunctions. In general, the Cauchy problem is not well-posed, and herein the Hadamard-instability occurs drastically. Generalizing the kernel-based regularization method, we propose a stable approximate solution. Confirmed by the error estimate, this solution strongly converges to the exact solution in L 2 -norm.The main concern of this paper is also with the way to compute the regularized solution formed by an alike integral equation. We employ the proposed techniques that successfully approximated the highly oscillatory integral, and apply the Picard-like iteration to organize an efficient and reliable tool of computations.The results are viewed as the improvement as well as the generalization of many previous works. The paper is also accompanied by a numerical example that demonstrates the potential of this idea.

Recently, embedded systems have become popular because of the rising demand for portable, low-power devices. A common task for these devices is object tracking, which is an essential part of various applications. Until now, object tracking in video sequences remains a challenging problem because of the visual properties of objects and their surrounding environments. Among the common approaches, particle filter has been proven effective in dealing with difficulties in object tracking. In this research, we develop a particle filter based object tracking method using color distributions of video frames as features, and deploy it in an embedded system. Because particle filter is a high-complexity algorithm, we utilize computing power of embedded systems by implementing a parallel version of the algorithm. The experimental results show that parallelization can enhance the performance of particle filter when deployed in embedded systems.

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