Coastal countries began to develop green energy, and offshore wind power equipment in coastal areas was gradually built. Since coastal wind power generation often requires carrying out maintenance between wind turbines with the assistance of service operation vessels, this situation may cause coastal areas to be prone to people falling into the water. However, traditional maritime search and rescue plans take a long time to gather information from man overboard incidents. In order to minimize injuries to people in distress, the maritime search and rescue process must be as short as possible. Despite that all the search and rescue plans are based on the concept of the shortest path, the efficient plans must not only consider the distance but also consider the cost of search and rescue. Therefore, this study established a set of practices applicable to the on-site commander (OSC) to dispatch rescue ships, as well as the planning of maritime search and rescue route models. Based on the easy-to-observe state of the target in distress, the model is analyzed and calculated by Floyd–Warshall algorithm and Grey relational analysis so as to sort the rescue plan and optimize the effect of the search and rescue route at sea. According to the simulation analysis, when the man overboard incident occurs in the coastal area, the OSC can immediately use this model to plan the best search and rescue route and dispatch a reasonable number of rescue ships.

In the new inspection regime (NIR) of port state control (PSC), the criteria for being judged as a standard risk ship (SRS) is too broad. Some ships are classified as SRS even though they have a large number of ship deficiencies. This paper develops a selection system to identify the hidden risk of target ships in the SRS category using PSC inspection records. This system allows the target ship to be used to help reduce cases of flags being greylisted or blacklisted, which can cause huge shipping losses. This study analyzes ship deficiency data in the Tokyo memorandum of understanding (Tokyo MoU) database. It adopts the multiple criteria decision making (MCDM) model as a data processing technique to build a risk assessment scale. It uses fuzzy importance performance analysis (F-IPA) and technology for order preference by similarity to the ideal solution (TOPSIS) for its analysis. Subsequently, the weights of F-IPA and TOPSIS are adopted into the MCDM model. This article also consulted the Tokyo MoU database. It has been verified that the next time PSC inspection, the system hits 83.3% of the hidden risk ships in the SRS category. Thus, this system will help inspectors be more insightful for target ships.

In the paper, we transform the general Sturm–Liouville problem (SLP) into two canonical forms: one with the homogeneous Dirichlet boundary conditions and another with the homogeneous Neumann boundary conditions. A boundary shape function method (BSFM) was constructed to solve the SLPs of these two canonical forms. Owing to the property of the boundary shape function, we could transform the SLPs into an initial value problem for the new variable with initial values that were given definitely. Meanwhile, the terminal value at the right boundary could be entirely determined by using a given normalization condition for the uniqueness of the eigenfunction. In such a manner, we could directly determine the eigenvalues as the intersection points of an eigenvalue curve to the zero line, which was a horizontal line in the plane consisting of the zero values of the target function with respect to the eigen-parameter. We employed a more delicate tuning technique or the fictitious time integration method to solve an implicit algebraic equation for the eigenvalue curve. We could integrate the Sturm–Liouville equation using the given initial values to obtain the associated eigenfunction when the eigenvalue was obtained. Eight numerical examples revealed a great advantage of the BSFM, which easily obtained eigenvalues and eigenfunctions with the desired accuracy.

In this paper, an efficient and straightforward numerical procedure is constructed to solve multi-dimensional linear and nonlinear elliptic partial differential equations (PDEs). Although the numerical procedure for the constraint-type fictitious time integration method overcomes the numerical stability problem, the parameter’s definition, numerical accuracy and computational efficiency have not been resolved, and the lack of initial guess values results in reduced computational efficiency. Therefore, the normalized two-point boundary value solution of the Lie-group shooting method is proposed and considered in the numerical procedure to avoid the problem of the initial guess value. Then, a space-time variable, including the minimal fictitious time step and convergence rate factor, is introduced to study the relationship between the initial guess value and convergence rate factor. Some benchmark numerical examples are tested. As the results show, this numerical procedure using the normalized boundary value solution can significantly converge within one step, and the numerical accuracy is better than that demonstrated in the previous literature.

The shooting method consists of guessing unknown initial values, transforming a second-order nonlinear boundary value problem (BVP) to an initial value problem and integrating it to obtain the values at the right end to match the specified boundary condition, which acts as a target equation. In the shooting method, the key issue is accurately solving the target equation to obtain highly precise initial values. Due to the implicit and nonlinear property, we develop a generalized derivative-free Newton method (GDFNM) to solve the target equation, which offers very accurate initial values. Numerical examples are examined to show that the shooting method together with the GDFNM can generate a very accurate solution. Additionally, the GDFNM can successfully solve the three-point nonlinear BVPs with high accuracy. A new splitting-linearizing method is developed to express the approximate analytic solutions of nonlinear BVPs in terms of elementary functions, which adopts the Lyapunov technique by inserting a dummy parameter into the governing equation and the power series solution. Then, linearized differential equations are sequentially solved to derive the analytic solution.

The paper solves the eigenvalues of a symmetric matrix by using three novel algorithms developed in the m-dimensional affine Krylov subspace. The n-dimensional eigenvector is superposed by a constant shifting vector and an m-vector. In the first algorithm, the m-vector is derived as a function of eigenvalue by maximizing the Rayleigh quotient to generate the first characteristic equation, which, however, is not easy to determine the eigenvalues since its roots are not of simple ones, exhibiting turning points, spikes, and even no intersecting point to the zero line. To overcome that difficulty by the first algorithm, we propose the second characteristic equation through a new quotient with the inner product of the shifting vector to the eigen-equation. The Newton method and the fictitious time integration method are convergent very fast due to the simple roots of the second characteristic equation. For both symmetric and nonsymmetric eigenvalue problems solved by the third algorithm, we develop a simple iterative detection method to maximize the Euclidean norm of the eigenvector in terms of the eigen-parameter, of which the peaks of the response curve correspond to the eigenvalues. Through a few finer tunings to the smaller intervals sequentially, a very accurate eigenvalue and eigenvector can be obtained. The efficiency and accuracy of the proposed iterative algorithms are verified and compared to the Lanczos algorithm and the Rayleigh quotient iteration method.