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We construct examples of exact solutions of the temperature problem for a square: the sides of the square are (i) free and (ii) firmly clamped. Initially, we solve the inhomogeneous problem for an infinite plane. The known exact solutions for a square, with which the boundary conditions on the sides of the square are satisfied, are added to this solution. The solutions are represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from simple formulas.

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

In the paper, in the form of series in the Papkovich–Fadle eigenfunctions, exact solutions of two inhomogeneous boundary value problems of elasticity theory for a half-strip with free long sides are constructed: (1) the half-strip end is free, (2) the half-strip end is rigidly clamped. To solve the inhomogeneous problem in a free strip, the Papkovich orthogonality relation is used. The exact solution for the half-strip is then added to it.

We derive the formulas that describe the exact solution of the boundary value problem in the theory of elasticity for a rectangle in which two opposite (horizontal) sides are free and stresses are specified (all cases of symmetry relative to the central axes) on the other two sides (rectangle ends). The formulas for a half-strip are also given. The solutions are represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from simple formulas. The obtained formulas remain the same for other types of homogeneous boundary conditions, for example, when the horizontal sides of the rectangle are firmly clamped, have stiffening ribs that work in tension–compression and/or bending, etc. Obviously, in this case, the Papkovich–Fadle eigenfunctions will change, as well as the corresponding biorthogonal functions and normalizing factors. To solve a specific boundary value problem, it is enough to find the Lagrange coefficients, which are determined from simple formulas, as Fourier integrals of boundary functions specified at the ends of the rectangle, and then substitute them into the necessary formulas. Examples of solving two problems (even-symmetric deformation relative to the central coordinate axes) are given: (a) The normal stresses are known at the rectangle ends, and the tangential ones are zero; and (b) the longitudinal displacements conditioned by the action of some normal stresses are known at the rectangle ends (the tangential stresses are zero). These solutions are compared with the known solutions in trigonometric Fourier series. The basis of the exact solutions obtained is the theory of expansions in Papkovich–Fadle eigenfunctions based on the Borel transform in the class of quasi-entire functions of exponential type (developed by the authors in their previous studies).

We propose a method of solving mixed boundary value problems in the theory of elasticity in an infinite horizontal strip that has two points of change in the type of boundary conditions located on the upper and lower sides of the strip and lying at the ends of a vertical segment—the line of the joint between the left and right half-strips. The solutions to the right and the left of the joint line are represented as series in eigenfunctions that correspond to particular boundary conditions on the horizontal sides of the right and left half-strips. Solution continuity conditions or, on the contrary, discontinuities of the displacements or stresses can be specified at the joint between the half-strips. We illustrate the main idea of the method initially using a mixed boundary value problem for the harmonic equation as an example. Here, biorthogonal functions are constructed, their properties and the properties of the coefficients of expansions are investigated, and examples are given to illustrate the correctness of the obtained solutions. Subsequently we solve a mixed boundary value problem in which two trigonometric systems of functions with different eigenvalues are involved at the joint between the half-strips. We consider examples where (1) displacement and stress continuity conditions, (2) a discontinuity of the longitudinal displacements, and (3) a discontinuity of the transverse displacements are specified at the joint between the half-strips.

The integrated study on the evolution of the regional social-ecological system (RSES) is one of many complex and classical research topics. An innovative approach of conjugate analysis is introduced to describe the RSES evolution in this paper. We select land use intensity (L), bilateral dynamic change of land use types (S), landscape pattern index (P), ecological security index (Q), and the RSES composite index (T) to describe the RSES. We analyze and express the conjugacy of the RSES evolution: for the spatial correlations, they are revealed by variance analysis, spatial autocorrelation, and regionalized variable analysis; and for the temporal correlations, they are described by the analysis of spatiotemporal correlation. We use a case study of the Hanjiang River basin in Hubei province (China) to test the conjugate evolution of the RSES. We find that the RSES evolution in this region is driven by dual-source forces, both resource driven and economic power driven. These driving forces result in the spiral rise of the RSES evolution, where L, S, Q, P, and T have high self-autocorrelations, and there are remarkable and highly positive correlations and inheritance between the RSES and their subsystems. These results can corroborate the hypothesis about conjugate evolvement of the RSES. The spatial patterns of the RSES evolution are controlled by physical factors, especially geomorphology, where as its direction is guided by human activities, and its progress is pushed forward by human-environment interactions. The analysis of the RSES conjugate evolution can provide a new perspective for the RSES management, that is, the RSES management decisions should consider conjugate effects, because these effects can directly influence regional sustainable development.

Using the boundary value problem on the bending of a thin elastic semi-infinite plate in which the long sides are free, while a self-balanced bending moment and a generalized shearing force are specified at its end as an example, we consider the main steps in constructing exact solutions to the boundary value problems of bending of thin elastic rectangular plates. The solutions are constructed in the form of series in Papkovich–Fadle eigenfunctions. The unknown expansion coefficients are determined in the same way as in classical periodic solutions in trigonometric series and have the same structure, i.e., are expressed via the Fourier integrals of the boundary functions specified at the half-strip end. The systems of functions biorthogonal to the Papkovich–Fadle eigenfunctions constructed here are used in this case. The exact solutions possess properties that are not inherent in any of the known solutions in the theory of plate bending. Some of them are discussed in the paper. The final formulas describing the exact solution of the boundary value problem are simple and can be easily used in engineering practice. The results that we have obtained previously when solving the boundary values problems of the plane theory of elasticity in a rectangular region form the basis for our work.

In this paper, in the form of expansions in Papkovich–Fadle eigenfunctions, an exact solution to the boundary value problem of the theory of elasticity is constructed for a rectangle whose horizontal sides are rigidly clamped. The expansion coefficients are determined with the help of biorthogonal functions by simple formulas.

This paper provides an exact solution to the boundary value problem of elasticity theory for a square loaded with tangential stresses along all its sides (even-symmetric deformation with respect to the central axes). The solution is represented as series in Papkovich–Fadle eigenfunctions. The coefficients of the series are determined by simple closed formulas.

In the paper, we construct an exact solution to a boundary value problem of the theory of elasticity for a rectangle in which the longitudinal sides are free, while normal and tangential stresses are given at the ends (even-symmetric deformation with respect to the central axes). The solution is represented in the form of series in Papkovich–Fadle eigenfunctions. The coefficients are determined explicitly by using functions biorthogonal to the Papkovich–Fadle eigenfunctions. We give the final formulas which have a simple appearance and can easily be used in engineering practice. The obtained solution is compared with the solution to the corresponding boundary value problem for a half-strip.

The article focuses on proposing a new hybrid time-Laplace domain response calculation method for symmetrical floating structures by solving Cummins equation, which is depending on the state–space model identified from transfer function. Different from a time or frequency domain method, the proposed approach estimates the state–space model of a floating structure system from the transfer function in the Laplace domain and calculates the response by considering the exciting forces as an input of the state–space model. Implementing complex exponential decomposition to the retardation functions, the retardation functions in the Laplace domain are expressed using a series of poles and the corresponding residues, which avoids the numerical integral of the convolution terms in the Cummins equation and greatly improves the computational efficiency during the process of a dynamic response analysis. Three examples are applied to investigate the validity of the proposed method. The first is a simple single degree of freedom mathematical model excited by an irregular wave. Studies have shown that the response calculated by the proposed method matches well with that of a traditional Newmark- β method. Meanwhile, this approach is insensitive to the interval time of the calculation and consumes less calculation time, which means that the proposed method has higher precision and computational efficiency. The last two examples are a spar-type offshore wind turbine and a semi-submersible platform model (SEMI) in SESAM, which extend the proposed method to solve the response estimation problem of marine structures. The results of the spar-type floating structure show that the estimated responses when using the proposed method are in good agreement with the results of the traditional time domain method. By studying the response calculation of SEMI, the following conclusions can be obtained: (1) the estimated responses match well with the traditional time domain method and WASIM code of SESAM, and (2) the calculation time of the new approach is reduced significantly compared with the Newmark- β method especially under a long simulation time.

Land Use/Cover Change (LUCC) has gradually became an important direction in the research of global changes. LUCC is a complex system, and entropy is a measure of the degree of disorder of a system. According to land use information entropy, this paper analyzes changes in land use from the perspective of the system. Research on the entropy of LUCC structures has a certain “guiding role” for the optimization and adjustment of regional land use structure. Based on the five periods of LUCC data from the year of 1990 to 2010, this paper focuses on analyzing three types of LUCC entropies among counties in China—namely, Shannon, Renyi, and Tsallis entropies. The findings suggest that: (1) Shannon entropy can reflect the volatility of the LUCC, that Renyi and Tsallis entropies also have this function when their parameter has a positive value, and that Renyi and Tsallis entropies can reflect the extreme case of the LUCC when their parameter has a negative value., (2) The entropy of China’s LUCC is uneven in time and space distributions, and that there is a large trend during 1990–2010, the central region generally has high entropy in space.