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We consider the inverse problem of recovering the source function with pointwise overdetermination conditions. The source is represented as a finite segment of a series with Fourier coefficients depending on time. We prove that the solution is uniquely determined and continuously depends on the data. [ABSTRACT FROM AUTHOR]

Many papers have been recently devoted to the study of the radial behavior as z → 1− of transcendental r-Mahler functions holomorphic in the open unit disk. In particular, Bell and Coons showed in 2017 that, in a generic sense, r-Mahler functions behave like (1 + o(1))C(z)/(1 − z)ρ for some ρ ∈ ℂ and C(z) is a real analytic function of z ∈ (0, 1) such that C(z) = C(zr). They did not provide a formula for C(z) which was made explicit only in a few examples of r-Mahler functions of orders 1 and 2, and for specific values of r. In this paper, we first provide an explicit expression of C(z) as an exponential of a Fourier series in the variable loglog(1/z)/log(r) for every r-Mahler function of order 1. Then, extending to a large setting a method introduced by Brent–Coons–Zudilin in 2016 to compute C(z) associated to the Dilcher–Stolarsky function (a 4-Mahler function of order 2 in ℚ[[z]]), we provide an explicit expression of C(z) for every r-Mahler function of order 2 under mild assumptions on the coefficients in ℝ(z) of the underlying r-Mahler equations. This applies in particular to the generating function of the Baum–Sweet sequence. We do the same for r-Mahler functions solutions of inhomogeneous Mahler equations of order 1. [ABSTRACT FROM AUTHOR]

The current investigation address a novel generalized elasto‐thermodiffusion model for a thermoelastic porous half‐space incorporating the nonlocal stress theory proposed by Eringen. Modeling of the problem is performed by adopting Moore‐Gibson‐Thompson (MGT) thermoelasticity theory defined in an integral form of a common derivative on a slipping interval, well known as the memory‐dependent derivative. The bounding plane of the medium is subjected to time‐dependent thermal and chemical shocks and there is no change in the volume fraction field. Laplace transform and the Fourier transform techniques have been adopted to represent the analytical solutions in the transformed domain. The distributions of the physical fields such as the temperature, stress, chemical potential, mass concentration and the volume fraction field were found in the real space‐time domain adopting suitable numerical scheme based on the Fourier series expansion. According to the discussion of the computational results and the respective graphical representations, the prominent role of different parameters such as the effect of nonlocality, effect of void and thermodiffusion is analyzed. Moreover, the superiority of a nonlinear kernel function compared to a linear form is also reported. [ABSTRACT FROM AUTHOR]

Determining the stress state in a circular ring has been a classical topic in the stress analysis literature. Based on the principle of superposition, the results may be obtained by adding known solutions to an associated ring problem, where the boundary stresses on the inner and outer walls of the ring are represented in Fourier series. In this work, the coefficients of the Fourier series are generated through an algorithm based on the fast Fourier transform (FFT). In the case of concentrated loading, the required additional fundamental solutions are derived in closed-form. The presented numerical method allows for accurate and efficient computations of the stress distributions in a circular ring in static equilibrium under arbitrary in-plane loading; and generally, the FFT-based algorithm provides a convenient and versatile tool for handing some two-dimensional problems involving circular boundaries. [ABSTRACT FROM AUTHOR]

Accurate prediction of employment rate of graduated students can greatly help education authorities to make informed decisions as well as for universities to adjust their teaching plans. Unfortunately, prediction of the employment rate of graduated students is still a difficult problem because the historical employment rate data exhibits fluctuations. In this paper, a new method is proposed for employment rate prediction using fractional gray GM(1,1|$$ \mid $$sin) model, which aims to alleviate the effect of data fluctuation on prediction accuracy and increase the contribution of new data in the prediction procedure. To further decrease prediction error, a Fourier series is adopted to model the residual series. The proposed model, called Fourier Fractional GM(1,1|$$ \mid $$sin) Model (FFGMsin), is used to predict the employment rate of graduated students of Yanshan University from 2010 to 2019. Results show that the proposed method can obtain more accurate prediction results than GM(1,1) model and its variants. © 2024 Institute of Electrical Engineer of Japan and Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

Carbon mitigation and net-zero targets have an important place on the political agenda for all countries. Countries have been seeking to decouple economic growth and carbon emissions to achieve decarbonization. However, there is a cost of the decarbonization of economies, which can be called as Green Sacrifice Ratio (GSR), and this is critical for developing countries because they have been in the growing stage. By considering the critical role of GSR in achieving net-zero emissions by decarbonizing economies, the study comparatively examines how much environmental quality should be sacrificed for per capita economic growth in China and India. In line with this purpose, the study uses the Fourier-Shin cointegration test and various time series estimators with Fourier approximations for the period from 1989/Q1 to 2022/Q4. Estimation results are as follows: (i) all variables follow the non-stationary process, (ii) the long-run relationship exists between variables, and (iii) Estimation results show that GSR is 0.65% for China and 0.78% for India. Therefore, China and India sacrifice a significant amount of environmental quality for a 1% increase in per capita income level. Also, the ecological cost of increasing economic welfare in India is higher with regard to China. Besides, the results are robust according to the alternative method. Thus, the study recommends that both Chinese and Indian policymakers should revise the economic expansion structure with green growth strategies to prevent irreversible environmental degradation due to higher economic growth. [ABSTRACT FROM AUTHOR]

Low-thrust trajectory design in the context of the three-body problem, particularly in the vicinity of liberation points, is addressed in this paper with a focus on efficiency and accuracy. The Fourier series, previously successful in approximating near-coplanar low-thrust trajectories within two-body dynamics, is applied to the challenging landscape of the three-body problem, marked by instability and nonlinearity. This paper introduces a novel approach for constructing an initial guess trajectory, that combines coast arcs from periodic orbits and intermediate trajectory arcs from other natural dynamical structures. The resulting initial guess trajectory is instrumental in generating efficient initial estimates for the coefficients of the Fourier series. The study encompasses the analysis of homoclinic and heteroclinic transfers in both two-dimensional and three-dimensional scenarios. The approximated trajectories derived from this methodology serve as invaluable starting points for high-fidelity optimization solvers, offering promise for enhancing the precision and efficiency of low-thrust transfers in cislunar space. • The shape method based on Fourier series is applied to the initial design of the transfer trajectories between periodic orbits in cislunar space. • An innovative approach to construct a continuous initial guess trajectory is introduced to derive initial guess values for the coefficients of Fourier terms. • An accurate approximation of the optimal transfer trajectory can be rapidly obtained. [ABSTRACT FROM AUTHOR]

The mathematical formulations for transverse compression of a thin elastic disc are considered, including various boundary conditions along the faces of the disc. The mixed boundary conditions corresponding to the loading by normal stresses in absence of sliding are studied in detail. These conditions support an explicit solution in a Fourier series for the boundary layers localised near the edge of the disc and also do not assume making use of the Saint-Venant principle underlying the traditional asymptotic theory for thin elastic structures. As an example, an axisymmetric problem is studied. Along with the leading order solution for a plane boundary layer, a two-term outer expansion is derived. The latter is expressed through the derivatives of the prescribed stresses. Generalisations of the developed approach are addressed. [ABSTRACT FROM AUTHOR]

An approach for solving the plane nonaxisymmetric thermoelasticity problems in multilayer cylinders is suggested from the single-solid standpoint, which implies considering a multilayer structure as a single unity with a step-wise variation of material properties. The direct integration method was used to the advantage of our approach by applying toward the compatibility equation in terms of strains instead of the one in terms of stresses. As a result, the governing equation is derived without involving generalized derivatives by the radial coordinate and then reduced to an integral equation of the second kind. Using the resolvent-kernel for solving the latter equation, the thermal stresses are computed as the Fourier series whose constituents are explicitly determined by the corresponding constituents of the thermal loading. [ABSTRACT FROM AUTHOR]

We consider nonharmonic Fourier series defined in terms of arbitrarily close exponentials. Our aim is to use the positivity of quadratic polynomials in order to get new Ingham-type weighted inequalities. The proof relies on an Ingham proof technique inspired by Jaffard et al. (J Fourier Anal Appl 3:577–582, 1997). As applications, we consider families of frequencies with relevance in control approximation theory, for which we can prove the uniform (with respect to the mesh-size) controllability property of the semi-discrete model, when the spurious frequencies (the gap between them tends to zero when the mesh size goes to zero) are eliminated. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to show that in certain classes of real (or complex) functions, Bernoulli polynomials are essentially the only ones satisfying the Raabe functional equation. For the class of real 1-periodic functions which are expandable as Fourier series, we point out new solutions of the Raabe functional equation, not related to Bernoulli polynomials. Furthermore, we will give for the considered classes various proofs, making the mathematical content of the paper quite rich. [ABSTRACT FROM AUTHOR]

This paper values fixed-income (discrete- and continuous-time) European Asian and Australian options. We assume that the term structure of interest rates is modelled by the specification proposed in Moreno et al. (Econ Model 72:140–150, 2018, https://doi.org/10.1016/j.econmod.2018.01.015). We obtain closed-form expressions for the premiums of geometric average options and, for arithmetic average options, premiums are computed by numerical methods. We also perform a sensitivity analysis with respect to different parameters for both (geometric and arithmetic) options. [ABSTRACT FROM AUTHOR]

The full-field responses and high-resolution mode shapes are of paramount importance in the detection of bridge damage and finite element model updating. As enlightened by the Fourier series expansion of the mode shapes for a bridge, this study proposes an approach for compressive sensing-based reconstruction of the full-field responses and identification of high-resolution mode shapes, in which a general basis matrix is newly formulated, with particular application to beam-like continuous or non-uniform bridges. The merits of the proposed approach are model-free, output-only, and easy to implement, especially the need of a few sensors. The effectiveness of the approach is validated through both numerical and experimental examples by considering various bridge boundary conditions and non-uniform cross-sections, from which the potential application of this general approach is foreseeable. [ABSTRACT FROM AUTHOR]

AbstractThe paper investigates the potential of the full-sinusoidal Fourier series as a solution form for the deflection of layered functionally graded beams associated with smart actuators, drawing on the fundamental principles of classical Fourier series theory. The current method simplifies the difficult beam problem to a set of linear algebraic equations by using the Duhamel integration technique and the orthogonality of the trial function. The study takes into account generalized boundary conditions and moving forces, which are seldom discussed in previous research. Under the action of a moving load, the boundary value problem for a functionally graded beam integrated with Terfenol-D is efficiently addressed using the approach of combined differential and integral quadrature. The generalized boundary conditions may be easily achieved by adjusting the stiffness of the restraining springs. The significant agreement between the differential quadrature solution and the Fourier series solution underlines the efficiency and accuracy of both methods. Furthermore, the influences of various crucial physical characteristics on the natural frequencies and the essential flow velocities are explored, including boundary stiffness, foundation parameters, and geometric parameters. [ABSTRACT FROM AUTHOR]

In this paper the Walsh system will be considered in the Kaczmarz rearrangement. We estimate the difference between matrix transform means of Walsh–Kaczmarz–Fourier series and the corresponding function in norm, and the upper estimation is given by the modulus of continuity of the function. We also prove norm convergence with similar conditions. [ABSTRACT FROM AUTHOR]

New exact solutions of the Charney–Obukhov equation for the ocean are obtained in the form of a partial superposition of elementary solutions with different wave numbers. The boundary conditions for the ocean are satisfied due to the presence of a carrier zonal flow in the solution. The existing arbitrariness in the choice of wave numbers and other solution parameters makes it possible to simulate an arbitrary stream function profile at a fixed ocean depth on an interval of a fixed length using a Fourier series or in a circle of a fixed radius using a Fourier–Bessel series. An example of modeling a Gaussian stream function profile on the ocean surface in the presence of circular symmetry is considered. [ABSTRACT FROM AUTHOR]

This paper introduces a new lower-limb rehabilitation machine that meets the rehabilitation needs of hemiplegic patients. First, a left–right independent rotary pedal mechanism was selected to facilitate rehabilitation and adapt to the user's physical condition. Then, a half model of the lower-limb rehabilitation machine is designed and manufactured with ergonomics in mind. As analytical tools, we combine non-negative matrix factorization and non-negative double singular value decomposition to calculate muscle synergy of the walking muscle surface electromyography (sEMG) signal, and use cosine similarity to evaluate the similarity between walking and pedaling activities. By comparing the results of the walking and pedaling experiments, the effectiveness of pedaling in gait rehabilitation is revealed. To further improve the similarity between walking and pedaling, double integration of the sEMG signal is introduced, and the relationship between load input and rotation angle is described for the first time using Fourier series. The results of the experiment confirmed that more than half of the 10 subjects performed pedaling exercises similar to walking using Fourier series loading compared to pedaling exercises with normal constant loading. This loading parameter may have the potential to improve rehabilitation efficiency for many subjects compared to the usual exercise. [ABSTRACT FROM AUTHOR]

Cropland mapping using remote sensing data is the basis for effective crop monitoring, crop rotation control, and the detection of irrational land use. Classification using Normalized Difference Vegetation Index (NDVI) time series from multi-year data requires additional time costs, especially when sentinel data are sparse. Approximation by nonlinear functions was proposed to solve this problem. Time series of weekly NDVI composites were plotted using multispectral Sentinel-2 (Level-2A) images at a resolution of 10 m for sites in Khabarovsk District from April to October in the years 2021 and 2022. Missing values due to the lack of suitable images for analysis were recovered using cubic polynomial, Fourier series, and double sinusoidal function approximation. The classes that were considered included crops, namely, soybean, buckwheat, oat, and perennial grasses, and fallow. The mean absolute percentage error (MAPE) of each class fitting was calculated. It was found that Fourier series fitting showed the highest accuracy, with a mean error of 8.2%. Different classifiers, such as the support vector machine (SVM), random forest (RF), and gradient boosting (GB), were comparatively evaluated. The overall accuracy (OA) for the site pixels during the cross-validation (Fourier series restored) was 67.3%, 87.2%, and 85.9% for the SVM, RF, and GB classifiers, respectively. Thus, it was established that the best result in terms of combined accuracy, performance, and limitations in cropland mapping was achieved by composite construction using Fourier series and machine learning using GB. Similar results should be expected in regions with similar cropland structures and crop phenological cycles, including other regions of the Far East. [ABSTRACT FROM AUTHOR]

This paper introduces an innovative approach for generating re-entry trajectories for a reusable winged spacecraft. This approach utilizes Fourier series parametrized control profiles. The Fourier coefficients are derived through a combination of the Improved Search Space Reduction (ISSR) technique and Sequential Quadratic Programming (SQP) optimization methods, which effectively limit the maximum heat rate experienced by the spacecraft. These re-entry trajectories and control profiles are then used to train artificial neural networks, enabling the controller to provide optimal control inputs based on the spacecraft's current altitude and velocity. To validate this methodology, Hardware-in-loop (HIL) simulations are conducted, integrating the designed neural network-based controller with a Texas Instruments TI Delfino TMSF28335 controller and a real-time simulator, the OPAL-RT OP4510. The results of the HIL simulation demonstrate that the generated re-entry trajectory accurately adheres to heat rate and terminal constraints. Additionally, the Fourier series parametrization of control profiles is applied to a high lift-to-drag ratio CAV-H vehicle, showcasing the method's versatility. Furthermore, resilience of proposed method to uncertainties in aerodynamic coefficients and atmospheric density is also demonstrated. The results show that the proposed method is generic and exhibits robustness to uncertainties. [ABSTRACT FROM AUTHOR]

This work attempts to propose a practical methodology for the one-dimensional heat conduction problems with unknown boundary conditions of the first kind at both ends. The methodology can be effectively applied when the remote boundary condition is inaccessible or internal temperature is required, such as a cutting process with cooling lubricant, and ablative thermal protection system. The proposed methodology utilizes the shifting function method in conjunction with the least squares error method to transform the inverse heat conduction problem into an approximate "well-posed" problem. Consequently, the temperature and the heat flux distributions over the entire time and space domains are determined by using half-range Fourier cosine series solutions. Experimental examples of spray cooling problems are provided to illustrate the advantages of the proposed methodology, including fast convergence of the temperature function, fewer discrete measured times in numerical analysis, and regardless of two interior temperature probes positions. [ABSTRACT FROM AUTHOR]

We prove that among all 1-periodic configurations |$\Gamma $| of points on the real line |$\mathbb{R}$| the quantities |$\min _{x \in \mathbb{R}} \sum _{\gamma \in \Gamma } e^{- \pi \alpha (x - \gamma)^{2}}$| and |$\max _{x \in \mathbb{R}} \sum _{\gamma \in \Gamma } e^{- \pi \alpha (x - \gamma)^{2}}$| are maximized and minimized, respectively, if and only if the points are equispaced and whenever the number of points |$n$| per period is sufficiently large (depending on |$\alpha $|⁠). This solves the polarization problem for periodic configurations with a Gaussian weight on |$\mathbb{R}$| for large |$n$|⁠. The first result is shown using Fourier series. The second result follows from the work of Cohn and Kumar on universal optimality and holds for all |$n$| (independent of |$\alpha $|⁠). [ABSTRACT FROM AUTHOR]

The geometrically nonlinear stability analysis of restrained nanobeams under an elastic medium is considered in the presented manuscript. The investigation is based on the nonlocal strain gradient elasticity theory, Euler-Bernoulli beam theory and geometrical nonlinearity which is a significant change in geometry. To the best of the authors' knowledge, the nonlinear stability response of a nanobeam with elastic boundary conditions and on an elastic foundation has not been presented before via the theory of nonlocal strain gradient. The aim of this paper is to fill this gap in the literature by offering a method that can give a solution under general elastic boundary conditions. Using a Fourier sine series, a Fourier coefficient suitable for constructing an eigenvalue problem is computed. The constructed problem has been treated in the transformed region with the help of the Stokes' transform, then the stability analysis is executed for arbitrary boundary conditions (rigid or restrained) based on the nonlocal strain gradient elasticity. The influences of various nonlinear parameters, deformable boundary conditions, small-scale parameters and elastic foundation parameters on the stability of the constrained nanobeam are studied. It is observed that the investigated variables produce significant changes in buckling loads. [ABSTRACT FROM AUTHOR]

To estimate the tunnel response induced by a new tunnel excavation underneath, theoretical solutions are proposed in this study. The overlying tunnel is idealized as an infinite Timoshenko beam resting on the Kerr foundation model, then the vertical force balance equation is established. The unloading stress can be expressed as Fourier cosine series and a theoretical solution can be derived. The effectiveness of the proposed method is verified by a centrifuge test and an actual field measurement, which are extracted from previous investigations. Calculation results from the proposed method show good accord to centrifuge test results and field‐measured data. Meantime the predictions given by the proposed method are more accurate to estimate tunnel response compared with the previous method. Parametric analysis indicates that the volume loss ratio and the vertical clearance between two tunnels, are both significantly alleviate the tunnel response, but the effect of tunnel shear stiffness is slight. The theoretical solutions can be adopted to predict the overlying tunnel response induced by tunneling underneath in real projects. [ABSTRACT FROM AUTHOR]

The composite-laminated rotationally stiffened plate is widely applied in aviation, aerospace, ship, machinery, and other fields. For structural design and optimization, to investigate the vibration characteristics is important. In this paper, a modeling method of composite-laminated rotationally plate is established. The first-order shear deformation theory (FSDT) and the modified Fourier series are applied to construct the admissible displacement function of the stiffened plate-coupled systems. On this basis, the energy function of composite-laminated rotationally stiffened plate is established. Combined with the artificial virtual spring technology, the proposed theory could be used to analyze the vibration characteristics of composite-stiffened plate-coupled systems with various classical boundary conditions or arbitrary elastic boundary conditions. The Rayleigh–Ritz method is used to solve the energy function. Thus, the vibration characteristics of the composite-laminated rotationally stiffened plate are obtained and analyzed. The correctness of the theoretical analysis model was verified through modal experiments. On this basis, the effect of some important parameters on the vibration characteristics of stiffened plate structures is studied, such as the number, thickness, and width of the laminated stiffener, varying structural parameters, and different boundary conditions. This study can provide the theoretical basis for the vibration and noise reduction of such structures. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach for assessing the uncertainty in vibration and static responses of laminated composite beams resulting from uncertainty in material properties and distributed loads. The proposed method utilizes surrogate models developed using polynomial chaos expansion (PCE) based on a relatively small sample size. These training samples are computed using a high-order shear beam model in which the governing equations are derived using Hamilton's principle, and solved by Ritz's approach using a trigonometric series approximation. The proposed PCE model's coefficients are estimated using the spectral projection and linear regression techniques. The first four statistical moments and probability distributions of the mid-span displacement and the fundamental frequency of laminated composite beams are predicted. Global sensitivity analysis is also conducted to assess how material property variation and stochastic loads affect the beam's deflection and the fundamental frequency. The accuracy and efficiency of the proposed PCE models are compared with those from Monte Carlo simulation (MCS). A remarkable reduction in the computational cost of PCE models compared to MCS is observed without compromising the predictions' accuracy. As most real-world systems are subjected to multiple sources of uncertainty, this study provides a state-of-the-art method to quantify such uncertain parameters more efficiently and allow for a better reliability assessment in composite beam design. [ABSTRACT FROM AUTHOR]

This paper examines the relationship between the overconvergence of Fourier series and the existence of Hadamard–Ostrowski gaps. Ostrowski's result on the overconvergence of power series serves as a motivating factor for obtaining a natural generalization: the overconvergence of Fourier series. The connection between Hadamard–Ostrowski gaps and the overconvergence of Fourier series is clarified by applying the Hadamard three-circle theorem and the theory of orthogonal polynomials. Our main result is obtained by applying the Hadamard three-circle theorem. [ABSTRACT FROM AUTHOR]

In this study, the space-time fractional generalised reaction duffing model is investigated analytically, which is a generalization for a collection of prominent fractional models describing various key phenomenon in science and engineering. The governing equation is converted to a nonlinear ODE by the compatible travelling wave transform. The investigation established that for analysing nonlinear evolution equations of fractional order, the recommended approach is more effective and realistic. The findings are given extensively in rational forms of trigonometric function series or clearly in powers of specific trigonometric functions. A collection of bright, dark, periodic, and optical solitons is discovered. Mathematica is used to flourish the presence of some obtained solutions in 3D graphs with different fractional orders. The results show that the recommended methods are more practical and effective ways to illustrate the dynamics of several complex wave structures in modern science and technology. [ABSTRACT FROM AUTHOR]

A method based on the hypersingular integral equation approach and the modal analysis is presented to consider the effects of the motion of a submerged elastic disk on the incoming waves. Initially, the governing boundary value problem is reduced to a two-dimensional integral equation with a hypersingular kernel. This integral equation is further reduced to a one-dimensional Fredholm integral equation of the second kind with the help of Fourier series expansions and a newly defined function. As a part of modal analysis, eigenfunction expansion based on natural modes of structural motion is considered to describe the motion of a thin circular elastic disk. Physical quantities, such as hydrodynamic force, added mass, damping coefficient, and surface elevation, are numerically evaluated. The computed numerical results are verified by comparing them with those for the rigid disk horizontally submerged in deep water. Apart from this, as a part of the analytical verification of our present analysis, the reciprocity relation has been included. The effects of different parameters (disk's rigidity, radius, submergence depth, and mode of vibrations) on the aforementioned physical quantities have been studied. The maximum hydrodynamic force occurs around Ka = 0.5, while the maximum added mass and damping coefficient occur around the wavenumber Ka = 0.3 and Ka = 0.5, respectively. The peaks of the hydrodynamic force and free surface elevation become sharper with the increasing values of the disk's size. The numerical results emphasize that the wave focusing can be controlled by changing the submergence depth, size, and rigidity of the disk. [ABSTRACT FROM AUTHOR]

The propagation characteristics of Lamb waves in a functionally graded material (FGM) plate with periodic gratings have been studied. The power series technique is employed to solve the governing differential equations with variable coefficients. In the propagation direction, the displacements of Lamb waves are expanded in the Fourier series due to the periodicity of the structure. The convergences of the power series and Fourier series are proved and the method presented in the article is verified by the finite element method (FEM). The band gaps are obtained by the couplings between the different modes of Lamb waves induced by FGM. The effects of the structural parameters (such as the periodicity, mass, and length of the gratings) and gradient coefficient on band gaps are investigated. Numerical results show that the band gaps shift up as the gradient coefficient increases and the second band gap (SBG) is closed when the gradient coefficient is large enough. The conclusions are of practical significance for designing elastic wave filters with high-performance. [ABSTRACT FROM AUTHOR]

A new approach for the solution of transient eddy-current problems involving pieces with one direction of symmetry is presented. The approach is applicable to pieces with translational or rotational symmetry. A Fourier series decomposition of the solution is introduced in the direction of symmetry, which converts the original 3D numerical problem into a series of independent 2D problems. The decomposition has significant benefits in terms of computational time and numerical noise reduction, and is inherently parallelisable. [ABSTRACT FROM AUTHOR]

This paper focuses on the valuation of variable annuities with a guaranteed minimum maturity benefit under a regime-switching Lévy model. The model allows policyholders to surrender their annuities and receive a surrender benefit at predetermined tenor times before maturity. Additionally, we consider a state-dependent periodic fee structure where fees are deducted from the policyholder's account if it exceeds a certain level at discrete time points. Incorporating this fee structure, the Fourier cosine series expansion method based on characteristic functions is employed to determine the values and optimal surrender strategies for variable annuity contracts. Finally, we provide a comprehensive set of numerical examples to demonstrate and assess the effectiveness of our approach thoroughly. [ABSTRACT FROM AUTHOR]

In data analysis using a nonparametric regression approach, we are often faced with the problem of analyzing a set of data that has mixed patterns, namely, some of the data have a certain pattern and the rest of the data have a different pattern. To handle this kind of datum, we propose the use of a mixed estimator. In this study, we theoretically discuss a developed estimation method for a nonparametric regression model with two or more response variables and predictor variables, and there is a correlation between the response variables using a mixed estimator. The model is called the multiresponse multipredictor nonparametric regression (MMNR) model. The mixed estimator used for estimating the MMNR model is a mixed estimator of smoothing spline and Fourier series that is suitable for analyzing data with patterns that partly change at certain subintervals, and some others that follow a recurring pattern in a certain trend. Since in the MMNR model there is a correlation between responses, a symmetric weight matrix is involved in the estimation process of the MMNR model. To estimate the MMNR model, we apply the reproducing kernel Hilbert space (RKHS) method to penalized weighted least square (PWLS) optimization for estimating the regression function of the MMNR model, which consists of a smoothing spline component and a Fourier series component. A simulation study to show the performance of proposed method is also given. The obtained results are estimations of the smoothing spline component, Fourier series component, MMNR model, weight matrix, and consistency of estimated regression function. In conclusion, the estimation of the MMNR model using the mixed estimator is a combination of smoothing spline component and Fourier series component estimators. It depends on smoothing and oscillation parameters, and it has linear in observation and consistent properties. [ABSTRACT FROM AUTHOR]

This paper considers the inverse problem of determining the time‐dependent coefficient in the time‐fractional diffusion‐wave equation. In this case, an initial boundary value problem was set for the fractional diffusion‐wave equation, and an additional condition was given for the inverse problem of determining the coefficient from this equation. First of all, it was considered the initial boundary value problem. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag‐Leffler function and the generalized singular Gronwall inequality, we get a priori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved. The stability estimate is also obtained. [ABSTRACT FROM AUTHOR]

This study focuses on the rate of uniform approximation for continuous functions g(x,y),2π$$ g\left(x,y\right),2\pi $$‐periodic in each variable, using double matrix means of the rectangular partial sums of double Fourier series. The importance of the weighted integral modulus of symmetric smoothness lies in its ability to capture the local behavior of a function. So we obtain the results in terms of the weighted integral modulus of symmetric smoothness. In addition, we introduce the concept of the weighted Lipschitz and Zygmund classes as an extension of the existing Lipschitz classes Lip(α,β)$$ Lip\left(\alpha, \beta \right) $$ and Lip(α,β;p)$$ Lip\left(\alpha, \beta; p\right) $$ and Zygmund classes Z(α,β)$$ Z\left(\alpha, \beta \right) $$ and Z(α,β;p)$$ Z\left(\alpha, \beta; p\right) $$, respectively. These weighted classes allow us a more detailed analysis of the approximation rates of functions by assigning the weights to different regions of the domain of function. Further, we prove two theorems about the degree (error) of approximation of functions belonging to these classes using double matrix means. We also discuss some corollaries from our results. [ABSTRACT FROM AUTHOR]

In order to meet the motion characteristics requirements of multi-objective fusion of execution components, a synthesis method of open-chain coupled serial linkage mechanism fusing trajectory and posture is proposed. According to periodic change characteristics of the execution component trajectory and posture of the linkage mechanism, the motion characteristics equation including trajectory and posture information is established by using Fourier series. The harmonic parameters of the link trajectory and posture are calculated according to the principle of discrete Fourier transform. The complex vector theory is applied to determine the functional relationship between the harmonic parameters of the trajectory and posture and the independent variable of the linkage mechanism, and furthermore the motion synthesis equation of the open-chain linkage mechanism fusing trajectory and posture information is established. The linkage mechanism parameters are calculated using Mathcad Prime software. The single-degree-of-freedom open-chain coupled serial linkage mechanism fusing trajectory and posture information is obtained by using a noncircular gear planetary gear train to couple the connection points of the linkage mechanism. The correctness of the method is verified through an example. [ABSTRACT FROM AUTHOR]

In this paper, we study the inversion of the inner boundary data for Poisson equation in a doubly connected domain by fast Fourier ultraspherical spectral solver. The solver depends on the truncated Fourier series expansion, where the differential equations of the Fourier coefficients are solved by an ultraspherical spectral method. Because this problem is seriously ill‐posed, the Cauchy data with noise will lead to ill‐conditioned linear system. Hence, we apply Tikhonov regularization to solve the obtained linear system and use generalized cross‐validation (GCV) criterion to select regularization parameters. The accuracy and efficiency of the proposed method are illustrated by several numerical results of different regions. [ABSTRACT FROM AUTHOR]