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A linear sixth-order partial differential equation (PDE) of "parabolic" type describes the dynamics of thin liquid films beneath surfaces with elastic bending resistance when deflections from the equilibrium film height are small. On a finite domain, the associated sixth-order eigenvalue problem is self-adjoint for the boundary conditions corresponding to a thin film in a closed trough, and the eigenfunctions form a complete orthonormal set. Using these eigenfunctions, we derive the Green's function for the governing sixth-order PDE on a finite interval and compare it to the known infinite-line solution. Further, we propose a Galerkin spectral method based on the constructed sixth-order eigenfunctions and their derivative expansions. The system of ordinary differential equations for the time-dependent expansion coefficients is solved by standard numerical methods. The numerical approach is applied to versions of the governing PDE with a second-order spatial derivative (in addition to the sixth-order one), which arises from gravity acting on the film. In the absence of gravity, we demonstrate the self-similar intermediate asymptotics of initially localized disturbances on the film surface, at least until the disturbances "feel" the finite boundaries, and show that the derived Green's function is an attractor for such solutions. In the presence of gravity, we use the proposed Galerkin numerical method to demonstrate that self-similar behavior persists, albeit for shortened intervals of time, even for large values of the gravity-to-bending ratio. [ABSTRACT FROM AUTHOR]

A linear sixth-order partial differential equation (PDE) of "parabolic" type describes the dynamics of thin liquid films beneath surfaces with elastic bending resistance when deflections from the equilibrium film height are small. On a finite domain, the associated sixth-order eigenvalue problem is self-adjoint for the boundary conditions corresponding to a thin film in a closed trough, and the eigenfunctions form a complete orthonormal set. Using these eigenfunctions, we derive the Green's function for the governing sixth-order PDE on a finite interval and compare it to the known infinite-line solution. Further, we propose a Galerkin spectral method based on the constructed sixth-order eigenfunctions and their derivative expansions. The system of ordinary differential equations for the time-dependent expansion coefficients is solved by standard numerical methods. The numerical approach is applied to versions of the governing PDE with a second-order spatial derivative (in addition to the sixth-order one), which arises from gravity acting on the film. In the absence of gravity, we demonstrate the self-similar intermediate asymptotics of initially localized disturbances on the film surface, at least until the disturbances "feel" the finite boundaries, and show that the derived Green's function is an attractor for such solutions. In the presence of gravity, we use the proposed Galerkin numerical method to demonstrate that self-similar behavior persists, albeit for shortened intervals of time, even for large values of the gravity-to-bending ratio. [ABSTRACT FROM AUTHOR]

This study deals with an oblique wave interaction by a floating rigid rectangular structure in the presence of a porous barrier which is placed in front of the floating structure. By considering the bottom topography as an elevated-type and a trench-type bottom, it is assumed that the train of water waves interacts with the porous barrier and the floating structure due to which it undergoes partial reflection. The boundary value problem in the fluid domain, which is split into five sub-regions, is solved by the utilization of separation of variables technique and eigenfunction expansion. The impact of the porous barrier and bottom topography on the reflection coefficient and hydrodynamic forces acting on the floating structure is discussed. As the distance of the porous barrier from the floating structure increases, an oscillatory pattern in the reflection coefficient is observed. Further, when the height of the elevated bottom or trench-bottom increases, the reflection coefficient increases in both cases; the reflection is higher in the presence of a trench than that due to the elevated sea-bed. The obtained results are validated against an established result which shows a very close match. • Water wave scattering by a rigid rectangular floating structure in the presence of a thin porous barrier is studied. • An elevated-type and a trench-type bottom topography are considered to represent the sea-bed. • The study aims to examine the variation of reflection and mitigation of forces acting on the rigid structure. • Significant variations in wave reflection and forces occur with changes in the porous-effect parameter of the barrier. • The effect of the various parameters on the reflection coefficient and hydrodynamic forces is also investigated. [ABSTRACT FROM AUTHOR]

A modification of Dean's method is proposed for determining the impedance in the case of a nonuniform sound field on the front and bottom surfaces of a resonator. Instead of acoustic pressures in Dean's formula, the modification uses the coefficients of eigenfunctions, which correspond to a uniform acoustic pressure distribution on the front and bottom surfaces of the resonator. The eigenproblem is solved by the finite element method; the coefficients of the eigenfunctions are found by the least squares method. At the current stage of research, the full-scale experiment has been replaced by numerical simulation in a linear formulation of sound propagation in an impedance tube with normal wave incidence with a honeycomb resonator attached to it. The inhomogeneity of the pressure field over the cross section of the resonator is created from the different positions of holes in the resonator face plate. The study is done for a different number of acoustic pressure measurement points at the bottom of the resonator. Calculations show that the proposed method is efficient and provides good agreement with the straight method for determining impedance. However, the possibilities of using modification of Dean's method in full-scale measurements are limited, because accurate resonator impedance determination requires a large number of measurement points. [ABSTRACT FROM AUTHOR]

In this study, a singular impulsive Hahn–Dirac system is studied. A spectral function has been established for this type of system. With the help of this function, the Parseval equation and eigenfunction expansion were obtained. [ABSTRACT FROM AUTHOR]

Thin porous plates serve as an effective model for the construction of breakwater. Thus, the problem involving oblique wave interaction with a tunnel in the presence of a submerged horizontal porous plate over a trench-type bottom is investigated. In this article, for the mathematical formulation of the physical model, water wave potentials are defined using Havelock's expansions and flow past over porous structure is modeled based on Darcy's law. The advantage of the trench type of bottom and horizontal plate is studied through the numerical results of forces on the tunnel. The study reveals that more energy loss and less force on the tunnel are obtained if the porous effect parameter of the plate or the length of the plate is increased up to a moderated value of these parameters. Compared to the case without porous plate and trench-type bottom topography, there are significant changes in forces due to this porous breakwater and trench-type bottom topography. In addition, from the present results, it may be noted that the load on the submerged tunnel is reduced by adding a submerged horizontal porous plate and asymmetric trench, which is helpful in understanding the role of porous breakwaters and trenches in applications to ocean and coastal engineering. [ABSTRACT FROM AUTHOR]

In this article, we study a conformable fractional heat conduction equation. Applying the method of separation variables to this problem, we get a conformable fractional Sturm-Liouville eigenvalue problem. Later, we prove the existence of a countably infinite set of eigenvalues and eigenfunctions. Finally, we establish uniformly convergent expansions in the eigenfunctions. [ABSTRACT FROM AUTHOR]

In the process of designing the new NEPTUN pulsed reactor, the phenomenon of dynamic bending was discovered; it is a feature of the dynamics of pulsed reactors that does not appear in reactors of other types; however, it plays a significant role in evaluating the stability of reactor operation. Movements of fuel elements (fuel rods) under the influence of periodically changing temperature can lead to significant changes in the reactivity and fluctuations of the reactor power. To accurately simulate the operation of the reactor, it is necessary to know the shape of the fuel element at the time of the next power pulse. Since numerical methods for solving this dynamic problem of thermoelasticity require a computation time that is too large, an analytical method was applied. In this study, the first results of the numerical–analytical solution of the equation of induced oscillations of a fuel rod, as well as the prospects for using the method in studying the dynamics of the NEPTUN reactor, are presented. [ABSTRACT FROM AUTHOR]

In this work, we take into consideration a two-layer fluid system made up of a cylindrical buoy which is floating vertically in the upper fluid layer and a coaxial bottom-mounted cylindrical caisson placed in the bottom fluid layer. The analytical solution for the water wave diffraction caused by the proposed device has obtained by using the separation of variable method. In order to evaluate the undetermined coefficients to the diffracted velocity potential, we use matched eigenfunction expansion method. In two-layer flow, the incident wave propagates in two modes, namely mode of surface and internal-wave, and the hydrodynamic force occurs in both modes of motion. Further, we obtain hydrodynamic forces, namely, exciting force and moment acting on the device in each wave-mode of motion. We present the results of surge exciting force, heave exciting force and moment by the cylinders in surface and internal wave modes of motion along with various parameters of the device in wide range of frequencies. The significant effect of depth and draft ratios on the exciting forces and moments are observed in both modes of motion. The influence of surge and heave exciting forces by the lower cylinder exhibit almost similar trends and oscillations can be observed in moment for the surface wave mode. The obtained results are compared with the available established results. It can be seen that the results of the present work are in good agreement with established results. It validates the present results [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel model for pricing double barrier options, where the asset price is modeled as a threshold geometric Brownian motion time changed by an integrated activity rate process, which is driven by the convolution of a fractional kernel with the CIR process. The new model both captures the leverage effect and produces rough paths for the volatility process. The model also nests the threshold diffusion, Heston and rough Heston models. We can derive analytical formulas for the double barrier option prices based on the eigenfunction expansion method. We also implement the model and numerically investigate the sensitivities of option prices with respect to the parameters of the model.

In this paper, water waves interaction with two thin vertical barriers over shelf bottom topography is analysed using linearised wave theory. The associated mixed boundary value problem is solved with the aid of method involving eigenfunction expansions of the velocity potential and orthogonality relation of the eigenfunctions. Further, the resulting system of algebraic equations is solved using the least square method to find the physical quantities, that is, reflection and transmission coefficients, free surface elevation and non-dimensional horizontal force experienced by the barriers. The energy balance relation is derived from Green's identity which ensures the correctness of the present results. The obtained results are also compared with the results available in the literature for validation purpose. With the help of different plots, the effect of depth ratios, length of the barriers, angle of incidence and gap between the barriers is investigated for various values of physical parameters. The study reveals that the phenomena of zero reflection, that is, full transmission can be avoided by using non-identical barriers or asymmetric shelf bottom topography. Also, it is highlighted that the presence of two barriers instead of a single barrier over shelf topography will help to reduce the transmitted wave energy near the seashore. A generalisation of number of surface piercing barriers over the shelf bottom topography is also demonstrated. [ABSTRACT FROM AUTHOR]

For a weight function r ∈ L 1 [ - 1 , 1 ] with sign change at 0 we consider two indefinite regular Sturm–Liouville problems: - f ′ ′ = λ r f with Neumann boundary conditions and - (u ′ r) ′ = λ u with Dirichlet boundary conditions. It is known that the eigenfunctions may (or may not) form a Riesz basis in the naturally associated weighted Hilbert spaces only simultaneously. Here, we study the situation where this property fails to hold. Nevertheless, we obtain eigenfunction expansions on smaller domains or in spaces with stronger or weaker topology. As the main result, we identify explicit functions in the "original" spaces which do not allow an eigenfunction expansion. Furthermore, we find a sequence of functions which violates a certain HELP inequality and indicate limits of other related results in Kreĭn spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the basic theory of regular Hahn-Hamiltonian systems. In this context, we establish an existence and uniqueness result. We introduce the corresponding maximal and minimal operators for this system and some properties of these operators are investigated. Moreover, we give a criterion under which these operators are self-adjoint. Finally, an expansion theorem is proved. [ABSTRACT FROM AUTHOR]

A weak form quadrature element formulation is established to obtain the coefficients of Mode III crack-tip asymptotic fields. The problem domain is divided into circular subdomains near the crack tips and several integrable normal subdomains elsewhere. The total potential energy of the crack problem is given using the displacement expressions in circular subdomains and nodal displacements in normal subdomains. Accordingly, the minimum potential energy principle is employed to establish algebraic equations. Coefficients of any order, which are part of the unknowns of the algebraic equations, are solved for from the equations. Several benchmark examples are examined to validate the present formulation, showing high effectiveness in evaluation of coefficients of Mode III eigenfunction expansion. [ABSTRACT FROM AUTHOR]

Hydroelastic wave interaction with a circular crack of an ice-cover in a channel together with some related problems is considered, based on the linearized velocity potential theory and Kirchhoff plate theory. The domain decomposition method is adopted in the solution procedure. Two sub-domains are divided by the crack, one below the inner ice sheet and the other below the outer ice sheet. By using the Green function of an ice-covered channel, the velocity potential in the outer domain is established from the source distribution formula over an artificial vertical surface extended from the crack. The source distribution is expanded in both vertical and circumferential directions, which allows the velocity potential to be obtained in an explicit form with unknown coefficients. The velocity potential in the inner domain is expanded into a double series. An orthogonal inner product is used to impose continuity conditions on the artificial vertical surface and the edge conditions at the crack. The derived formulation is not just limited to the circular crack problem but can also be readily used in a variety of other problems, including wave diffraction by a surface-piercing vertical cylinder, polynya and circular disc floating on the free surface in a channel. Extensive results are provided for the forces on the inner ice sheet, the transmission and reflection coefficients. In particular, a detailed analysis is made on their behaviours near the natural frequencies of the channel, and the natural frequencies corresponding to the motion of the inner ice sheet. [ABSTRACT FROM AUTHOR]

This paper develops a semi-analytical solution of a water waves problem concerning the interaction between linear waves and an Oscillating Water Column (OWC) embedded into a circular platform. Unlike similar studies available in the literature, the proposed investigation concerns a realistic situation involving the use of an OWC having limited angular width. Such a configuration is likely to be employed in the arrangement of platforms in which the use of a large OWC is neither efficient nor feasible. The proposed solution is developed in the framework of the linear potential flow theory. The solution relies on the use of a domain decomposition approach involving eigen-function expansions of velocity potentials with unknown coefficients, which are calculated via a matching technique. The solution is utilized for conducting relevant parameter studies concerning the effects of the main geometrical parameters on the overall system performance and on the system hydrodynamic parameters. • Integration between a vertical cylinder and an OWC with limited angular width. • Analytical solution throughout the matching of eigenfunctions is obtained. • The results show an increased wave energy production. • Parameter study involving the main geometrical parameters is undertaken. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give a direct proof of an eigenfunction expansion formula for one-dimensional two-state quantum walks, which is an analog of that for Sturm–Liouville operators due to Weyl, Stone, Titchmarsh, and Kodaira. In the context of the theory of CMV matrices, it had been already established by Gesztesy–Zinchenko. Our approach is restricted to the class of quantum walks mentioned above, whereas it is direct and it gives some important properties of Green functions. The properties given here enable us to give a concrete formula for a positive-matrix-valued measure, which gives directly the spectral measure, in a simplest case of the so-called two-phase model. [ABSTRACT FROM AUTHOR]

The velocity and the displacement time history obtained by directly integrating the acceleration time history will drift unrealistically because of the over-determinacy of the integration constants. Applying baseline corrections to resolve drift disturbs the frequency content and renders the corrected processes mutually inconsistent. In this study, eigenfunctions of sixth-order eigenvalue problem are introduced as basis functions for decomposing recorded ground motion time history. Taking advantage of the eigenfunctions and their first two order of differentiations are drift-free and consistent, the decomposed and reconstructed acceleration, velocity, and displacement time histories can be drift-free and mutually consistent without direct integration and baseline correction. Two earthquake ground motions are presented as examples, which show that the decomposed acceleration, velocity, and displacement time histories are drift-free, mutually consistent, and almost the same as the seed motion. The new approach can replace integral in the generation of velocity and displacement time histories. [ABSTRACT FROM AUTHOR]

Viscoelastic pipes are commonly found in various industry applications. When undergoing deformations, a viscoelastic material combines both viscous and elastic behaviours, by exhibiting time-dependent strains. Hamilton's principle is employed to derive the governing equation for the slightly curved viscoelastic pipe resting on an elastic medium. The pipe's viscoelasticity is assumed using the Maxwell model. Three trigonometric function are used to describe the initial curvatures, while Eigenfunction expansion method was used to convert the partial differential equation to a set of four ODE's in time. The Eigenvalues, corresponding to the natural frequencies, were obtained up to four modes. Numerical results show the influence of the initial curvature, viscoelastic coefficient and the linear elastic medium on the frequency and fluid velocity of the pipe. It is noticeable that the initial curvature and the elastic medium cause the bifurcations point of the slightly curved pipe to be extended. For low viscoelasticity coefficient, the results obtained show two dissimilar imaginary frequencies. This is unlike what is previously reported in the literature. For higher coefficient, the pipe behaves like a steel pipe. At higher velocities, coupled mode flutter are absent for some modes depending on the boundary condition. [ABSTRACT FROM AUTHOR]

We investigate the effectiveness of a porous box in attenuating the structural response of a very large floating structure (VLFS). Assuming the water depth to be finite and small amplitude water wave theory, the physical problem is formulated by employing Darcy's law for flow past a porous structure. The boundary value problem is reduced to a system of linear algebraic equations with the aid of matched eigenfunction expansion method. Further, these simultaneous equations are solved numerically to compute physical quantities. The mathematical model is validated through a comparison with the theoretical results available in the literature. The reflection, transmission, and dissipation coefficients, elastic plate deflection, forces acting on the box, and free-surface elevation are computed. The reflection, transmission, and dissipation coefficients exhibit an oscillatory pattern for large values of wavenumber irrespective of the structural parameters of the box. It is highlighted that a porous box with moderate values of the porous-effect parameter is effective in reducing strain and shear force on the VLFS. Further, the results on elevation and VLFS deflection depict that the suitable porous-effect parameter values for the box reduce the resultant wave amplitude of the elastic plate deflection as well as the amplitude in the lee side of the VLFS. The study reveals that the width and height of the porous box are critical toward the trapping of incident waves inside the box and dissipating the maximum amount of incident wave energy in reduction of wave transmission in the lee side of the structure and thereby attenuating the structural response of the VLFS. [ABSTRACT FROM AUTHOR]

An optimal switching heating control strategy based on minimum margin (OS-MM) is proposed for high efficiency, low energy consumption, and high-uniformity fine control in a large-scale vertical quench furnace. This work was stimulated by the need for solving the contradiction between the heating rate and temperature overshoot to achieve a rapid rise in temperature with no overshoot. Based on a three-dimensional (3-D) transient temperature field model of the quenching furnace, the temperature field optimization control problem is transformed into a boundary optimization control problem of rapid heating, whereby it is rigorously proved that the fastest rise in temperature is attained by the full-power heating, which satisfies the bang-bang characteristics. The key parameter that determines the overshoot in the full-power heating mode is introduced and defined as the minimum margin in the temperature-rising process. The analytical expression for solving the minimum margin is calculated by the eigenfunction expansion method and using the strong metal thermal inertia of the internal temperature field, the OS-MM method is designed to stop the heating in advance at a calculated switching point to optimize full-power heating, thus conserving energy by reducing its consumption. The results of simulation and industrial experiments show that this strategy greatly shortens the temperature adjusting time, significantly reduces energy consumption, and effectively improves the control performance indices, such as overshoot and static error in the temperature holding period of the thermal treatment process, as compared to the existing heating methods. [ABSTRACT FROM AUTHOR]

The behavior of Cattaneo-Hristov heat diffusion moving in a line segment under the influence of specified initial and source temperatures has been investigated. The Fourier method has been applied to determine the eigenfunctions thus allowing reducing the problem to a set of time-fractional ordinary differential equations. Analytical solutions by applying the Laplace transform method have been developed.

The interaction of a linear wave with a dual cylindrical caisson breakwater with dual porous ring plates fixed inside is investigated using the matched eigenfunction expansion method. Both the inner impermeable cylinders and the outer porous cylinders are rigidly fixed to the flat seabed. The dual porous ring plates are fixed between the inner and outer cylinders, below the still water level. All porous surfaces are assumed to be thin, which allows the use of Darcy's law. By matching the boundary conditions between different regions, a set of linear equations with unknown coefficients is obtained and solved to get the velocity potential. After verifying the computational results of this paper with published literature, the effects of the structural parameters and wave incident angle on the hydrodynamic performance of a tandem 5-caisson breakwater are investigated. The results show that it is meaningful to conduct wave diffraction studies on the caisson array as a whole, influenced by the inter-cylinder interactions. • The dual cylindrical caisson breakwater is optimized by dual plates. • The hydrodynamic performance of a breakwater with dual plates was investigated. • Dual plates can cope with different tide levels of sea conditions. [ABSTRACT FROM AUTHOR]

In this paper, oblique water wave interaction with a surface piercing box-type porous structure placed in front of a vertical rigid wall is investigated in the presence of current, which flows parallel to vertical wall. The current speed is considered distinct in different subdomains. However, in a particular subdomain current speed is uniform throughout the water depth. In this analysis, Sollitt and Cross model is used for the study of wave motion inside the porous structure. Employing small amplitude linear water wave theory, the present model of our interest is developed into a boundary value problem and solved analytically using the eigenfunction expansion method. Using the developed solution, the reflection coefficient and hydrodynamic wave forces on porous structure are computed for different values of parameters: the dimension of porous structure, the friction coefficient of porous structure, the current speed of following current and opposing current. This research paper suggests that suitable arrangement of rigid walls and surface-piercing partially extended porous structures can be an effective and economical approach for guiding ocean currents and protecting marine facilities against wave attacks. • Oblique wave trapping in the presence of current and porous structure is discussed. • Boundary value problem solved analytically using the eigenfunction expansion method. • Wave forces are higher for following current speed than the opposing current speed. • Node of standing waves are formed on the rear face of the porous breakwater. [ABSTRACT FROM AUTHOR]

In this paper, we consider an impulsive Sturm-Lioville problem on Sturmian time scales. We investigate the existence and uniqueness of the solution of this problem. We study some spectral properties and self-adjointness of the boundary-value problem. Later, we construct the Green function for this problem. Finally, an eigenfunction expansion is obtained. [ABSTRACT FROM AUTHOR]

A linearized model is presented for the scattering problem of flexural gravity waves by the bottom with a finite number of rectangular steps in the ocean. The matched eigenfunction expansion method is used to solve the boundary-value problem, and the results are expressed in terms of a set of integral equations solved by the multiterm Galerkin's approximation technique. Numerical values of the reflection and transmission coefficients are calculated and graphed to display the influence of various parameters of the problem. The accuracy of the present method is verified by using the energy balance relation for the reflection and transmission coefficients. The results are compared with those available in some earlier works. [ABSTRACT FROM AUTHOR]

A general unified solution of the plane Couette–Poiseuille–Stokes–Womersley incompressible linear fluid flow in a slit in the presence of oscillatory pressure gradients with periodic synchronous vibrating boundaries is presented. Oscillatory flow remains stable and laminar with no-slip boundary conditions applied. Eigenfunction expansion method is used to obtain the exact analytical solution of the general linear inhomogeneous boundary value problem. Fourier expansion of arbitrary harmonic pressure gradient and non-harmonic wall oscillations was used to calculate arbitrary driving of the fluid. In-house developed optimized computational fluid dynamics marching-in-time finite-volume method was used to test and verify all analytical results. A number of particular transients, steady-state and combined flows were obtained from the general analytical result. Generalized Stokes and Womersley flows were solved using the analytical computations and numerical experiments. The combined effects of periodic non-harmonic wall movements with oscillatory pressure gradients offers rich and interesting flow patterns even for a linear Newtonian fluid and may be particularly interesting for pumping-assist microfluidic devices. The main motivation for developing a unified solution of the unsteady laminar planar Couette–Stokes–Poiseuille–Womersley flow originates in a need for, but is not limited to, in-depth exploration of flow patterns in hemodynamic and microfluidic pumping applications. [ABSTRACT FROM AUTHOR]

We derive tail asymptotics for the running maximum of the Cox-Ingersoll-Ross process. The main result is proved by the saddle point method, where the tail estimate uses a new monotonicity property of the Kummer function. This auxiliary result is established by a computer algebra assisted proof. Moreover, we analyse the coefficients of the eigenfunction expansion of the running maximum distribution asymptotically. [ABSTRACT FROM AUTHOR]

In this work, we consider one dimensional Dirac operators on unbounded time scales. We construct a spectral function of such an operator. Using this function, we establish a Parseval equality and an expansion formula in eigenfunctions for the Dirac operator on unbounded time scales. [ABSTRACT FROM AUTHOR]

The present work deals with the structure of the thermo-mechanical stress field near the crack-tip in a graded material, where properties vary along the crack front in an exponential manner. With assumptions of steady-state conditions and insulated crack faces, expansion of the temperature field around the crack-tip is obtained. Results indicate that the heat flux in the radial and angular direction has inverse square root singularity near the crack-tip. The heat flux in the direction of gradation does not have any singularity. Navier's equation for a thermo-mechanical system is solved using the Eigenfunction expansion approach to obtain the near crack-tip stress field. The explicit form of the first four terms in the expansion of the displacement field is obtained. It is observed that terms corresponding to r n / 2 − 1 (n > 1) in the stress field are directly affected by the temperature or heat flux. Effect of temperature is observed in out-of-plane normal stress only for n ≥ 2. Considering the first two terms in the asymptotic expansion, a parametric analysis, shows that thermal stresses help relax the constraint ahead of the crack-tip when the structure is constrained in the thickness direction. It helps in the suppression of crack growth mechanisms, which delays the failure of structures. [ABSTRACT FROM AUTHOR]

This study examines whether an alternative calculation approach exists for the asymptotic expansion of the probability density function of risky asset prices. We focus on the partial differential equation by using an eigenfunction expansion related to Hermite polynomials. We show that the expansion coefficients can be obtained by solving ordinary differential equations recursively and which order of Hermite polynomials appear in asymptotic expansion explicitly. [ABSTRACT FROM AUTHOR]

In this paper we employ a time-changed Ornstein-Uhlenbeck (OU) process for modeling temperature and pricing weather derivatives, where the time change process is a Lévy subordinator time changed by a deterministic clock with seasonal activity rate. The drift, diffusion volatility and jumps under the new model are all seasonal, which are supported by the observed temperature time series. An important advantage of our model is that we are able to derive the analytical pricing formulas for temperature futures and future options based on eigenfunction expansion technique. Our empirical study indicates the new model has the potential to capture the main features of temperature data better than the competing models.

In this paper, we study some spectral properties of the one-dimensional Hahn-Dirac boundary-value problem, such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green's function, the existence of a countable sequence of eigenvalues, eigenfunctions forming an orthonormal basis of L² ?,q((?0, a); E). [ABSTRACT FROM AUTHOR]

• Groundwater flow in a vertically N -layered sloping unconfined aquifer was analytically built. • A comparison with previous studies was performed and verified the present solution. • Three kinds of error analysis was performed to validate the analytical solution. This study is aimed at proposing an analytical approach of modeling groundwater flows in an unconfined aquifer with multiple vertical soil layers. First, a mathematical problem was built describing groundwater flow in an unconfined aquifer of N -layered vertical soils with an inclined base. Then, an analytical solution to the problem was proposed after a solution of separable form was sought and the eigenfunction expansion technique was used. This solution can address any vertically multilayered soil stratification with temporally varied rainfall recharge. Finally, the presented solution was verified by employing previous analytical solutions and numerical results modeled using Modflow software. Through this study, the presented analytical mathematical model, instead of a numerical mathematical model, of groundwater in a multilayered aquifer is an effective tool for estimating the groundwater level. [ABSTRACT FROM AUTHOR]

In this work, we study certain spectral properties of the one dimensional Dirac systems on time scales, such as formally self-adjointness, orthogonality of eigenfunctions, Green's function, the existence of a countable sequence of eigenvalues. Later, we give an expansion formula in eigenfunctions for Dirac operators on time scales. These results could provide an important contribution to the spectral theory of such operators on time scales. [ABSTRACT FROM AUTHOR]

This is a survey of accumulated spectral analysis observations spanning more than a century, referring to the double layer potential integral equation, also known as Neumann-Poincaré operator. The very notion of spectral analysis has evolved along this path. Indeed, the quest for solving this specific singular integral equation, originally aimed at elucidating classical potential theory problems, has inspired and shaped the development of theoretical spectral analysis of linear transforms in XX-th century. We briefly touch some marking discoveries into the subject, with ample bibliographical references to both old, sometimes forgotten, texts and new contributions. It is remarkable that applications of the spectral analysis of the Neumann-Poincaré operator are still uncovered nowadays, with spectacular impacts on applied science. A few modern ramifications along these lines are depicted in our survey. [ABSTRACT FROM AUTHOR]

In this paper we develop a new class of models for pricing autocallables based on multivariate subordinate Orstein Uhlenbeck (OU) processes. Starting from d independent OU processes and an independent d-dimensional Lévy subordinator, we construct a new process by time changing each of the OU processes with a coordinate of the Lévy subordinator. The prices of underlying assets are then modeled as an exponential function of the subordinate processes. The new models introduce state-dependent jumps in the asset prices and the dependence among jumps is governed by the Lévy measure of the d-dimensional subordinator. By employing the eigenfunction expansion technique, we are able to derive the analytical formulas for the worst-of autocallable prices. We also numerically implement a specific model and test its sensitivity to some of the key parameters of the model.

A class of quadratic irregular pencils of the second order ordinary differential operators with constant coefficients is considered in this paper. It is assumed that the boundary conditions are splitting and the characteristic equation has positive roots. Both amounts of two-fold expansions in the series in terms of eigenfunctions of such pencils and necessary and sufficient conditions for convergence of these expansions to the decomposed vector-valued function are found. For this class of irregular pencils, the question of a two-fold expansion in terms of eigenfunctions was not considered earlier. [ABSTRACT FROM AUTHOR]

In this article, a coupled model is proposed for wave interaction with a pair of submerged floating tunnels in the presence of an array of bottom-standing trapezoidal porous breakwaters. The theory of Sollitt and Cross is adopted to govern the fluid flow inside the porous medium. For constant water-depth, the eigenfunction expansion method is employed, whereas for varying water-depth, the eigenfunction expansion method along with the mild-slope approximation is employed. The solutions, thus derived, are matched at the shared boundaries under defined physical conditions. First, the performance of a single breakwater of impermeable and permeable type in reducing wave forces on tunnels is analyzed. Next, the performance of two and three submerged breakwaters is studied. The reflection and transmission coefficients of waves are high in the absence of the submerged breakwater and in the presence of an impermeable breakwater. These coefficients significantly reduce in the presence of the submerged porous breakwater. As a result, the horizontal and vertical forces acting on bridges and tunnels are substantially subsided. Wave forces on tunnels reduce with an increase in the angle of incidence. Multiple porous breakwaters show better performance in mitigating wave forces on tunnels. Higher wave force on tunnels is noticed in intermediate water-depth. The findings can enhance the knowledge of submerged porous breakwaters' performance in reducing wave loads on bridges and tunnels. [ABSTRACT FROM AUTHOR]

The behavior of Cattaneo-Hristov heat diffusion moving in a line segment under the influence of specified initial and source temperatures has been investigated. The Fourier method has been applied to determine the eigenfunctions thus allowing reducing the problem to a set of time-fractional ordinary differential equations. Analytical solutions by applying the Laplace transform method have been developed. [ABSTRACT FROM AUTHOR]

A general unified solution of the plane Couette–Poiseuille–Stokes–Womersley incompressible linear fluid flow in a slit in the presence of oscillatory pressure gradients with periodic synchronous vibrating boundaries is presented. Oscillatory flow remains stable and laminar with no-slip boundary conditions applied. Eigenfunction expansion method is used to obtain the exact analytical solution of the general linear inhomogeneous boundary value problem. Fourier expansion of arbitrary harmonic pressure gradient and non-harmonic wall oscillations was used to calculate arbitrary driving of the fluid. In-house developed optimized computational fluid dynamics marching-in-time finite-volume method was used to test and verify all analytical results. A number of particular transients, steady-state and combined flows were obtained from the general analytical result. Generalized Stokes and Womersley flows were solved using the analytical computations and numerical experiments. The combined effects of periodic non-harmonic wall movements with oscillatory pressure gradients offers rich and interesting flow patterns even for a linear Newtonian fluid and may be particularly interesting for pumping-assist microfluidic devices. The main motivation for developing a unified solution of the unsteady laminar planar Couette–Stokes–Poiseuille–Womersley flow originates in a need for, but is not limited to, in-depth exploration of flow patterns in hemodynamic and microfluidic pumping applications.

Analytical solutions of transient concentration of degraded components inside cylindrical and slab-type PLGA particles immersed in infinite medium were derived by solving reaction-diffusion equations of autocatalytic reaction using eigenfunction expansion method. The resulting average concentrations were compared with the modeling results of spherical PLGA particles by Versypt and her colleagues to study the effect of particle morphology on the autocatalytic reaction. Mass transfer resistance inside and outside of the particles was also considered using Biot number, and its effects on the concentration inside particles with various morphologies were also studied by solving reaction-diffusion equation. To predict transient concentration in surrounding medium, coupled differential equations were solved for the three shapes of PLGA particles by assuming finite volume of the decomposition system. Mathematical solutions were obtained by Laplace transform, and the results were compared for the PLGA particles with different shapes depending on Thiele modulus and particle volume fraction. [ABSTRACT FROM AUTHOR]

Let A be a symmetric operator. By using the method of boundary triplets we parameterize in terms of a Nevanlinna parameter τ all exit space extensions A ˜ = A ˜ ⁎ of A with the discrete spectrum σ (A ˜) and characterize the Shtraus family of A ˜ in terms of abstract boundary conditions. Next we apply these results to the eigenvalue problem for the 2 r -th order differential equation l [ y ] = λ Δ (x) y on an interval [ a , b) , − ∞ < a < b ≤ ∞ , subject to λ -depending separated boundary conditions with entire operator-functions C 0 (λ) and C 1 (λ) , which form a Nevanlinna pair (C 0 , C 1). The weight Δ (x) is nonnegative and may vanish on some intervals (α , β) ⊂ I. We show that in the case when the minimal operator of the equation has the discrete spectrum (in particular, in the case of the quasiregular equation) the set of eigenvalues of the eigenvalue problem is an infinite subset of R without finite limit points and each function y ∈ L Δ 2 (I) admits the eigenfunction expansion y (x) = ∑ k = 1 ∞ y k (x) converging in L Δ 2 (I). Moreover, we give an explicit method for calculation of eigenfunctions y k in this expansion and specify boundary conditions on y implying the uniform convergence of the eigenfunction expansion of y. These results develop the known ones obtained for the case of the positive weight Δ and the more restrictive class of λ -depending boundary conditions. [ABSTRACT FROM AUTHOR]

In this work, we show the existence of a spectral function for a singular Sturm-Liouville problem with transmission condition. Also we establish a Parseval equality and expansion formula in eigenfunctions in terms of the spectral function.

Analytical solutions of transient concentration profile inside particles were calculated by solving rection-diffusion equation in spherical, cylindrical, and slab-type porous particles, assuming first order irreversible reaction. Time-dependent concentration in bulk fluid phase was assumed as exponential decay for each particle, and convective boundary conditions were taken using arbitrary Biot number to obtain a general solution by eigenfunction expansion or Laplace transform method. Factors affecting average transient concentration inside particles were studied by adjusting Biot number, reaction time, and Thiele modulus as well as the position inside particles. To predict transient bulk concentration in batch photocatalytic reactor containing porous particles, coupled differential equations were solved by Laplace transform to obtain analytical solutions of bulk concentration as well as average concentration inside porous particles as a function of reaction time. The factors affecting transient concentrations were investigated by adjusting the concentration and porosity of the catalytic particles, morphology of the particles, and Thiele modulus in batch-mode reactor to study reduction speed of reactant concentration during photocatalytic reaction. The solutions from coupled differential equations were useful for the prediction of transient behavior in batch-type photocatalytic reactor and were compared with the results from CSTR containing slab-type photocatalytic particles with various space time. [ABSTRACT FROM AUTHOR]

The present article investigates a new mathematical model of generalized thermo-viscoelasticity based on memory-dependent derivative in the context of three-phase-lag heat transfer subjected to an instantaneous point heat source. The governing coupled equations associated with kernel function and time delay are constituted for a two-dimensional problem of semi-infinite space in an isotropic medium. The bounding surface of the half-space is assumed to be subjected to a time-dependent thermal shock and rigidly fixed. The Eigenfunction expansion approach is proposed to analyze the problem and, consequently, the analytical solutions of the thermodynamic field variables viz. temperature, displacement and stresses are obtained in the Laplace-Fourier transformed domain. The effects of the kernel function and time-delay factor on the distributions of the non-dimensional thermodynamic field variables are presented graphically and compared with the several well-known generalized thermoelasticity theories present in the literature. Finally, some remarkable points are mentioned in conclusion. [ABSTRACT FROM AUTHOR]