Series Expansion Technique Research Articles

Approximate analytic solution for Asian options with stochastic volatility

The valuation of Asian options is complicated because the arithmetic average of lognormal random variables is no longer lognormal. Furthermore, the stochastic volatility inherent in financial asset prices is easily observed. However, few academic studies consider the pricing and hedging of Asian options with stochastic volatility, despite the popularity of such options. This study extends the work of Hull and White (1987) and integrates the Taylor series expansion technique to derive an approximate analytic solution for Asian options with stochastic volatility. Numerical experiments show that the proposed approximate analytic solution performs favorably and is computationally efficient compared with large-sample simulations. The approximate analytic solution provides a practical approach for pricing and hedging Asian options with stochastic volatility and is both easy to implement and desirable in terms of computing speed.

Transient responses in a two‐temperature thermoelastic infinite medium having cylindrical cavity due to moving heat source with memory‐dependent derivative

AbstractThis work deals with the study of transient responses in a two‐temperature generalized thermoelastic infinite medium having a cylindrical cavity due to a time‐dependent moving heat source where the conventional Fourier's law of heat conduction is modified by introducing a new Taylor's series expansion using memory‐dependent derivative (MDD). The resulting non‐dimensional equations are applied to a specific problem. A direct approach is introduced to obtain the analytical expressions of the physical quantities in the Laplace transform domain. The inversion of the Laplace transforms are carried out using the methods based on the Fourier series expansion technique. Numerical results for the dynamical and conductive temperatures, stress, strain, and displacement are presented graphically. The effects of time‐delay and kernel function have been studied on the thermo‐physical quantities.

Analysis of the influence of boundary pressure and friction on determining fracture toughness of shale using cracked Brazilian disc test

As the exact distributions of the pressure and friction on the disc-jaw interface are not known in the cracked Brazilian disc test, we analyze their influence on determining mode I fracture toughness. The analytic solution of stress intensity factor is derived for the arbitrarily distributed pressure and friction by using the power series expansion technique and weight function method. This investigation shows that such influence should be considered for the case of large contact angle and long crack. The mode I fracture toughness of shale is determined indirectly from the test.

A semi-analytical method for the dynamic analysis of cylindrical shells with arbitrary boundaries

The dynamic behavior of cylindrical shells with arbitrary boundaries is studied in this paper. Love's shell theory and Hamilton's principle are employed to derive the motion equations for cylindrical shells. A semi-analytical methodology, which incorporates Durbin's inverse Laplace transform, differential quadrature method and Fourier series expansion technique, is proposed to investigate this phenomenon. The use of the differential quadrature method provides a solution in terms of the axial direction whereas the use of Durbin's numerical inversion method generates a solution in the time domain. Comparison of calculated frequency parameters to that derived from the literature illustrates the effectiveness of the method. Specifically, convergence tests indicate that the present approach has a rapid convergence, the time-history response and the Navier's solution are in great agreement. Comparisons between time-history responses derived by two shell theories show that the results fit well with each other when the thickness-radius ratios are small enough. An analysis of the influences of boundaries on the time-history response of cylindrical shells indicates that the peak displacement is closely related to the degrees of freedom of boundaries. The influences of the length-radius ratios and the thickness-radius ratios on the peak displacement are further investigated.

Analytical Solutions for Water Infiltration into Unsaturated–Semi-Saturated Soils Under Different Water Content Distributions on the Top Boundary

Modeling and simulation of water infiltration from the ground surface into unsaturated–saturated soil is very important in the study of the hydrological cycle process. Transient fluid flow through unsaturated–saturated soil is generally described by Richards’ equation. The equation is a nonlinear parabolic partial differential equation for which many numerical and limited analytical solutions exist. In this paper, exponential models for water content and hydraulic conductivity are selected so the nonlinear equation becomes linear and then separation of variables and Fourier series expansion techniques are used to derive analytical solutions to 2D water infiltration into unsaturated–semi-saturated soils with different distributions of water content on a part and/or parts of the top boundary. 2D water infiltrations are investigated in a rectangular soil domain, where high water content zone is considered as semi-saturated zone and residual water content zone is considered as unsaturated zone at initial time. A total of three cases are investigated in this paper. In case 1, a uniform distribution of water content on a part at the top boundary and no-flow boundaries at side edges are considered. Also, horizontal water table in unsaturated–semi-saturated domain is assumed as initial condition. In case 2, an inclined water table initial condition and a uniform distribution of water content on a part at the top boundary are investigated. Finally, in case 3, an analytical solution for ramp distributions of water content on two parts at the top boundary and an inclined water table initial condition is obtained. To illustrate the use of the derived equations, water content values are obtained from a numerical solution and compared to those from analytical solutions for two cases, showing less than 2% errors. These analytical solutions may be used as a benchmark for verification and efficiency assessment of numerical approaches.

Magnetic Properties of Diluted Spinels Mg<i><sub>x</sub></i>Ni<sub>1−<i>x</i></sub>Fe<sub>2</sub>O<sub>4</sub> Systems: Are Studied by Green’s Functions and Mean Field Theories

Many-body Green's function, and mean field theories have been developed for magnetic systems MgxNi1−xFe2O4. We apply this theory to evaluate thermal magnetization and magnetic susceptibility for different values of magnetic field and dilution x, by considering all components of the magnetisation when an external magnetic field is applied in (x-z)-plane. The critical temperatures of MgxNi1−xFe2O4 systems in the range 0 ≤ x ≤ 1 have been deduced. The Green's function results are compared with the results of high temperature series expansion technique, and experimental magnetic measurements.

Interface tension-induced stress field around periodic nano-inclusions of arbitrary shape

This paper investigates the plane deformation of periodic nano-inclusions of arbitrary shape embedded in a homogeneous isotropic material. A representative unit cell (RUC) with periodic boundary conditions imposed on its edges is used to represent the periodicity of the structure. Residual interface tension is incorporated into the deformation model so that the normal and tangential stresses have to jump across the matrix–inclusion interface, despite that the displacement can generally be treated as continuous across that interface. The stress field in the entire RUC is obtained by using the complex variable methods with the assistance of conformal mapping, series expansion, and collocation techniques. Numerical examples are presented for three different inclusion shapes. The results show that the interface tension-induced stress field can be greatly influenced by the shape, elastic modulus, and volume fraction of the inclusions.

Taylor-series approximations for queues with arrival correlation

• We propose a queueing model with session-based arrivals, where sessions are not always active. • We analyse the system in the light traffic regime, by Taylor series expansion techniques. • We find explicit expressions for the 3rd order expansions of the first two moments of the queue content. • Combining light and heavy traffic approximations covers a large part of parameter space. • We formally prove the existence of the Taylor series expansions . This paper studies a discrete-time single-server queueing system with correlated arrivals. Arrivals at the queue stem from a number of active sessions, each generating a packet in a slot with a fixed probability q . Since an exact queueing analysis is not feasible for q ≠ 1, we rely on Taylor-series expansions of the joint probability generating functions of the number of active sessions and the queue content around q = 0 . These expansions are then either combined with the known generating function for q = 1 if the system is stable for q = 1 , or with heavy-traffic results if this is not the case. In either case, we obtain expressions for the moments of queue content and packet delay and assess the accuracy of our approximations by a simulation study.

A novel method for numerical simulation of sand motion model in beach formation based on fractional Taylor–Jumarie series expansion and piecewise interpolation technique

In this paper, for the first time, fractional Taylor–Jumarie series expansion is used to solve a glass of time-fractional delay partial differential equation by piecewise interpolation reproducing kernel method (RKM), this class of equations describe sand motion model in beach formation. The aim of this work is to obtain more accurate numerical solution by fractional Taylor–Jumarie series expansion and piecewise interpolation technique. Three numerical experiments are provided to show the advantage of this method, the results show the characteristics of sand motion. The research in this paper is the theoretical basis for further sand motion study.

Influence of pressure distribution and friction on determining mechanical properties in the Brazilian test: Theory and experiment

The improved Brazilian test is a popular method for indirectly measuring rock mechanical parameters, in which the current assumption of a uniform contact pressure distribution on the disc-jaw interface would result in measurement deviations because the real distribution is non-uniform. This investigation examines the influence of pressure distribution and friction on the determination of mechanical properties such as tensile strength σt and Young's modulus E by both theoretical analyses and experiments. For a Brazilian disc under arbitrary distributions of normal and tangential loads, the power series expansion technique is used to find the analytical solutions of the full-field displacements and stresses. The related formulae for calculating the tensile strength σt and Young's modulus E are derived. Eight load distributions with identical resultant forces are considered. The results show that the series solution converges in the form of a negative power law. Based on the Griffith failure criterion, the minimum contact angles that make the crack initiate at the center of the disc are given for different load distributions. The analysis indicates that these distributions have a significant influence on Young's modulus E and, if the contact angle is smaller, on tensile strength σt. The deviation caused by ignoring friction is also discussed. The Brazilian test is implemented for shale disc samples. The tensile strength σt and Young's modulus E of shale are determined indirectly using the theoretical formulae discussed here and experimental data.

Transient response in a thermoelastic half-space solid due to a laser pulse under three theories with memory-dependent derivative

Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of a non-Gaussian pulsed laser type heat source in a stress free isothermal half-space in the context of Lord–Shulman (LS), dual-phase lag (DPL), and three-phase lag (TPL) theories of thermoelasticity simultaneously. The memory-dependent derivative is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. Employing Laplace transform as a tool, the problem has been transformed to the space-domain, and it is then solved analytically. To get back all the thermophysical quantities as a function of real time, we use two Laplace inversion formulas, viz. Fourier series expansion technique (Honig in J Comput Appl Math10(1):113–132, 1984) and Zakian method (Electron Lett 6(21):677–679, 1970). According to the graphical representations corresponding to the numerical results, a comparison among LS, DPL, and TPL model has been studied in the presence and absence of a memory effect simultaneously. Moreover, the effects of a laser pulse have been studied in all the thermophysical quantities for different kernels (randomly chosen) and different delay times. Then, the results are depicted graphically. Finally, a comparison of results, deriving from the two numerical inversion formulas, has been made.

From implicit to recursive equations

The technique of relaxed power series expansion provides an efficient way to solve so called recursive equations of the form $$F = \varPhi (F)$$ , where the unknown F is a vector of power series, and where the solution can be obtained as the limit of the sequence $$0, \varPhi (0), \varPhi (\varPhi (0)), \ldots $$ . With respect to other techniques, such as Newton’s method, two major advantages are its generality and the fact that it takes advantage of possible sparseness of $$\varPhi $$ . In this paper, we consider more general implicit equations of the form $$\varPhi (F) = 0$$ . Under mild assumptions on such an equation, we will show that it can be rewritten as a recursive equation.

Wetting Front Analysis of the Richards’ Equation with Impervious Boundary

Abstract It is analyzed the wetting front of the Richards’ equation (RE) for horizontal infiltration problem with impervious layer in a finite medium, and obtained an approximate analytical solution by the series expansion technique. The present approximation is suitable for arbitrary diffusivity in RE and applied to simulate the changes of saturation after the front pass through the impervious point. Two examples about power law diffusivity are analyzed to confirm the accuracy of present solution.

Modeling of memory-dependent derivative in a fiber-reinforced plate under gravitational effect

ABSTRACTEnlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena for a fiber-reinforced thick plate due to the gravitational effects having a heat source, in the context of three-phase-lag model of generalized thermoelasticity, which is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. The upper surface of the plate is free of traction having a prescribed surface temperature while the lower surface rests in a rigid foundation and is thermally insulated. Employing Laplace and Fourier transforms as tools, the problem has been solved analytically in the transformed domain. The inversion of the Fourier transform is carried out using suitable numerical techniques while the numerical inversion of Laplace transform is done incorporating a method on Fourier series expansion technique. According to the graphical representations corresponding to the numerical results, conclusions about the new theory is constructed. Excellent predictive capability is demonstrated due to the presence of memory dependent derivative, reinforcement and gravitational effect.

One-dimensional dynamic equations of a piezoelectric semiconductor beam with a rectangular cross section and their application in static and dynamic characteristic analysis

Within the framework of continuum mechanics, the double power series expansion technique is proposed, and a series of reduced one-dimensional (1D) equations for a piezoelectric semiconductor beam are obtained. These derived equations are universal, in which extension, flexure, and shear deformations are all included, and can be degenerated to a number of special cases, e.g., extensional motion, coupled extensional and flexural motion with shear deformations, and elementary flexural motion without shear deformations. As a typical application, the extensional motion of a ZnO beam is analyzed sequentially. It is revealed that semi-conduction has a great effect on the performance of the piezoelectric semiconductor beam, including static deformations and dynamic behaviors. A larger initial carrier density will evidently lead to a lower resonant frequency and a smaller displacement response, which is a little similar to the dissipative effect. Both the derived approximate equations and the corresponding qualitative analysis are general and widely applicable, which can clearly interpret the inner physical mechanism of the semiconductor in the piezoelectrics and provide theoretical guidance for further experimental design.

An uncertainty model of acoustic metamaterials with random parameters

Acoustic metamaterials (AMs) are man-made composite materials. However, the random uncertainties are unavoidable in the application of AMs due to manufacturing and material errors which lead to the variance of the physical responses of AMs. In this paper, an uncertainty model based on the change of variable perturbation stochastic finite element method (CVPS-FEM) is formulated to predict the probability density functions of physical responses of AMs with random parameters. Three types of physical responses including the band structure, mode shapes and frequency response function of AMs are studied in the uncertainty model, which is of great interest in the design of AMs. In this computation, the physical responses of stochastic AMs are expressed as linear functions of the pre-defined random parameters by using the first-order Taylor series expansion and perturbation technique. Then, based on the linear function relationships of parameters and responses, the probability density functions of the responses can be calculated by the change-of-variable technique. Three numerical examples are employed to demonstrate the effectiveness of the CVPS-FEM for stochastic AMs, and the results are validated by Monte Carlo method successfully.

An implicit algorithm for the dynamic study of nonlinear vibration of spur gear system with backlash

In this work, we propose some regularization techniques to adapt the implicit high order algorithm based on the coupling of the asymptotic numerical methods (ANM) (Cochelin et al., Méthode Asymptotique Numérique, Hermès-Lavoisier, Paris, 2007; Mottaqui et al., Comput. Methods Appl. Mech. Eng. 199 (2010) 1701–1709; Mottaqui et al., Math. Model. Nat. Phenom. 5 (2010) 16–22) and the implicit Newmark scheme for solving the non-linear problem of dynamic model of a two-stage spur gear system with backlash. The regularization technique is used to overcome the numerical difficulties of singularities existing in the considered problem as in the contact problems (Abichou et al., Comput. Methods Appl. Mech. Eng. 191 (2002) 5795–5810; Aggoune et al., J. Comput. Appl. Math. 168 (2004) 1–9). This algorithm combines a time discretization technique, a homotopy method, Taylor series expansions technique and a continuation method. The performance and effectiveness of this algorithm will be illustrated on two examples of one-stage and two-stage gears with spur teeth. The obtained results are compared with those obtained by the Newton–Raphson method coupled with the implicit Newmark scheme.

The investigation of trapped thickness shear modes in a contoured AT-cut quartz plate using the power series expansion technique

The dynamic model about the anti-plane vibration of a contoured quartz plate with thickness changing continuously is established by ignoring the effect of small elastic constant c56. The governing equation is solved using the power series expansion technique, and the trapped thickness shear modes caused by bulge thickness are revealed. Theoretically, the proposed method is more general, which can be capable of handling various thickness profiles defined mathematically. After the convergence of the series is demonstrated and the correctness is numerically validated with the aid of finite element method results, systematic parametric studies are subsequently carried out to quantify the effects of the geometry parameter upon the trapped modes, including resonant frequency and mode shape. After that, the band structures of thickness shear waves propagation in a periodically contoured quartz plate, as well as the power transmission spectra, are obtained based on the power series expansion technique. It is revealed that broad stop bands below cut-off frequency exist owing to the trapped modes excited by the geometry inhomogeneity, which has little relationship with the structural periodicity, and its physical mechanism is different from the Bragg scattering effect. The outcome is widely applicable, and can be utilized to provide theoretical and practical guidance for the design and manufacturing of quartz resonators and wave filters.

An approximate analytical solution of Richards equation with finite boundary

We apply a series expansion technique to estimate the water content distribution and front position in finite boundary conditions. We derive an approximate analytical solution of the Richards equation (RE) for the horizontal infiltration problem. The solution is suitable for arbitrary hydraulic diffusivity in water infiltration. Compared with the finite element method, two examples in power law diffusivity and van Genuchten model are shown to test the accuracy of present approximation.

An Elementary Proof of <img src=image/tit1.png></img> and a Recurrence Formula for ζ(2<i>k</i>)

In this shortnote, a series expansion technique introduced recently by Dancs and He for generating Euler-type formulae for odd zeta values $\:\zeta{(2 k +1)}$, $\zeta{(s)}$ being the Riemann zeta function and $k$ a positive integer, is modified in a manner to furnish the even zeta values $ \zeta{(2k)}$. As a result, I find an elementary proof of $\sum_{n=1}^\infty{{1/{n^2}}} = {\pi^2/6}$, as well as a recurrence formula for $\zeta{(2k)}$ from which it follows that the ratio ${\zeta{(2k)} / \pi^{2k}}$ is a rational number, without making use of Euler's formula and Bernoulli numbers.