Slow axisymmetric rotation of a sphere in a circular tube with slip surfaces

The steady rotation of a slip spherical particle about a diameter lying along the longitudinal axis of a slip circular tube filled with an incompressible Newtonian fluid at low Reynolds numbers is analyzed. To solve the Stokes equations for the fluid flow, the solution is constituted by the summation of general solutions in both cylindrical and spherical coordinates. The boundary conditions are implemented first along the tube wall via the Fourier cosine transform and then over the particle surface through a collocation method. Results of the resisting torque acting on the particle are obtained for various values of the relevant dimensionless parameters. The effect of the confining tube on the axisymmetric rotation of the particle with slip surfaces is interesting. The torque increases monotonically with an increase in the stickiness of the tube wall, keeping the other parameters unchanged. When the stickiness of the tube wall is greater than a critical value, the torque is greater than that on the particle in an unbounded identical fluid and increases with increases in the stickiness of the particle surface and particle-to-tube radius ratio. When the stickiness of the tube wall is less than the critical value, conversely, the torque is smaller than that on the unconfined particle and decreases with increases in the particle stickiness and radius ratio.

RECURSIVE FORMULAE FOR DROPLETS TRANSIENT HEATING AND EVAPORATION MODELS VIA A COMBINED METHOD OF INTEGRAL TRANSFORMS

The transient heating of a spherical droplet at rest in a hot gas environment, is analysed when the temperature distribution is initially assumed to be non uniform inside the droplet. A combined method of integral transforms, namely the classical Fourier cosine transform together with the unilateral Laplace transform, is used in solving the resulting initial-boundary value problem, stated in the dimensionless form. Explicit solutions of the problem are first obtained in the Laplace domain, and then analytical approximations in short time limits (timessteps) are derived for the droplet internal and surface temperature fields. The analytical approximation for the droplet internal temperature during the time step is proven to be highly accurate, while the innovative recursive formula obtained for the droplet surface temperature may lead to computationally efficient droplets and sprays vaporization models.

An analytical approximation formula for the pricing of credit default swaps with regime switching

We derive an analytical approximation for the price of a credit default swap (CDS) contract under a regime-switching Black–Scholes model. To achieve this, we first derive a general formula for the CDS price, and establish the relationship between the unknown no-default probability and the price of a down-and-out binary option written on the same reference asset. Then we present a two-step procedure: the first step assumes that all the future information of the Markov chain is known at the current time and presents an approximation for the conditional price under a time-dependent Black–Scholes model, based on which the second step derives the target option pricing formula written in a Fourier cosine series. The efficiency and accuracy of the newly derived formula are demonstrated through numerical experiments. doi:10.1017/S1446181121000274

Computation of Fourier transform representations involving the generalized Bessel matrix polynomials

Motivated by the recent studies and developments of the integral transforms with various special matrix functions, including the matrix orthogonal polynomials as kernels, in this article we derive the formulas for Fourier cosine and sine transforms of matrix functions involving generalized Bessel matrix polynomials. With the help of these transforms several results are obtained, which are extensions of the corresponding results in the standard cases. The results given here are of general character and can yield a number of (known and new) results in modern integral transforms.

AN ANALYTICAL APPROXIMATION FORMULA FOR THE PRICING OF CREDIT DEFAULT SWAPS WITH REGIME SWITCHING

We derive an analytical approximation for the price of a credit default swap (CDS) contract under a regime-switching Black–Scholes model. To achieve this, we first derive a general formula for the CDS price, and establish the relationship between the unknown no-default probability and the price of a down-and-out binary option written on the same reference asset. Then we present a two-step procedure: the first step assumes that all the future information of the Markov chain is known at the current time and presents an approximation for the conditional price under a time-dependent Black–Scholes model, based on which the second step derives the target option pricing formula written in a Fourier cosine series. The efficiency and accuracy of the newly derived formula are demonstrated through numerical experiments.

Estimating the Gerber-Shiu Function in Lévy Insurance Risk Model by Fourier-Cosine Series Expansion

In this paper, we propose an estimator for the Gerber–Shiu function in a pure-jump Lévy risk model when the surplus process is observed at a high frequency. The estimator is constructed based on the Fourier–Cosine series expansion and its consistency property is thoroughly studied. Simulation examples reveal that our estimator performs better than the Fourier transform method estimator when the sample size is finite.

Tích chập suy rộng với hàm trọng g(y) = sin ay đối với các phép biến đổi tích phân Fourier cosine và Fourier sine

Tích chập suy rộng mới với hàm trọng đối với hai phép biến đổi tích phân Fourier cosine và Fourier sine được chúng tôi xây dựng và nghiên cứu trong bài báo này. Chúng tôi chứng minh sự tồn tại của tích chập suy rộng mới này trong không gian L(R¬+). Đẳng thức nhân tử hóa cốt yếu với sự có mặt của hai phép biến đổi tích phân khác biệt là Fourier cosine, Fourier sine và hàm trọng cùng một số tính chất khác như tính không giao hoán, tính không kết hợp khác với các tích chập của một phép biến đổi tích phân được phát biểu và chứng minh. Cuối cùng là áp dụng tích chập suy rộng mới được xây dựng để giải hệ phương trình tích phân kiểu Toeplitz-Hankel và nhận được nghiệm dưới dạng đóng. Trong ba thập niên trở lại đây, tích chập suy rộng được các nhà toán học quốc tế và trong nước quan tâm nghiên cứu. Đồng thời các nhà toán học cũng ứng dụng chúng trong việc giải các bài toán về phương trình tích phân, phương trình vi tích phân,… Vì vậy, việc nghiên cứu tích chập suy rộng là vấn đề thời sự. Do đó, nhóm tác giả chúng tôi đã viết bài báo này.

The vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections

In this study, an investigation on the free vibration of the beam with material properties and cross section varying arbitrarily along the axis direction is studied based on the so-called Spectro-Geometric Method. The cross-section area and second moment of area of the beam are both expanded into Fourier cosine series, which are mathematically capable of representing any variable cross section. The Young’s modulus, the mass density and the shear modulus varying along the lengthwise direction of the beam, are also expanded into Fourier cosine series. The translational displacement and rotation of cross section are expressed into the Fourier series by adding some polynomial functions which are used to handle the elastic boundary conditions with more accuracy and high convergence rate. According to Hamilton’s principle, the eigenvalues and the coefficients of the Fourier series can be obtained. Some examples are presented to validate the accuracy of this method and study the influence of the parameters on the vibration of the beam. The results show that the first four natural frequencies gradually decrease as the coefficient of the radius [Formula: see text] increases, and decreases as the gradient parameter n increases under clamped–clamped end supports. The stiffness of the functionally Timoshenko beam with arbitrary cross sections is variable compared with the uniform beam, which makes the vibration amplitude of the beam have different changes.