Form Of Infinite Series Research Articles

An explicit analytic formula for pricing barrier options with regime switching

This paper investigates the valuation of a European-style barrier option in a Markovian, regime-switching, Black–Scholes–Merton economy, where the price process of an underlying risky asset is assumed to follow a Markov-modulated geometric Brownian motion. An explicit analytic solution in infinite series form for the price of a European-style barrier option in a two-state regime is presented.

Trapped modes and scattering for oblique waves in a two-layer fluid

AbstractThe system describing time-harmonic motions of a two-layer fluid in the linearised shallow-water approximation is considered. It is assumed that the depth is constant, with a cylindrical protrusion (an underwater ridge) of small height. For obliquely incident waves, the system reduces to a pair of coupled ordinary differential equations. The values of frequency for which wave propagation in the unperturbed system is possible are bounded from below by a cutoff, to the left of which no propagating modes exist. Under the perturbation, a trapped mode appears to the left of the cutoff and, if a certain geometric requirement is imposed upon the shape of the perturbation (for example, if the ridge is a rectangular barrier of certain width), a trapped mode appears whose frequency is embedded in the continuous spectrum. When these geometric conditions are not satisfied, the embedded trapped mode transforms into a complex pole of the reflection and transmission coefficients of the corresponding scattering problem, and the phenomenon of almost total reflection is observed when the frequency coincides with the real part of the pole. Exact formulae for the trapped modes are obtained explicitly in the form of infinite series in powers of the small parameter characterising the perturbation. The results provide a theoretical understanding of analogous phenomena observed numerically in the literature for the full problem for the potentials in a two-layer fluid in the presence of submerged cylinders, and furnish explicit formulae for the frequencies at which total reflection occurs and the trapped modes exist.

The revised method of solving the contact problem for ring layer, taking into account the forces. Message 1. Mathematical model of contact interaction of ring layer with rigid base

The article proposes revised method of solving the contact problem for ring layer, taking into account frictional forces in the contact zone. The law of changes radial contact pressures is presented in the form of infinite series containing two groups of unknown constants. To determine unknown constants there was obtained the system of two functional equations.

A Study of Multiple Integrals with Maple

The multiple integral problem is closely related with probability theory and quantum field theory. This paper uses the mathematical software Maple for the auxiliary tool to study two types of multiple integrals. We can obtain the infinite series forms of these two types of multiple integrals by using integration term by term theorem. In addition, we provide some examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. For this reason, Maple provides insights and guidance regarding problem-solving methods.

Generalization of Two Types of Improper Integrals

This article uses the mathematical software Maple for the auxiliary tool to study two types of improper integrals. We can obtain the infinite series forms of these two types of integrals by using geometric series, differentiation term by term, and differentiation with respect to a parameter. On the other hand, we provide some examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. For this reason, Maple provides insights and guidance regarding problem-solving methods.

Solving Multiple Integrals Using Maple

The multiple integral problem is closely related to probability theory and quantum field theory. This paper uses the mathematical software Maple for the auxiliary tool to study three types of multiple integrals. We can obtain the infinite series forms of these three types of multiple integrals by using differentiation with respect to a parameter, differentiation term by term, and integration term by term. In addition, we provide some examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. For this reason, Maple provides insights and guidance regarding problem-solving methods.

Evaluating Multiple Integrals Using Maple

This paper uses the mathematical software Maple for the auxiliary tool to study two types of multiple integrals. We can obtain the infinite series forms of these two types of multiple integrals by using binomial series and integration term by term theorem. On the other hand, we propose some examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple.

Stress state around cylindrical cavities in transversally isotropic rock mass

The present paper is dealing with the investigation of the stress field around the infinitely long cylindrical cavity, of a circular cross section, contained in the transversally isotropic elastic continuum. Investigation is based upon the determination of the stress function that satisfies the biharmonic equation, for the given boundary conditions and for rotationaly symmetric loading. The solution of the partial differential equation of the problem is given in the form of infinite series of Bessel's functions. Determination of the stress-strain field around cavities is a common requirement for estimation of safety of underground rock excavations.

Scattering problems in the fractional quantum mechanics governed by the 2D space-fractional Schrödinger equation

The 2D space-fractional Schrödinger equation in the time-independent and time-dependent cases for the scattering problems in the fractional quantum mechanics is studied. We define the Green's functions for the two cases and give the mathematical expression of them in infinite series form and in terms of some special functions. The asymptotic formulas of the Green's functions are also given, and applied to get the approximate wave functions for the fractional quantum scattering problems. These results contain those in the standard (integer) quantum mechanics as special cases, and can be applied to study the complex quantum systems.

Evaluating the Derivatives of Some Functions Using Maple

This article uses the mathematical software Maple for the auxiliary tool to study the differential problem of two types of functions. We can obtain the infinite series forms of any order derivatives of these two types of functions by using binomial series, Leibniz differential rule and differentiation term by term theorem, and hence greatly reduce the difficulty of calculating their higher order derivative values. On the other hand, we propose two examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. For this reason, Maple provides insights and guidance regarding problem-solving methods.

Solving Some Types of Integrals Using Maple

This paper uses the mathematical software Maple for the auxiliary tool to study six types of integrals. We can obtain the infinite series forms of these six types of integrals by using integration term by term theorem. In addition, we propose some examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. Therefore, Maple provides insights and guidance regarding problem-solving methods. As information technology advances, whether computers can become comparable with human brains to perform abstract tasks, such as abstract art similar to the paintings of Picasso and musical compositions similar to those of Beethoven, is a natural question. Currently, this appears unattainable. In addition, whether computers can solve abstract and difficult mathematical problems and develop abstract mathematical theories such as those of mathematicians also appears unfeasible. Nevertheless, in seeking for alternatives, we can study what assistance mathematical software can provide. This study introduces how to conduct mathematical research using the mathematical software Maple. The main reasons of using Maple in this study are its simple instructions and ease of use, which enable beginners to learn the operating techniques in a short period. By employing the powerful computing capabilities of Maple, difficult problems can be easily solved. Even when Maple cannot determine the solution, problem-solving hints can be identified and inferred from the approximate values calculated and solutions to similar problems, as determined by Maple. For this reason, Maple can provide insights into scientific research. Inquiring through an online support system provided by Maple or browsing the Maple website (www.maplesoft.com) can facilitate further understanding of Maple and might provide unexpected insights. For the instructions and operations of Maple, (1-7) can be adopted as references. In calculus and engineering mathematics courses, we learnt many methods to solve the integral problems including change of variables method, integration by parts method, partial fractions method, trigonometric substitution method, and so on.In this paper, we study the following six types of integrals which are not easy to obtain their answers using the methods mentioned above.

Partial Derivatives of Three Variables Functions

This paper takes the mathematical software Maple as the auxiliary tool to study the partial differential problem of two types of three variables functions. We can obtain the infinite series forms of any order partial derivatives of these two types of functions by using differentiation term by term theorem, and hence greatly reduce the difficulty of calculating higher order partial derivative values of these functions. On the other hand, we propose two examples to do calculation practically.

Evaluation of Two Types of Integrals Using Maple

This paper uses the mathematical software Maple for the auxiliary tool to study two types of integrals. We can obtain the infinite series forms of these two types of integrals by using binomial series and integration term by term theorem. On the other hand, we provide some examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. For this reason, Maple provides insights and guidance regarding problem-solving methods.

English

It is well known that initial-value problem for the Cauchy-Riemann system is ill-posed and the problem with such Hadamard instability cannot be solved unless the initial data are analytic. In this paper, we present the vectorial reduced differential transform (VRDT) method to solve initial-value problem for the inhomogeneous Cauchy-Riemann system with analytic data. The VRDTM solution vector achieved is in the form of infinite series whose compact form is in agreement with the exact solution vector. Key words: Inhomogeneous Cauchy-Riemann system, initial-value problem, vectorial reduced differential transform.

Application of Differentiation Term by Term Theorem on the Partial Differential Problems

This paper takes the mathematical software Maple as the auxiliary tool to study the partial differential problems of two types of two-variables functions. We can obtain the infinite series forms of any order partial derivatives of these two types of functions by using differentiation term by term theorem, and hence greatly reduce the difficulty of calculating their higher order partial derivative values. On the other hand, we propose two examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying our answers by using Maple.

Energy Detection in Multihop Cooperative Diversity Networks: An Analytical Study

This work presents a study of detection performance of energy detector in relay-based multihop cooperative diversity networks operating over independent Rayleigh fading channels. In particular, upper bound average detection probability expressions are obtained for three scenarios: (i) multihop cooperative relay communication; (ii) multi-hop cooperative relay communication with direct link and maximum ratio combiner (MRC) at destination; and (iii) multi-hop cooperative relay communication with direct link and selection combiner (SC) at destination. Classical method of performance evaluation based on probability density function (PDF) of received signal-to-noise ratio (SNR) is used. Furthermore, an alternative series form representation of generalized Marcum- Q function is employed in order to simplify the ensuing mathematical derivations. Because the obtained detection probability expressions are in the form of infinite summation series, their respective truncation error bounds are also derived. In the end, analyses are validated with the help of simulations and several observations regarding the impact of different parameters on detector's performance are outlined.

On the Closed-Form Performance Analysis of Maximal Ratio Combining in Shadowed-Rician Fading LMS Channels

In this paper, the maximal ratio combining (MRC) scheme in Shadowed-Rician (SR) fading land mobile satellite (LMS) channels is studied. The MRC scheme for SR fading LMS channels has been studied in existing literature; however, most of the existing analytical results are in the form of infinite power series, which are not in closed-form. In this paper, we derive approximate closed-form expressions of the probability density function and cumulative distribution function of the received signal-to-noise ratio of the MRC based receiver in SR fading LMS channels. Then we provide approximate closed-form expressions of the bit error rate (BER), outage probability, and capacity of the considered scheme. One of the derived closed-form BER expressions is found useful for obtaining the analytical diversity order and coding gain of the considered MRC scheme.

Application of Maple on the Differential Problem

This paper uses the mathematical software Maple for the auxiliary tool to study the differential problem of two types of functions. We can obtain the infinite series forms of any order derivatives of these two types of functions by using Leibniz differential rule, differentiation term by term, and integration term by term. And hence greatly reduce the difficulty of calculating the higher order derivative values of these functions. On the other hand, we propose two examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. For this reason, Maple provides insights and guidance regarding problem-solving methods.

Studying the Partial Differential Problem Using Maple

This paper uses the mathematical software Maple for the auxiliary tool to study the partial differential problem of four types of multivariable functions. We can obtain the infinite series forms of any order partial derivatives of these four types of multivariable functions by using differentiation term by term theorem, and hence greatly reduce the difficulty of calculating their higher order partial derivative values. On the other hand, we propose some examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. For this reason, Maple provides insights and guidance regarding problem-solving methods.

Using Maple to Study the Partial Differential Problems

This paper uses the mathematical software Maple for the auxiliary tool to study the partial differential problem of two types of multivariable functions. We can obtain the infinite series forms of any order partial derivatives of these two types of multivariable functions by using differentiation term by term theorem, and hence greatly reduce the difficulty of calculating their higher order partial derivative values. On the other hand, we propose two examples of multivariable functions to evaluate their any order partial derivatives, and some of their higher order partial derivative values practically. At the same time, we employ Maple to calculate the approximations of these higher order partial derivative values and their infinite series forms for verifying our answers.