Functions Of First Kind Research Articles

Bessel functions of first kind and their Anuj transforms

Various problems of Statistics, Mathematics, Radio Physics, Nuclear Physics, Atomic Physics, Fluid Mechanics, Engineering and Science can easily handle by applying integral transform techniques on their mathematical models. Problems of heat equation, Schrodinger equation, Laplace equation, Helmholtz equation and wave equation have solutions in terms of Bessel functions. To solve such equations by integral transform methods, we need to know the integral transform of Bessel functions. In this paper, authors discuss Bessel functions of first kind and determine their Anuj transforms.

On certain subclasses of uniformly spirallike functions associated with struve functions

The main object of this paper is to find necessary and sufficient conditions for generalized Struve functions of first kind to be in the classes SP p (α,β) and UCSP(α,β) of uniformly spirallike functions and also give necessary and sufficient conditions for z(2 − up(z)) to be in the above classes. Furthermore, we give conditions for the integral operator L(m,c,z) = \int_{0}^{z} (2 − u p (t))dt to be in the class UCSPT(α,β). Several corollaries and consequences of the main results are also considered.

Study on the Coincident-Loop Transient Electromagnetic Method in Seafloor Exploration—Taking Jiaodong Polymetallic Mine as a Model

The transient electromagnetic (TEM) method becomes more urgent than ever for marine exploration due to abundant resource reserves and the increasing undersea engineering construction activities, especially in the offshore exploration of mineral deposits such as Sanshandao gold mine. However, the research and application of TEM method in marine environment are still challenged by many problems. Such contradiction motivates our study on the coincident-loop TEM in seafloor exploration. The TEM response of coincident loops is firstly derived in the integral form, based on the potential functions in Helmholtz equations for a magnetic source locating in the whole-space layered model. The frequency-domain vertical magnetic field is described as the Hankel integral with double first-order Bessel functions of first kind. Secondly, the time-domain induced voltage is obtained by transforming the frequency-domain response through the cosine transform and then taking the derivative of time. To simultaneously solve the Hankel transform and the cosine transform, a novel algorithm is introduced by adapting the fixed-point quadrature and extrapolation via the Shanks transformation. Finally, a typical conductivity model for marine polymetallic deposit is designed to investigate the characteristic of TEM response under various conditions. Numerical results demonstrate that existence of conductive seawater causes the TEM response to increase significantly and decay slower. The air-sea reflected electromagnetic waves lead to a significantly large fake negative response (NR) in shallower seawater with depth less than 300 m. Increase in the height of loops will weaken and delay the anomaly response and shorten the observation time-window. The height of configuration should be no more than 100 m for shallower targets and 50 m for deeper targets, respectively. The observation time-window should cover 10-1 000 ms. Increase in the radius of loops only enhances the TEM response proportionally but hardly improves the relative anomaly. The vertical resolution on the low-resistivity target approximates 20 m for the configuration considered in the study. Decreases in D.C. resistivity and chargeability cause the positive response (PR) to increase significantly and decay more rapidly. Meanwhile, the NR is advanced and enlarged significantly and decays slower compared with the PR. The influence of time constant is not monotony and there exists an optimal value for producing the maximum NR. As the frequency parameter increases, the PR is caused to decay more rapidly without magnitude change and the NR is advanced and decays more rapidly with significant increase in magnitude. The influence of frequency parameter is more pronounced than that of time constant.

On certain subclasses of uniformly spirallike functions associated with struve functions

The main object of this paper is to find necessary and sufficient conditions for generalized Struve functions of first kind to be in the classes SP p (α,β) and UCSP(α,β) of uniformly spirallike functions and also give necessary and sufficient conditions for z(2 − up(z)) to be in the above classes. Furthermore, we give conditions for the integral operator L(m,c,z) = \int_{0}^{z} (2 − u p (t))dt to be in the class UCSPT(α,β). Several corollaries and consequences of the main results are also considered.

Radii of Starlikeness and Convexity of Some Entire Functions

A normalized analytic function f is lemniscate starlike if the quantity $$zf'(z)/f(z)$$ lies in the region bounded by the right half of the lemniscate of Bernoulli $$|w^2-1|=1$$ . It is Janowski starlike if the quantity $$zf'(z)/f(z)$$ lies in the disk whose diametric end points are $$(1-A)/(1-B)$$ and $$(1+A)/(1+B)$$ for $$-1\le B<A\le 1$$ . The radii of lemniscate starlikeness and Janowski starlikeness have been determined for normalizations of q-Bessel functions, Bessel functions of first kind of order $$\nu $$ and Lommel functions of first kind. Corresponding convexity radii are also determined.

Averaging Gaussian functionals

This paper consists of two parts. In the first part, we focus on the average of a functional over shifted Gaussian homogeneous noise and as the averaging domain covers the whole space, we establish a Breuer-Major type Gaussian fluctuation based on various assumptions on the covariance kernel and/or the spectral measure. Our methodology for the first part begins with the application of Malliavin calculus around Nualart-Peccati's Fourth Moment Theorem, and in addition we apply the Fourier techniques as well as a soft approximation argument based on Bessel functions of first kind. The same methodology leads us to investigate a closely related problem in the second part. We study the spatial average of a linear stochastic heat equation driven by space-time Gaussian colored noise. The temporal covariance kernel $\gamma_0$ is assumed to be locally integrable in this paper. If the spatial covariance kernel is nonnegative and integrable on the whole space, then the spatial average admits Gaussian fluctuation; with some extra mild integrability condition on $\gamma_0$, we are able to provide a functional central limit theorem. These results complement recent studies on the spatial average for SPDEs. Our analysis also allows us to consider the case where the spatial covariance kernel is not integrable: For example, in the case of the Riesz kernel, the first chaotic component of the spatial average is dominant so that the Gaussian fluctuation also holds true.

Potentials of a Family of Arrangements of Hyperplanes and Elementary Subarrangements

We consider the Frobenius algebra of functions on the critical set of the master function of a weighted arrangement of hyperplanes in $\C^k$ with normal crossings. We construct two potential functions (of first and second kind) of variables labeled by hyperplanes of the arrangement and prove that the matrix coefficients of the Grothendieck residue bilinear form on the algebra are given by the $2k$-th derivatives of the potential function of first kind and the matrix coefficients of the multiplication operators on the algebra are given by the $(2k+1)$-st derivatives of the potential function of second kind. Thus the two potentials completely determine the Frobenius algebra. The presence of these potentials is a manifestation of a Frobenius like structure similar to the Frobenius manifold structure. We introduce the notion of an elementary subarrangement of an arrangement with normal crossings. It turns out that our potential functions are local in the sense that the potential functions are sums of contributions from elementary subarrangements of the given arrangement. This is a new phenomenon of locality of the Grothendieck residue bilinear form and multiplication on the algebra. It is known that this Frobenius algebra of functions on the critical set is isomorphic to the Bethe algebra of this arrangement. (That Bethe algebra is an analog of the Bethe algebras in the theory of quantum integrable models.) Thus our potential functions describe that Bethe algebra too.

Inverses of generators of integrated fractional resolvent operator functions

Abstract This paper is devoted to the inverse generator problem in the setting of generators of integrated resolvent operator functions. It is shown that if the operator A is the generator of a tempered β-times integrated α-resolvent operator function ((α, β)-ROF) and it is injective, then the inverse operator A −1 is the generator of a tempered (α, γ)-ROF for all γ &gt; β + 1/2, by means of an explicit representation of the integrated resolvent operator function based in Bessel functions of first kind. Analytic resolvent operator functions are also considered, showing that A −1 is in addition the generator of a tempered (δ, 0)-ROF for all δ &lt; α. Moreover, the optimal decay rate of (α, β)-ROFs as t → ∞ is given. These result are applied to fractional Cauchy problem unsolved in the fractional derivative.

Approximate analytical treatment of annular fins of rectangular profile for teaching fin heat transfer: Utilization of the mean value theorem for integrals

In the teaching of annular fins of rectangular profile, the main obstacle lies in solving the quasi one-dimensional heat conduction equation, the modified Bessel equation of first kind. In the modified Bessel equation, the variable coefficient [Formula: see text] multiplying the first order derivative of temperature [Formula: see text] is problematic. The principal objective of the present paper on engineering education is to solve the modified Bessel equation of first kind in approximate analytical form. Specifically, we seek to apply the mean value theorem for integrals to the variable coefficient [Formula: see text], viewed as an auxiliary function in the domain of the annular fin extending from the inner radius [Formula: see text] to the outer radius [Formula: see text]. It is demonstrated in a convincing manner that approximate analytical temperature distributions having exponential functions are easy to obtain without resorting to the exact analytical temperature distribution embodying four modified Bessel functions of first kind. Furthermore, the easiness in calculating heat transfer rates in annular fins of rectangular profile for realistic combinations of the two controlling parameters: the normalized radius ratio and the dimensionless thermo-geometrical parameter is verifiable.

The Evaluation of Derivatives of Double Barrier Options of the Bessel Processes by Methods of Spectral Analysis

The paper deals with the spectral methods to calculate the value of the double barrier option generated by the Bessel diffusion process. This technique enables us to calculate the option price in the form of a Fourier-Bessel series with the corresponding ratio. The authors propose a simple method to estimate options using the Green’s expansion function for boundary value problem for a singular parabolic equation. Thus, the accuracy of the estimation coincides with the accuracy of the convergence of the Fourier-Bessel series. In this paper, the authors use the spectral theory to calculate the price of derivatives of financial assets considering that the processes are described by Markov and can be considered in Hilbert spaces. In this work, the authors use the diffusion process to find derivatives prices by introducing them through the Bessel functions of first kind. They also examine the Sturm-Liouville problem where the boundary conditions utilize the Bessel functions and their derivatives. All assumptions lead to analytical formulae that are consistent with the empirical evidence and, when implemented in practice, reflect adequately the passage of processes on stock markets. The authors also focus on the financial flows generated by Bessel diffusion processes which are presented in the system of Bessel functions of the first order under the condition that the linear combination of the flow and its spatial derivative are taken into account. Such a presentation enables us to calculate the market value of a share portfolio, provides the measurement of internal volatility in the market at any given time, and allows us to investigate the dynamics of the stock market. The splitting of Green’s function in the system of Bessel functions is presented by an analytical formula which is convenient for calculating the price level of options.

Radii Problems for Normalized Bessel Functions of First Kind

For three different normalizations of Bessel functions of first kind, the radius of k-parabolic starlikeness and k-uniform convexity of order $$\alpha $$ are determined. The radius of strong starlikeness and other related radius are also obtained for these functions. We also find optimal parameters for which these functions are k-parabolic starlike and k-uniformly convex in the open unit disk.

The evaluation of derivatives of double barrier options of the Bessel processes by methods of spectral analysis

The paper deals with the spectral methods to calculate the value of the double barrier option generated by the Bessel diffusion process. This technique enables us to calculate the option price in the form of a Fourier-Bessel series with the corresponding ratio. The autors propose a simple method to estimate options using the Green’s expansion function for boundary value problem for a singular parabolic equation. Thus, the accuracy of the estimation coincides with the accuracy of the convergence of the Fourier-Bessel series. In this paper, the authors use the spectral theory to calculate the price of derivatives of financial assets considering that the processes described are by Markov and can be considered in Hilbert spaces. In this work, the authors use the diffusion process to find derivatives prices by introducing them through the Bessel functions of first kind. They also examine the Sturm-Liouville problem where the boundary conditions utilize the Bessel functions and their derivatives. All assumptions lead to analytical formulae that are consistent with the empirical evidence and, when implemented in practice, reflect adequately the passage of processes on stock markets. The authors also focus on the financial flows generated by Bessel diffusion processes which are presented in the system of Bessel functions of the first order under the condition that the linear combination of the flow and its spatial derivative are taken into account. Such a presentation enables us to calculate the market value of a share portfolio, provides the measurement of internal volatility in the market at any given time, and allows us to investigate the dynamics of the stock market. The splitting of Green’s function in the system of Bessel functions is presented by an analytical formula which is convenient for calculating the price level of options.

Klein–Gordon equation for a charged particle in space-varying electromagnetic fields–A systematic study via the Laplace transform

Exact solutions of the Klein–Gordon equation for a charged particle in the presence of three spatially varying electromagnetic fields, namely, (i) E→=αβ0e−αx2x^2,B→=αβ1e−αx2x^3 (ii) E→=β0′x22x^2,B→=β1′x22x^3, and (iii) E→=2β0′x23x^2,B→=2β1′x23x^3, are studied. All these fields are generated from a systematic study of a particular type of differential equation whose coefficients are linear in the independent variable. The Laplace transform approach is used to find the solutions, and the corresponding eigenfunctions are expressed in terms of the hypergeometric functions 1F1(a′, b′; x) for the first two cases of the above configurations, while the same are expressed in terms of the Bessel functions of first kind, Jn(x), for the last case.

Numerical solutions of hyperbolic telegraph equation by using the Bessel functions of first kind and residual correction

In this study, a collocation method is presented to solve one-dimensional hyperbolic telegraph equation. The problem is given by hyperbolic telegraph equation under initial and boundary conditions. The method is based on the Bessel functions of the first kind. Using the collocation points and the operational matrices of derivatives, we reduce the problem to a set of linear algebraic equations. The determined coefficients from this system give the coefficients of the approximate solution. Also, an error estimation method is presented for the considered problem and the method. By using the residual function and the original problem, an error problem is constructed and thus the error function is estimated. By aid of the estimated function, the approximated solution is improved. Numerical examples are given to demonstrate the validity and applicability of the proposed method and also, the comparisons are made with the known results.

An Application of Generalized Bessel Functions on Certain Subclasses of Analytic Functions

The purpose of the present paper is to investigate some characterization for generalized Bessel functions of first kind to be in the new subclasses of β uniformly starlike and β uniformly convex functions of order α. Further we point out consequences of our main results.

Connections between certain classes of harmonic univalent mappings involving generalized Bessel functions

In the present paper, we obtain some sufficient conditions of certain convolution operator involving generalized Bessel functions of first kind belonging to various subclasses of harmonic univalent functions. To be more precise, we investigate such connections with harmonic g-uniformly convex and harmonic g-uniformly starlike mappings in the plane.

Redheffer type inequalities for modified Bessel functions

In this short note, we give new proofs of Redheffer’s inequality for modified Bessel functions of first kind published by Ling Zhu (2011). In addition, using the Grosswald formula we prove new Redheffer type inequality for the modified Bessel functions of the second kind.

Differential Subordinations Involving Generalized Bessel Functions

In this paper our aim is to present some subordination and superordination results, by using an operator, which involves the normalized form of the generalized Bessel functions of first kind. These results are obtained by investigating some appropriate classes of admissible functions. We obtain also some sandwich-type results and we point out various known or new special cases of our main results.

Numerical solutions of system of linear Fredholm–Volterra integro-differential equations by the Bessel collocation method and error estimation

In this study, the Bessel collocation method is presented for the solutions of system of linear Fredholm–Volterra integro-differential equations which includes the derivatives of unknown functions in integral parts. The Bessel collocation method transforms the problem into a system of linear algebraic equations by means of the Bessel functions of first kind, the collocation points and the matrix relations. Also, an error estimation is given for the considered problem and the method. Illustrative examples are presented to show efficiency of method and the comparisons are made with the results of other methods. All of numerical calculations have been made on a computer using a program written in Matlab.

HARMONIC STARLIKENESS AND CONVEXITY OF INTEGRAL OPERATORS GENERATED BY GENERALIZED BESSEL FUNCTIONS

The present paper establishes connections between various subclasses of harmonic univalent functions by applying certain integral operator involving the generalized Bessel functions of first kind. To be more precise, we investigate such connections with harmonic starlike and harmonic convex mappings in the plane.