Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 2010

# Search results for: hypergeometric functions

##### 2010 Nonhomogeneous Linear Fractional Differential Equations Will Bessel Functions of the First Kind Giving Hypergeometric Functions Solutions

Authors: Fernando Maass, Pablo Martin, Jorge Olivares

Abstract:

Fractional derivatives have become very important in several areas of Engineering, however, the solutions of simple differential equations are not known. Here we are considering the simplest first order nonhomogeneous differential equations with Bessel regular functions of the first kind, in this way the solutions have been found which are hypergeometric solutions for any fractional derivative of order α, where α is rational number α=m/p, between zero and one. The way to find this result is by using Laplace transform and the Caputo definitions of fractional derivatives. This method is for values longer than one. However for α entire number the hypergeometric functions are Kumer type, no integer values of alpha, the hypergeometric function is more complicated is type ₂F₃(a,b,c, t2/2). The argument of the hypergeometric changes sign when we go from the regular Bessel functions to the modified Bessel functions of the first kind, however it integer seems that using precise values of α and considering no integers values of α, a solution can be obtained in terms of two hypergeometric functions. Further research is required for future papers in order to obtain the general solution for any rational value of α. Downloads 111
##### 2009 Hypergeometric Solutions to Linear Nonhomogeneous Fractional Equations with Spherical Bessel Functions of the First Kind

Authors: Pablo Martin, Jorge Olivares, Fernando Maass

Abstract:

The use of fractional derivatives to different problems in Engineering and Physics has been increasing in the last decade. For this reason, we have here considered partial derivatives when the integral is a spherical Bessel function of the first kind in both regular and modified ones simple initial conditions have been also considered. In this way, the solution has been found as a combination of hypergeometric functions. The case of a general rational value for α of the fractional derivative α has been solved in a general way for alpha between zero and two. The modified spherical Bessel functions of the first kind have been also considered and how to go from the regular case to the modified one will be also shown. Downloads 156
##### 2008 Linear fractional differential equations for second kind modified Bessel functions

Authors: Jorge Olivares, Fernando Maass, Pablo Martin

Abstract:

Fractional derivatives have been considered recently as a way to solve different problems in Engineering. In this way, second kind modified Bessel functions are considered here. The order α fractional differential equations of second kind Bessel functions, Kᵥ(x), are studied with simple initial conditions. The Laplace transform and Caputo definition of fractional derivatives are considered. Solutions have been found for ν=1/3, 1/2, 2/3, -1/3, -1/2 and (-2/3). In these cases, the solutions are the sum of two hypergeometric functions. The α fractional derivatives have been for α=1/3, 1/2 and 2/3, and the above values of ν. No convergence has been found for the integer values of ν Furthermore when α has been considered as a rational found m/p, no general solution has been found. Clearly, this case is more difficult to treat than those of first kind Bessel Function. Downloads 243
##### 2007 Energy States of Some Diatomic Molecules: Exact Quantization Rule Approach

Authors: Babatunde J. Falaye

Abstract:

In this study, we obtain the approximate analytical solutions of the radial Schrödinger equation for the Deng-Fan diatomic molecular potential by using exact quantization rule approach. The wave functions have been expressed by hypergeometric functions via the functional analysis approach. An extension to rotational-vibrational energy eigenvalues of some diatomic molecules are also presented. It is shown that the calculated energy levels are in good agreement with the ones obtained previously E_nl-D (shifted Deng-Fan). Downloads 411
##### 2006 On the Fractional Integration of Generalized Mittag-Leffler Type Functions

Authors: Christian Lavault

Abstract:

In this paper, the generalized fractional integral operators of two generalized Mittag-Leffler type functions are investigated. The special cases of interest involve the generalized M-series and K-function, both introduced by Sharma. The two pairs of theorems established herein generalize recent results about left- and right-sided generalized fractional integration operators applied here to the M-series and the K-function. The note also results in important applications in physics and mathematical engineering. Downloads 333
##### 2005 The Behavior of The Zeros of Bargmann Analytic Functions for Multiple-Mode Systems

Authors: Muna Tabuni

Abstract:

The paper contains an investigation of the behavior of the Zeros of Bargmann functions for one and two-mode systems. A brief introduction to Harmonic oscillator formalism for one and two-mode is given. The Bargmann analytic representation for one and two-mode has been studied. The zeros of Bargmann analytic function for one-mode are considered. The Q Husimi functions are introduced. The Bargmann functions and the Husimi functions have the same zeros. The Bargmann functions f(z) have exactly q zeros. The evolution time of the zeros are discussed. The zeros of Bargmann analytic functions for two-mode are introduced. Various examples have been given. Downloads 395
##### 2004 Some Inequalities Related with Starlike Log-Harmonic Mappings

Authors: Melike Aydoğan, Dürdane Öztürk

Abstract:

Let H(D) be the linear space of all analytic functions defined on the open unit disc. A log-harmonic mappings is a solution of the nonlinear elliptic partial differential equation where w(z) ∈ H(D) is second dilatation such that |w(z)| < 1 for all z ∈ D. The aim of this paper is to define some inequalities of starlike logharmonic functions of order α(0 ≤ α ≤ 1). Downloads 436
##### 2003 RAFU Functions in Robotics and Automation

Authors: Alicia C. Sanchez

Abstract:

This paper investigates the implementation of RAFU functions (radical functions) in robotics and automation. Specifically, the main goal is to show how these functions may be useful in lane-keeping control and the lateral control of autonomous machines, vehicles, robots or the like. From the knowledge of several points of a certain route, the RAFU functions are used to achieve the lateral control purpose and maintain the lane-keeping errors within the fixed limits. The stability that these functions provide, their ease of approaching any continuous trajectory and the control of the possible error made on the approximation may be useful in practice. Downloads 137
##### 2002 Subclasses of Bi-Univalent Functions Associated with Hohlov Operator

Abstract:

The coefficients estimate problem for Taylor-Maclaurin series is still an open problem especially for a function in the subclass of bi-univalent functions. A function f ϵ A is said to be bi-univalent in the open unit disk D if both f and f-1 are univalent in D. The symbol A denotes the class of all analytic functions f in D and it is normalized by the conditions f(0) = f’(0) – 1=0. The class of bi-univalent is denoted by  The subordination concept is used in determining second and third Taylor-Maclaurin coefficients. The upper bound for second and third coefficients is estimated for functions in the subclasses of bi-univalent functions which are subordinated to the function φ. An analytic function f is subordinate to an analytic function g if there is an analytic function w defined on D with w(0) = 0 and |w(z)| < 1 satisfying f(z) = g[w(z)]. In this paper, two subclasses of bi-univalent functions associated with Hohlov operator are introduced. The bound for second and third coefficients of functions in these subclasses is determined using subordination. The findings would generalize the previous related works of several earlier authors. Downloads 202
##### 2001 Geometric Properties of Some q-Bessel Functions

Authors: İbrahim Aktaş, Árpád Baricz

Abstract:

In this paper, the radii of star likeness of the Jackson and Hahn-Exton q-Bessel functions are considered, and for each of them three different normalizations is applied. By applying Euler-Rayleigh inequalities for the first positive zeros of these functions tight lower, and upper bounds for the radii of starlikeness of these functions are obtained. The Laguerre-Pólya class of real entire functions plays an important role in this study. In particular, we obtain some new bounds for the first positive zero of the derivative of the classical Bessel function of the first kind. Downloads 199
##### 2000 Approximation of Analytic Functions of Several Variables by Linear K-Positive Operators in the Closed Domain

Authors: Tulin Coskun

Abstract:

We investigate the approximation of analytic functions of several variables in polydisc by the sequences of linear k-positive operators in Gadjiev sence. The approximation of analytic functions of complex variable by linear k-positive operators was tackled, and k-positive operators and formulated theorems of Korovkin's type for these operators in the space of analytic functions on the unit disc were introduced in the past. Recently, very general results on convergence of the sequences of linear k-positive operators on a simply connected bounded domain within the space of analytic functions were proved. In this presentation, we extend some of these results to the approximation of analytic functions of several complex variables by sequences of linear k-positive operators. Downloads 252
##### 1999 Unconventional Calculus Spreadsheet Functions

Abstract:

The spreadsheet engine is exploited via a non-conventional mechanism to enable novel worksheet solver functions for computational calculus. The solver functions bypass inherent restrictions on built-in math and user defined functions by taking variable formulas as a new type of argument while retaining purity and recursion properties. The enabling mechanism permits integration of numerical algorithms into worksheet functions for solving virtually any computational problem that can be modelled by formulas and variables. Several examples are presented for computing integrals, derivatives, and systems of deferential-algebraic equations. Incorporation of the worksheet solver functions with the ubiquitous spreadsheet extend the utility of the latter as a powerful tool for computational mathematics. Downloads 270
##### 1998 Fuzzy Control and Pertinence Functions

Authors: Luiz F. J. Maia

Abstract:

This paper presents an approach to fuzzy control, with the use of new pertinence functions, applied in the case of an inverted pendulum. Appropriate definitions of pertinence functions to fuzzy sets make possible the implementation of the controller with only one control rule, resulting in a smooth control surface. The fuzzy control system can be implemented with analog devices, affording a true real-time performance. Downloads 322
##### 1997 A New Approach for Generalized First Derivative of Nonsmooth Functions Using Optimization

Abstract:

In this paper, we define an optimization problem corresponding to smooth and nonsmooth functions which its optimal solution is the first derivative of these functions in a domain. For this purpose, a linear programming problem corresponding to optimization problem is obtained. The optimal solution of this linear programming problem is the approximate generalized first derivative. In fact, we approximate generalized first derivative of nonsmooth functions as tailor series. We show the efficiency of our approach by some smooth and nonsmooth functions in some examples. Downloads 470
##### 1996 Numerical Solution for Integro-Differential Equations by Using Quartic B-Spline Wavelet and Operational Matrices

Abstract:

In this paper, semi-orthogonal B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of integro-differential equations.The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this function are presented to reduce the solution of integro-differential equations to the solution of algebraic equations. Here we compute B-spline scaling functions of degree 4 and their dual, then we will show that by using them we have better approximation results for the solution of integro-differential equations in comparison with less degrees of scaling functions. Downloads 254
##### 1995 High Accuracy Analytic Approximation for Special Functions Applied to Bessel Functions J₀(x) and Its Zeros

Authors: Fernando Maass, Pablo Martin, Jorge Olivares

Abstract:

The Bessel function J₀(x) is very important in Electrodynamics and Physics, as well as its zeros. In this work, a method to obtain high accuracy approximation is presented through an application to that function. In most of the applications of this function, the values of the zeros are very important. In this work, analytic approximations for this function have been obtained valid for all positive values of the variable x, which have high accuracy for the function as well as for the zeros. The approximation is determined by the simultaneous used of the power series and asymptotic expansion. The structure of the approximation is a combination of two rational functions with elementary functions as trigonometric and fractional powers. Here us in Pade method, rational functions are used, but now there combined with elementary functions us fractional powers hyperbolic or trigonometric functions, and others. The reason of this is that now power series of the exact function are used, but together with the asymptotic expansion, which usually includes fractional powers trigonometric functions and other type of elementary functions. The approximation must be a bridge between both expansions, and this can not be accomplished using only with rational functions. In the simplest approximation using 4 parameters the maximum absolute error is less than 0.006 at x ∼ 4.9. In this case also the maximum relative error for the zeros is less than 0.003 which is for the second zero, but that value decreases rapidly for the other zeros. The same kind of behaviour happens for the relative error of the maximum and minimum of the functions. Approximations with higher accuracy and more parameters will be also shown. All the approximations are valid for any positive value of x, and they can be calculated easily. Downloads 169
##### 1994 A Qualitative Case Study Exploring Zambian Mathematics Teachers' Content Knowledge of Functions

Abstract:

The relevance of what is content is taught in tertiary teacher training has long been in question. This study attempts to understand how advanced mathematics courses equip student teachers to teach functions at secondary school level. This paper reports on an investigation that was conducted in an African university, where preservice teachers were purposefully selected for participation in individual semi-structured interviews after completing a test on functions as taught at secondary school. They were asked to justify their reasoning in the test and to explain functions in a way that might bring about understanding of the topic in someone who did not know how functions work. These were final year preservice mathematics teachers who had studied advanced mathematics courses for three years. More than 50% of the students were not able to explain concepts or to justify their reasoning about secondary school functions in a coherent way. The results of this study suggest that the study of advanced mathematics does not automatically enable students to teach secondary school functions, and that, although these students were able to do advanced mathematics, they were unable to explain the working of functions in a way that would allow them to teach this topic successfully. Downloads 175
##### 1993 Bernstein Type Polynomials for Solving Differential Equations and Their Applications

Authors: Yilmaz Simsek

Abstract:

In this paper, we study the Bernstein-type basis functions with their generating functions. We give various properties of these polynomials with the aid of their generating functions. These polynomials and generating functions have many valuable applications in mathematics, in probability, in statistics and also in mathematical physics. By using the Bernstein-Galerkin and the Bernstein-Petrov-Galerkin methods, we give some applications of the Bernstein-type polynomials for solving high even-order differential equations with their numerical computations. We also give Bezier-type curves related to the Bernstein-type basis functions. We investigate fundamental properties of these curves. These curves have many applications in mathematics, in computer geometric design and other related areas. Moreover, we simulate these polynomials with their plots for some selected numerical values. Downloads 195
##### 1992 Exact Formulas of the End-To-End Green’s Functions in Non-hermitian Systems

Authors: Haoshu Li, Shaolong Wan

Abstract:

The recent focus has been on directional signal amplification of a signal input at one end of a one-dimensional chain and measured at the other end. The amplification rate is given by the end-to-end Green’s functions of the system. In this work, we derive the exact formulas for the end-to-end Green's functions of non-Hermitian single-band systems. While in the bulk region, it is found that the Green's functions are displaced from the prior established integral formula by O(e⁻ᵇᴸ). The results confirm the correspondence between the signal amplification and the non-Hermitian skin effect. Downloads 29
##### 1991 Certain Results of a New Class of Meromorphic Multivalent Functions Involving Ruscheweyh Derivative

Authors: Kassim A. Jassim

Abstract:

In the present paper, we introduce and discuss a new class Kp(λ,α) of meromorphic multivalent functions in the punctured unit disk U*={z∈¢:0<|z|<1} defined by Ruscheweyh derivative. We obtain some sufficient conditions for the functions belonging to the class Kp(λ,α). Downloads 243
##### 1990 Some Integral Inequalities of Hermite-Hadamard Type on Time Scale and Their Applications

Authors: Artion Kashuri, Rozana Liko

Abstract:

In this paper, the authors establish an integral identity using delta differentiable functions. By applying this identity, some new results via a general class of convex functions with respect to two nonnegative functions on a time scale are given. Also, for suitable choices of nonnegative functions, some special cases are deduced. Finally, in order to illustrate the efficiency of our main results, some applications to special means are obtained as well. We hope that current work using our idea and technique will attract the attention of researchers working in mathematical analysis, mathematical inequalities, numerical analysis, special functions, fractional calculus, quantum mechanics, quantum calculus, physics, probability and statistics, differential and difference equations, optimization theory, and other related fields in pure and applied sciences. Downloads 69
##### 1989 Duality in Multiobjective Nonlinear Programming under Generalized Second Order (F, b, φ, ρ, θ)− Univex Functions

Authors: Meraj Ali Khan, Falleh R. Al-Solamy

Abstract:

In the present paper, second order duality for multiobjective nonlinear programming are investigated under the second order generalized (F, b, φ, ρ, θ)− univex functions. The weak, strong and converse duality theorems are proved. Further, we also illustrated an example of (F, b, φ, ρ, θ)− univex functions. Results obtained in this paper extend some previously known results of multiobjective nonlinear programming in the literature. Downloads 261
##### 1988 The Role of Eclectic Approach to Teach Communicative Function at Secondary Level

Authors: Fariha Asif

Abstract:

The main purpose of this study was to investigate the effectiveness of eclectic approach in teaching of communicative functions. The objectives of the study were to get the information about the use of communicative functions through eclectic approach and to point out the most effective way of teaching functional communication and social interaction with the help of communicative activities through eclectic approach. The next step was to select sample from the selected population. As the research was descriptive so a questionnaire was developed on the basis of hypothesis and distributed to different selected schools of Lahore, Pakistan. Then data was tabulated, analyzed and interpreted through computer by finding percentages of different responses given by teachers to see the results. It was concluded that eclectic approach is effective in teaching communicative functions and communicative functions are better when taught through eclectic approach and communicative activities are more appropriate way of teaching communicative functions. It was found those teachers who were qualified in ELT gave better opinions as compare to those who did not have this degree. Techniques like presentations, dialogues and roleplay proved to be effective for teaching functional communication through communicative activities and also motivate the students not only in learning rules but also in using them to communicate with others.

Keywords: methodology, functions, teaching, ESP

##### 1987 Sequential Covering Algorithm for Nondifferentiable Global Optimization Problem and Applications

Authors: Mohamed Rahal, Djaouida Guetta

Abstract:

In this paper, the one-dimensional unconstrained global optimization problem of continuous functions satifying a Hölder condition is considered. We extend the algorithm of sequential covering SCA for Lipschitz functions to a large class of Hölder functions. The convergence of the method is studied and the algorithm can be applied to systems of nonlinear equations. Finally, some numerical examples are presented and illustrate the efficiency of the present approach. Downloads 290
##### 1986 Virtual Routing Function Allocation Method for Minimizing Total Network Power Consumption

Authors: Kenichiro Hida, Shin-Ichi Kuribayashi

Abstract:

In a conventional network, most network devices, such as routers, are dedicated devices that do not have much variation in capacity. In recent years, a new concept of network functions virtualisation (NFV) has come into use. The intention is to implement a variety of network functions with software on general-purpose servers and this allows the network operator to select their capacities and locations without any constraints. This paper focuses on the allocation of NFV-based routing functions which are one of critical network functions, and presents the virtual routing function allocation algorithm that minimizes the total power consumption. In addition, this study presents the useful allocation policy of virtual routing functions, based on an evaluation with a ladder-shaped network model. This policy takes the ratio of the power consumption of a routing function to that of a circuit and traffic distribution between areas into consideration. Furthermore, the present paper shows that there are cases where the use of NFV-based routing functions makes it possible to reduce the total power consumption dramatically, in comparison to a conventional network, in which it is not economically viable to distribute small-capacity routing functions. Downloads 258
##### 1985 Jensen's Inequality and M-Convex Functions

Authors: Yamin Sayyari

Abstract:

In this paper, we generalized the Jensen's inequality for m-convex functions and also we present a correction of Jensen's inequality which is a better than the generalization of this inequality for m-convex functions. Finally, we have found new lower and new upper bounds for Jensen's discrete inequality. Downloads 75
##### 1984 A Survey on Positive Real and Strictly Positive Real Scalar Transfer Functions

Abstract:

Positive real and strictly positive real transfer functions are important concepts in the control theory. In this paper, the results of researches in these areas are summarized. Definitions together with their graphical interpretations are mentioned. The equivalent conditions in the frequency domain and state space representations are reviewed. Their equivalent electrical networks are explained. Also, a comprehensive discussion about a difference between behavior of real part of positive real and strictly positive real transfer functions in high frequencies is presented. Furthermore, several illustrative examples are given. Downloads 303
##### 1983 Exact Solutions of K(N,N)-Type Equations Using Jacobi Elliptic Functions

Authors: Edamana Krishnan, Khalil Al-Ghafri

Abstract:

In this paper, modified K(n,n) and K(n+1,n+1) equations have been solved using mapping methods which give a variety of solutions in terms of Jacobi elliptic functions. The solutions when m approaches 0 and 1, with m as the modulus of the JEFs have also been deduced. The role of constraint conditions has been discussed. Downloads 220
##### 1982 Generalized Hyperbolic Functions: Exponential-Type Quantum Interactions

Authors: Jose Juan Peña, J. Morales, J. García-Ravelo

Abstract:

In the search of potential models applied in the theoretical treatment of diatomic molecules, some of them have been constructed by using standard hyperbolic functions as well as from the so-called q-deformed hyperbolic functions (sc q-dhf) for displacing and modifying the shape of the potential under study. In order to transcend the scope of hyperbolic functions, in this work, a kind of generalized q-deformed hyperbolic functions (g q-dhf) is presented. By a suitable transformation, through the q deformation parameter, it is shown that these g q-dhf can be expressed in terms of their corresponding standard ones besides they can be reduced to the sc q-dhf. As a useful application of the proposed approach, and considering a class of exactly solvable multi-parameter exponential-type potentials, some new q-deformed quantum interactions models that can be used as interesting alternative in quantum physics and quantum states are presented. Furthermore, due that quantum potential models are conditioned on the q-dependence of the parameters that characterize to the exponential-type potentials, it is shown that many specific cases of q-deformed potentials are obtained as particular cases from the proposal. Downloads 118
##### 1981 Degree of Approximation of Functions Conjugate to Periodic Functions Belonging to Lipschitz Classes by Product Matrix Means

Authors: Smita Sonker

Abstract:

Various investigators have determined the degree of approximation of conjugate signals (functions) of functions belonging to different classes Lipα, Lip(α,p), Lip(ξ(t),p), W(Lr,ξ(t), (β ≥ 0)) by matrix summability means, lower triangular matrix operator, product means (i.e. (C,1)(E,1), (C,1)(E,q), (E,q)(C,1) (N,p,q)(E,1), and (E,q)(N,pn) of their conjugate trigonometric Fourier series. In this paper, we shall determine the degree of approximation of 2π-periodic function conjugate functions of f belonging to the function classes Lipα and W(Lr; ξ(t); (β ≥ 0)) by (C1.T) -means of their conjugate trigonometric Fourier series. On the other hand, we shall review above-mentioned work in the light of Lenski. Downloads 325