Search results for: multidimensional functions

Authors: Diogo Silva, Fadul Rodor, Carlos Moraes

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This work compares the results of multidimensional function approximation using two algorithms: the classical Particle Swarm Optimization (PSO) and the Quantum Particle Swarm Optimization (QPSO). These algorithms were both tested on three functions - The Rosenbrock, the Rastrigin, and the sphere functions - with different characteristics by increasing their number of dimensions. As a result, this study shows that the higher the function space, i.e. the larger the function dimension, the more evident the advantages of using the QPSO method compared to the PSO method in terms of performance and number of necessary iterations to reach the stop criterion.

Keywords: PSO, QPSO, function approximation, AI, optimization, multidimensional functions

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Authors: Maja Andrić, Josip Pečarić

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Over the last five decades, an enormous amount of work has been done on Opial’s integral inequality, dealing with new proofs, various generalizations, extensions and discrete analogs. The Opial inequality is recognized as a fundamental result in the analysis of qualitative properties of solution of differential equations. We use submultiplicative convex functions, appropriate representations of functions and inequalities involving means to obtain generalizations and extensions of certain known multidimensional integral and discrete Opial-type inequalities.

Keywords: Opial's inequality, Jensen's inequality, integral inequality, discrete inequality

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Authors: Zouari Ezzeddine, Ghraieb Ikram

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From the results of the academic literature which evokes the limitations of previous studies, this article shows the reasons for multidimensionality Prediction of financial bubbles. A new framework for modeling study predicting financial bubbles by linking a set of variable presented on several dimensions dictating its multidimensional character. It takes into account the preferences of financial actors. A multicriteria anticipation of the appearance of bubbles in international financial markets helps to fight against a possible crisis.

Keywords: classical measures, predictions, financial bubbles, multidimensional, artificial neural networks

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Authors: Ouzayr Rabhi, Ibtissam Arrassen

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To provide a complete analysis of the organization and to help decision-making, leaders need to have relevant data; Data Warehouses (DW) are designed to meet such needs. However, designing DW is not trivial and there is no formal method to derive a multidimensional schema from heterogeneous databases. In this article, we present a Model-Driven based approach concerning the design of data warehouses. We describe a multidimensional meta-model and also specify a set of transformations starting from a Unified Modeling Language (UML) metamodel. In this approach, the UML metamodel and the multidimensional one are both considered as a platform-independent model (PIM). The first meta-model is mapped into the second one through transformation rules carried out by the Query View Transformation (QVT) language. This proposal is validated through the application of our approach to generating a multidimensional schema of a Balanced Scorecard (BSC) DW. We are interested in the BSC perspectives, which are highly linked to the vision and the strategies of an organization.

Keywords: data warehouse, meta-model, model-driven architecture, transformation, UML

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Authors: Ramya Rachel S.

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For women in a developing country like India, poverty is an ever-clinging problem which has rooted itself without any trace of absolute abolition. Poverty is a deprivation of many imminent needs and must be measured accordingly. Therefore, it is important to study the dimensions of education, health, and standard of living to understand the true nature of the impoverished. The study focused on studying the deprivation on these aspects using the Alkire-Foster methodology to estimate the Multidimensional Poverty Index. The study has utilized the individual data of women aged 15 to 49 of the National Family Health Survey (NFHS) for the year 2015-16. Findings reveal that women in India still face extreme levels of deprivation in various dimensions. More than one-third of the total women aged 15 to 24 in India were multidimensionally poor. Dimensional breakdown of the levels of multidimensional poverty indicates that the dimension of Education is the highest contributor to poverty. Decomposition of the multidimensional poverty among various demographic sub-groups, reveals that the multidimensional poverty level increases with age. Results point out that deprivations were higher among widowed and married women, and among women who lived alone. There was also a huge rural-urban divide with respect to poverty. The basic needs of these women must be targeted and met so that they are withdrawn from all forms of poverty.

Keywords: deprivations, multidimensional poverty, sub-group decomposition, women

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Authors: Bidyadhar Dehury, Sanjay Kumar Mohanty

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According to the Right to Education Act of India, education is the fundamental right of all children of age group 6-14 year irrespective of their status. Using the unit level data from India Human Development Survey (IHDS), we tried to understand the inter-relationship between the level of poverty and the academic performance of the children aged 8-11 years. The level of multidimensional poverty is measured using five dimensions and 10 indicators using Alkire-Foster approach. The weighted deprivation score was obtained by giving equal weight to each dimension and indicators within the dimension. The weighted deprivation score varies from 0 to 1 and grouped into four categories as non-poor, vulnerable, multidimensional poor and sever multidimensional poor. The academic performance index was measured using three variables reading skills, math skills and writing skills using PCA. The bivariate and multivariate analysis was used in the analysis. The outcome variable was ordinal. So the predicted probabilities were calculated using the ordinal logistic regression. The predicted probabilities of good academic performance index was 0.202 if the child was sever multidimensional poor, 0.235 if the child was multidimensional poor, 0.264 if the child was vulnerable, and 0.316 if the child was non-poor. Hence, if the level of poverty among the children decreases from sever multidimensional poor to non-poor, the probability of good academic performance increases.

Keywords: multidimensional poverty, academic performance index, reading skills, math skills, writing skills, India

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Authors: Joseph Siani

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Over the last decades, dissatisfaction with global indicators of economic performance, such as GDP (Gross Domestic Product) per capita, has shifted the attention to what is now referred to as multidimensional poverty. In this framework, poverty goes beyond income to incorporate aspects of well-being not captured by income measures alone. This paper applies the totally fuzzy approach to estimate the fuzzy index of poverty (FIP) in fourteen Sub-Saharan African (SSA) countries using Demographic and Health Survey (DHS) data and explores whether pictures created by the standard headcount ratio at $1.90 a day and the fuzzy index of poverty tell a similar story. The results suggest that there is indeed considerable mismatch between poverty headcount and the fuzzy index of multidimensional poverty, meaning that the majority of the most deprived people (as identified by the fuzzy index of multidimensional poverty) would not be identified by the poverty headcount ratio. Moreover, we find that poverty is distributed differently by colonial heritage (language). In particular, the most deprived countries in SSA are French-speaking.

Keywords: fuzzy set approach, multidimensional poverty, poverty headcount, overlap, Sub-Saharan Africa

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Authors: Fuad Al Sarari

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The purpose of the present paper is to study subclasses of analytic functions which generalize the classes of Janowski functions introduced by Polatoglu. We study certain convolution conditions. This leads to a study of the sufficient condition and the neighborhood results related to the functions in the class S (T; H; F ): and a study of new subclasses of analytic functions that are defined using notions of the generalized Janowski classes and -symmetrical functions. In the quotient of analytical representations of starlikeness and convexity with respect to symmetric points, certain inherent properties are pointed out.

Keywords: convolution conditions, subordination, Janowski functions, starlike functions, convex functions

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Authors: B. Schmid, C. Kexel, E. Djafarova

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Understanding consumers is elementary for practitioners in marketing. Consumers of sports events, the sports spectators, are a particularly complex consumer crowd. In order to identify and define their profiles different segmentation approaches can be found in literature, one of them being multidimensional segmentation. Multidimensional segmentation models correspond to the broad range of attitudes, behaviours, motivations and beliefs of sports spectators, other than earlier models. Moreover, in sports there are some well-researched disciplines (e.g. football or North American sports) where consumer profiles and marketing strategies are elaborate and others where no research at all can be found. For example, there is almost no research on athletics spectators. This paper explores the current state of research on sports spectators segmentation. An in-depth literature review provides the framework for a spectators segmentation in athletics. On this basis, additional potential consumer groups and implications for social media marketing will be explored. The findings are the basis for further research.

Keywords: multidimensional segmentation, social media, sports marketing, sports spectators segmentation

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Authors: Muna Tabuni

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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.

Keywords: Bargmann functions, two-mode, zeros, harmonic oscillator

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Authors: Ben Mansour Mouin, Elloumi Abdelkarim

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We will discuss the problem of routing a fleet of different vehicles from a central depot to different types of infrastructure-defaults with dynamic maintenance requests, modified time windows, and control of default maintained. For this reason, we propose a modified metaheuristicto to solve our mathematical model. SVRDP-CMTW is a variant VRP of an optimal vehicle plan that facilitates the maintenance task of different types of infrastructure-defaults. This task will be monitored after the maintenance, based on its priorities, the degree of danger associated with each default, and the neighborhood at the black-spots. We will present, in this paper, a multidimensional genetic algorithm “MGA” by detailing its characteristics, proposed mechanisms, and roles in our work. The coding of this algorithm represents the necessary parameters that characterize each infrastructure-default with the objective of minimizing a combination of cost, distance and maintenance times while satisfying the priority levels of the most urgent defaults. The developed algorithm will allow the dynamic integration of newly detected defaults at the execution time. This result will be displayed in our programmed interactive system at the routing time. This multidimensional genetic algorithm replaces N genetic algorithm to solve P different type problems of infrastructure defaults (instead of N algorithm for P problem we can solve in one multidimensional algorithm simultaneously who can solve all these problemsatonce).

Keywords: mathematical model, VRP, multidimensional genetic algorithm, metaheuristics

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Authors: Gebreegziabher Hailu Gebrecherkos

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This study explores the derivatives of functions defined by I-functions of two variables and their connections to generalized M-series. We begin by defining I-functions, which are generalized functions that encompass various special functions, and analyze their properties. By employing advanced calculus techniques, we derive new formulas for the first and higher-order derivatives of I-functions with respect to their variables; we establish some derivative formulae of the I-function of two variables involving generalized M-series. The special cases of our derivatives yield interesting results.

Keywords: I-function, Mellin-Barners control integral, generalized M-series, higher order derivative

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Authors: Nadereh Sohrabi Shegefti, Siamak Samani

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Level of adjustment is the first index to check mental health. The aim of this study was developing a valid and reliable Multidimensional Adjustment Scale (MAS). The sample consisted of 150 college students. Multidimensional adjustment scale and Depression, Anxiety, and stress scale (DASS) were used in this study. Principle factor analysis, Pearson correlation coefficient, and Cornbach's Alpha were used to check the validity and reliability of the MAS. Principle component factor analysis showed a 5 factor solution for the MAS. Alpha coefficients for the MAS sub scales were ranged between .69 to .83. Test-retest reliability for MAS was .88 and the mean of sub scales- total score correlation was .88. All these indexes revealed an acceptable reliability and validity for the MAS. The MAS is a short assessment instrument with good acceptable psychometric properties to use in clinical filed.

Keywords: psychological adjustment, psychometric properties, validity, Pearson correlation

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Authors: Melike Aydoğan, Dürdane Öztürk

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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).

Keywords: starlike log-harmonic functions, univalent functions, distortion theorem

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Authors: Alicia C. Sanchez

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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.

Keywords: automatic navigation control, lateral control, lane-keeping control, RAFU approximation

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Authors: Yang Xiong, Hua Cheng

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In this paper, for the basic problem of wireless sensor network topology control and deployment, the Roesser model in rectangular grid sensor networks is presented. In addition, a general constructive realization procedure will be proposed. The procedure enables a distributed implementation of linear systems on a sensor network. A non-trivial example is illustrated.

Keywords: grid sensor networks, Roesser model, state-space realization, multidimensional systems

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Authors: Rashidah Omar, Suzeini Abdul Halim, Aini Janteng

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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.

Keywords: analytic functions, bi-univalent functions, Hohlov operator, subordination

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Authors: İbrahim Aktaş, Árpád Baricz

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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.

Keywords: bessel function, lommel function, radius of starlikeness and convexity, Struve function

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Authors: Tulin Coskun

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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.

Keywords: analytic functions, approximation of analytic functions, Linear k-positive operators, Korovkin type theorems

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Authors: Ocen Walter Onen

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In 2018, the World Bank reported that approximately 8.6% of the global population was languishing in multidimensional poverty. This reality both challenged and motivated the research on the topic above. This research critically examined the validity of integrating the concept of servant leadership into the discourse of poverty eradication. The researcher applied documentary research methodology. Therefore, relevant literature, both printed and electronic, was analyzed, and desired data were obtained to enrich the discussion. The main finding from the research shows that; the concept of ‘servant leadership’, despite being paradoxical in nature, has the necessary potential to accelerate the effort of eliminating multidimensional poverty in any given context. Based on that, the researcher recommended that; state-actors, multi-national corporations, development organizations such as the United Nations, and other agencies working to make poverty history in our generation should both prioritize and promotes the integration of the concept of servant leadership in their policies’ formation, organizational leadership and management, and project design, implementation and evaluation of poverty-eradication initiatives.

Keywords: multidimensional poverty, poverty eradication, servant leadership, United Nations

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Authors: Chahid K. Ghaddar

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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.

Keywords: calculus, differential algebraic equations, solvers, spreadsheet

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Authors: Luiz F. J. Maia

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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.

Keywords: control surface, fuzzy control, Inverted pendulum, pertinence functions

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Authors: Mohammad Mehdi Mazarei, Ali Asghar Behroozpoor

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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.

Keywords: general derivative, linear programming, optimization problem, smooth and nonsmooth functions

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Authors: Aneri Mehta, Krunal Mehta

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The unpredictable and vacillating relationship between India and Pakistan since days of Independence struggle is known world over. And this instability has never lost its magnitude to decrease the tensions between the two countries. Since India aspires to run for the race of future superpower and Pakistan struggles to remove the tag of a highly fickle and under developed economy; ruined largely not by the outsiders, but its own people and systems; it becomes really important to gauge what steps would these neighbors take in years to come. The progress and stability of both countries heavily relies on the favorable equations between the two nations. Therefore the paper tries to trace some roots of their faltering relationship and attempts to predict their future in a multidimensional perspective.

Keywords: economy, faltering relationship, multidimensional perspective, international relations

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

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It is known that by introducing alternative forms of exponential and logarithmic functions, the Tsallis entropy Sᵩ is itself a generalization of Shannon entropy S. In this work, from a deformation through a scaling function applied to the differential operator, it is possible to generate a q-deformed calculus as well as a q-deformed arithmetic, which not only allows generalizing the exponential and logarithmic functions but also any other standard function. The updated q-deformed differential operator leads to an updated integral operator under which the functions are integrated together with a weight function. For each differentiable function, it is possible to identify its q-deformed partner, which is useful to generalize other algebraic relations proper of the original functions. As an application of this proposal, in this work, a generalization of exponential and logarithmic functions is studied in such a way that their relationship with the thermodynamic functions, particularly the entropy, allows us to have a q-deformed expression of these. As a result, from a particular scaling function applied to the differential operator, a q-deformed arithmetic is obtained, leading to the generalization of the Tsallis entropy.

Keywords: q-calculus, q-deformed arithmetic, entropy, exponential functions, thermodynamic functions

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Authors: Mary Banke Iyabo Omoniyi

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Children in the developing countries have complex social, economic, political and environmental contexts that create a wide range of challenges for school children to surmount as they journey through school from childhood to adolescent. Many of these children have little or no personal resources and social support to confront these challenges. This study employed a descriptive research design of survey type to investigate the psychological sense of school membership and coping skills as they relate to the multidimensional life satisfaction of the school children. The sample consists of 835 school children with the age range of 7-11 years who were randomly selected from twenty schools in Ondo state, Nigeria. The instrument for data collection was a questionnaire consisting of 4 sections A, B, C and D. Section A contained items on the children’s bio-data (Age, School, father’s and mother’s educational qualifications), section B is the Multidimensional Children Life Satisfaction Questionnaire (MCLSQ) with a 20 item Likert type scale. The response format range from Never= 1 to Almost always =4. The (MCLSQ) was designed to provide profile of children satisfaction with important domains of (school, family and friends). Section C is the Psychological Sense of School Membership Questionnaire (PSSMQ) with 18 items having response format ranging from Not at true=1 to completely true=5. While section D is the Self-Report Coping Questionnaire (SRCQ) which has 16 items with response ranging from Never =1 to Always=5. The instrument has a test-retest reliability coefficient of r = 0.87 while the sectional reliabilities for MCLSQ, PSSMQ and SRCQ are 0.86, 0.92 and 0.89 respectively. The results indicated that self-report coping skill was significantly correlated with multidimensional life satisfaction (r=592;p<0.05). However, the correlation between multidimensional life satisfaction and psychological sense of school membership was not significant (r=0.038;p>0.05). The regression analysis indicated that the contribution of mother’s education and father’s education to psychological sense of school member of the children were 0.923, Adjusted R2 is 0.440 and 0.730 and Adjusted R2 is 0.446. The results also indicate that contribution of gender to psychological sense of school for male and female has R= 0.782, Adjusted R2 = 0.478 and R = 0.998, Adjusted R2 i= 0.932 respectively. In conclusion, mother’s education qualification was found to contribute more to children psychological sense of membership and multidimensional life satisfaction than father’s. The girl child was also found to have more sense of belonging to the school setting than boy child. The counselling implications and recommendations among others were geared towards positive emotional gender sensitivity with regards to the male folk. Education stakeholders are also encouraged to make the school environment more conducive and gender friendly.

Keywords: multidimensional life satisfaction, psychological sense of school, coping skills, counselling implications

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Authors: Tamunotonye Mayowa Braide, Isaac Oluwatayo

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This study investigates multidimensional poverty, and its correlates among rural households in Sekhukhune and Capricorn District municipalities (SDM & CDM) in Limpopo Province, South Africa. Primary data were collected from 407 rural households selected through purposive and simple random sampling techniques. Analytical techniques employed include descriptive statistics, principal component analysis (PCA), and the Alkire Foster (A-F) methodology. The results of the descriptive statistics showed there are more females (66%) than males (34%) in rural areas of Limpopo Province, with about 45% of them having secondary school education as the highest educational level attained and only about 3% do not have formal education. In the analysis of deprivation, eight dimensions of deprivation, constructed from 21 variables, were identified using the PCA. These dimensions include type and condition of dwelling water and sanitation, educational attainment and income, type of fuel for cooking and heating, access to clothing and cell phone, assets and fuel for light, health condition, crowding, and child health. In identifying the poor with poverty cut-off (0.13) of all indicators, about 75.9% of the rural households are deprived in 25% of the total dimensions, with the adjusted headcount ratio (M0) being 0.19. Multidimensional poverty estimates showed higher estimates of poor rural households with 71%, compared to 29%, which fall below the income poverty line. The study conducted poverty decomposition, using sub-groups within the area by examining regions and household characteristics. In SDM, there are more multidimensionally poor households than in CDM. The water and sanitation dimension is the largest contributor to the multidimensional poverty index (MPI) in rural areas of Limpopo Province. The findings can, therefore, assist in better design of welfare policy and target poverty alleviation programs and as well help in efficient resource allocation at the provincial and local municipality levels.

Keywords: Alkire-Foster methodology, Limpopo province, multidimensional poverty, principal component analysis, South Africa

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Authors: Khosrow Maleknejad, Yaser Rostami

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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.

Keywords: ıntegro-differential equations, quartic B-spline wavelet, operational matrices, dual functions

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Authors: Richa Mishra, Sonia Munjal

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In this increasingly hectic and competitive climate, employees are expected to manage the resources available to them to perform their work. However, many are wasting the most precious and scarce resource at their disposal, time, by procrastinating on tasks and thereby costing themselves and their organizations. As timely performance is a requirement of most jobs, procrastination is particularly problematic in the workplace. Evidence suggests that procrastination and poor performance go hand-in-hand, as procrastinators miss more deadlines than non-procrastinators and make more errors and work at a slower speed than non-procrastinators when performing timed tasks. This research is hence an effort to add a little in the sparse knowledge base. It is an effort to throw light on the relationship of Levenson’s multi dimensions of locus of control and also an effort to identify if it is one of the causes and of employees procrastination which have not been explored earlier. The study also explores the effect and relationship of multidimensional locus of control and various levels of stress on procrastination. The results of the research have ascertained that there is significant impact of LOC dimensions on the procrastinating behavior of the employees. One of the major findings to emerge from the current research that managers with powerful others as their LOC dimensions were least procrastinating, contradicts the previous research results that external procrastinate more than internals.

Keywords: Multidimensional Locus of Control, workplace procrastination, employee behaviour, manufacturing industry

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Authors: Fernando Maass, Pablo Martin, Jorge Olivares

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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.

Keywords: analytic approximations, asymptotic approximations, Bessel functions, quasirational approximations

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