The Number of Real Zeros of the Single Parameter Mittag-Leffler Function for Parameter Values Between 1 and 2

2005 ◽  
Author(s):  
John W. Hanneken ◽  
David M. Vaught ◽  
B. N. Narahari Achar
Keyword(s):  
Fractional Calculus ◽  
Finite Number ◽  
Exponential Function ◽  
Significant Role ◽  
Single Parameter ◽  
Real Zeros ◽  
Iteration Formula ◽  
Physical Problems ◽  
Number Of Real Zeros ◽  
Parameter Values

The single parameter Mittag-Leffler function, which is a generalization of the exponential function, occurs naturally in the solution of physical problems involving fractional calculus and its zeros play a significant role in the dynamic solutions. It is known that the Mittag-Leffler function has a finite number of real zeros in the range of parameter values between 1 and 2, which is quite relevant for many physical problems. However, the number of real zeros for a given parameter value in this range is not known. An iteration formula for calculating the number of real zeros of the Mittag-Leffler function for any value of the parameter in the range between 1 and 2 is presented.

scholarly journals Some Applications of the Wright Function in Continuum Physics: A Survey

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 198
Author(s):  
Yuriy Povstenko
Keyword(s):  
Heat Conduction ◽  
Fractional Calculus ◽  
Exponential Function ◽  
Nonlocal Elasticity ◽  
Bessel Functions ◽  
Wright Function ◽  
Mainardi Function ◽  
Integral Relations

The Wright function is a generalization of the exponential function and the Bessel functions. Integral relations between the Mittag–Leffler functions and the Wright function are presented. The applications of the Wright function and the Mainardi function to description of diffusion, heat conduction, thermal and diffusive stresses, and nonlocal elasticity in the framework of fractional calculus are discussed.

scholarly journals Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mina Ketan Mahanti ◽  
Amandeep Singh ◽  
Lokanath Sahoo
Keyword(s):  
Random Variables ◽  
Random Coefficients ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Gaussian Random Variables ◽  
Hyperbolic Polynomials ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the formy=Pnt=n1a1cosh⁡t+n2a2cosh⁡2t+⋯+nnancosh⁡nt, wherea1,…,anis a sequence of standard Gaussian random variables, isn/2+op(1). It is shown that the asymptotic value of expected number of times the polynomial crosses the levely=Kis alson/2as long asKdoes not exceed2neμ(n), whereμ(n)=o(n). The number of oscillations ofPn(t)abouty=Kwill be less thann/2asymptotically only ifK=2neμ(n), whereμ(n)=O(n)orn-1μ(n)→∞. In the former case the number of oscillations continues to be a fraction ofnand decreases with the increase in value ofμ(n). In the latter case, the number of oscillations reduces toop(n)and almost no trace of the curve is expected to be present above the levely=Kifμ(n)/(nlogn)→∞.

scholarly journals Analysis of the spatial variation in the parameters of the SWAT model with application in Flanders, Northern Belgium

2004 ◽  
Vol 8 (5) ◽  
pp. 931-939 ◽  
Author(s):  
G. Heuvelmans ◽  
B. Muys ◽  
J. Feyen
Keyword(s):  
Stream Flow ◽  
Hydrological Model ◽  
Swat Model ◽  
Single Parameter ◽  
Model Parameters ◽  
Study Region ◽  
Entire Study ◽  
Time Periods ◽  
Parameter Values ◽  
Parameter Approach

Abstract. Operational applications of a hydrological model often require the prediction of stream flow in (future) time periods without stream flow observations or in ungauged catchments. Data for a case-specific optimisation of model parameters are not available for such applications, so parameters have to be derived from other catchments or time periods. It has been demonstrated that for applications of the SWAT in Northern Belgium, temporal transfers of the parameters have less influence than spatial transfers on the performance of the model. This study examines the spatial variation in parameter optima in more detail. The aim was to delineate zones wherein model parameters can be transferred without a significant loss of model performance. SWAT was calibrated for 25 catchments that are part of eight larger sub-basins of the Scheldt river basin. Two approaches are discussed for grouping these units in zones with a uniform set of parameters: a single parameter approach considering each parameter separately and a parameter set approach evaluating the parameterisation as a whole. For every catchment, the SWAT model was run with the local parameter optima, with the average parameter values for the entire study region (Flanders), with the zones delineated with the single parameter approach and with the zones obtained by the parameter set approach. Comparison of the model performances of these four parameterisation strategies indicates that both the single parameter and the parameter set zones lead to stream flow predictions that are more accurate than if the entire study region were treated as one single zone. On the other hand, the use of zonal average parameter values results in a considerably worse model fit compared to local parameter optima. Clustering of parameter sets gives a more accurate result than the single parameter approach and is, therefore, the preferred technique for use in the parameterisation of ungauged sub-catchments as part of the simulation of a large river basin. Keywords: hydrological model, regionalisation, parameterisation, spatial variability

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