Some new bounds for Sinc function by simultaneous approximation of the base and exponential functions

The present paper deals with the approximation properties for exponential functions of general Durrmeyer type operators having the weights of Sz?sz basis functions. Here we give explicit expressions for exponential type moments by means of which we establish, for the derivatives of the operators, the Voronovskaja formulas for functions of exponential growth and the corresponding weighted quantitative estimates for the remainder in simultaneous approximation.

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Convergence estimates of a family of approximation operators of exponential type

Filomat ◽

10.2298/fil2013329g ◽

2020 ◽

Vol 34 (13) ◽

pp. 4329-4341

Author(s):

Vijay Gupta ◽

Manuel López-Pellicer ◽

H.M. Srivastava

Keyword(s):

Asymptotic Formula ◽

Modulus Of Continuity ◽

Exponential Type ◽

Exponential Growth ◽

Simultaneous Approximation ◽

Type Result ◽

Exponential Functions ◽

Approximation Operators ◽

Convergence Estimates

The main object of this article is to consider a family of approximation operators of exponential type, which has presumably not been studied earlier due mainly to their seemingly complicated behavior. We estimate and establish a quantitative asymptotic formula in terms of the modulus of continuity with exponential growth, a Korovkin-type result for exponential functions and also a Voronovskaja-type asymptotic formula in the simultaneous approximation.

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Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽

10.2298/fil1809347k ◽

2018 ◽

Vol 32 (9) ◽

pp. 3347-3354 ◽

Cited By ~ 1

Author(s):

Nematollah Kadkhoda ◽

Michal Feckan ◽

Yasser Khalili

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Exact Solutions ◽

Present Article ◽

Analytical Solutions ◽

Nonlinear Differential Equations ◽

Nonlinear Partial Differential Equations ◽

Rational Functions ◽

Expansion Method ◽

Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

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Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

Open Mathematics ◽

10.1515/math-2020-0021 ◽

2020 ◽

Vol 18 (1) ◽

pp. 353-377 ◽

Cited By ~ 1

Author(s):

Zhien Li ◽

Chao Wang

Keyword(s):

Time Scales ◽

Quaternion Algebra ◽

Matrix Exponential ◽

Dynamic Equations ◽

Necessary Condition ◽

Quaternion Matrix ◽

Exponential Functions ◽

Cauchy Matrix ◽

Adjoint Matrix ◽

Sufficient And Necessary Condition

Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

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Using GEDI Waveforms for Improved TanDEM-X Forest Height Mapping: A Combined SINC + Legendre Approach

Remote Sensing ◽

10.3390/rs13152882 ◽

2021 ◽

Vol 13 (15) ◽

pp. 2882

Author(s):

Hao Chen ◽

Shane R. Cloude ◽

Joanne C. White

Keyword(s):

High Resolution ◽

Forest Canopy ◽

Test Site ◽

Convergent Series ◽

Canopy Height ◽

Ecosystem Dynamics ◽

Sinc Function ◽

Vertical Profiles ◽

Legendre Series ◽

Vertical Profiling

In this paper, we consider a new method for forest canopy height estimation using TanDEM-X single-pass radar interferometry. We exploit available information from sample-based, space-borne LiDAR systems, such as the Global Ecosystem Dynamics Investigation (GEDI) sensor, which offers high-resolution vertical profiling of forest canopies. To respond to this, we have developed a new extended Fourier-Legendre series approach for fusing high-resolution (but sparsely spatially sampled) GEDI LiDAR waveforms with TanDEM-X radar interferometric data to improve wide-area and wall-to-wall estimation of forest canopy height. Our key methodological development is a fusion of the standard uniform assumption for the vertical structure function (the SINC function) with LiDAR vertical profiles using a Fourier-Legendre approach, which produces a convergent series of approximations of the LiDAR profiles matched to the interferometric baseline. Our results showed that in our test site, the Petawawa Research Forest, the SINC function is more accurate in areas with shorter canopy heights (<~27 m). In taller forests, the SINC approach underestimates forest canopy height, whereas the Legendre approach avails upon simulated GEDI forest structural vertical profiles to overcome SINC underestimation issues. Overall, the SINC + Legendre approach improved canopy height estimates (RMSE = 1.29 m) compared to the SINC approach (RMSE = 4.1 m).

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Simultaneous Contour Method of a Trigonometric Integral by Prudnikov et al.

Mathematics ◽

10.3390/math9121453 ◽

2021 ◽

Vol 9 (12) ◽

pp. 1453

Author(s):

Robert Reynolds ◽

Allan Stauffer

Keyword(s):

Closed Form ◽

Contour Method ◽

Substantial Part ◽

Exponential Functions ◽

Closed Form Solutions ◽

Definite Integrals ◽

Transcendental Functions ◽

Trigonometric Integral

In this present work we derive, evaluate and produce a table of definite integrals involving logarithmic and exponential functions. Some of the closed form solutions derived are expressed in terms of elementary or transcendental functions. A substantial part of this work is new.

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A Class of Integral Operators that Fix Exponential Functions

Mediterranean Journal of Mathematics ◽

10.1007/s00009-021-01819-0 ◽

2021 ◽

Vol 18 (5) ◽

Author(s):

Carlo Bardaro ◽

Ilaria Mantellini ◽

Gumrah Uysal ◽

Basar Yilmaz

Keyword(s):

General Class ◽

Pointwise Convergence ◽

Type Theorem ◽

Integral Operators ◽

Exponential Functions ◽

Convergence Theorems ◽

Korovkin Type Theorem ◽

Type Formula

AbstractIn this paper we introduce a general class of integral operators that fix exponential functions, containing several recent modified operators of Gauss–Weierstrass, or Picard or moment type operators. Pointwise convergence theorems are studied, using a Korovkin-type theorem and a Voronovskaja-type formula is obtained.

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Energy Minimization Scheme for Split Potential Systems Using Exponential Variational Integrators

Applied Mechanics ◽

10.3390/applmech2030024 ◽

2021 ◽

Vol 2 (3) ◽

pp. 431-441

Author(s):

Odysseas Kosmas

Keyword(s):

Numerical Solutions ◽

Oscillatory Behavior ◽

Lagrangian Function ◽

Computational Time ◽

Variational Integrators ◽

Weighted Sum ◽

Lagrange Equations ◽

Exponential Functions ◽

Intermediate Time ◽

Definition Of

In previous works we developed a methodology of deriving variational integrators to provide numerical solutions of systems having oscillatory behavior. These schemes use exponential functions to approximate the intermediate configurations and velocities, which are then placed into the discrete Lagrangian function characterizing the physical system. We afterwards proved that, higher order schemes can be obtained through the corresponding discrete Euler–Lagrange equations and the definition of a weighted sum of “continuous intermediate Lagrangians” each of them evaluated at an intermediate time node. In the present article, we extend these methods so as to include Lagrangians of split potential systems, namely, to address cases when the potential function can be decomposed into several components. Rather than using many intermediate points for the complete Lagrangian, in this work we introduce different numbers of intermediate points, resulting within the context of various reliable quadrature rules, for the various potentials. Finally, we assess the accuracy, convergence and computational time of the proposed technique by testing and comparing them with well known standards.

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An elastic phenomenological material law of technical textile with a nonlinear shear behaviour

Journal of Reinforced Plastics and Composites ◽

10.1177/07316844211005842 ◽

2021 ◽

pp. 073168442110058

Author(s):

Dániel T Karádi ◽

András A Sipos ◽

Marianna Halász ◽

Viktor Hliva ◽

Dezső Hegyi

Keyword(s):

Service Level ◽

Shear Behaviour ◽

Nonlinear Behaviour ◽

Exponential Functions ◽

Normal Stiffness ◽

Material Parameters ◽

Fitting Method ◽

Highly Nonlinear ◽

Nonlinear Shear ◽

Magnitude Difference

In technical textile engineering, macro-level phenomenological modelling effectively describes the material’s highly nonlinear behaviour. However, existing material laws concentrate on the normal stiffness in the orthotropic yarns and simplify the shear effect because of the two orders of magnitude difference between shear and normal stiffness. This article introduces an improved phenomenological model that includes nonlinear shear behaviour, and it determines the material parameters with a previously applied data fitting method for exponential functions. The nonlinear shear behaviour is valid for the elastic state, that is, at the service level of the loads. Time-dependent, cyclic loading or plastic behaviour is not considered.

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