scholarly journals An Explicit Form of Heaviside Step Function

2021 ◽  
Author(s):  
John Venetis
Keyword(s):  
Explicit Form ◽  
Special Function ◽  
Special Functions ◽  
Error Function ◽  
Beta Function ◽  
Operational Calculus ◽  
Step Function ◽  
Heaviside Step Function ◽  
Computational Procedures ◽  
Engineering Practices

In this paper, the author derives an explicit form of Heaviside Step Function, which evidently constitutes a fundamental concept of Operational Calculus and is also involved in many other fields of applied and engineering mathematics.In particular, this special function is exhibited in a very simple manner as a summation of four inverse tangent functions. The novelty of this work is that the proposed exact formulae are not performed in terms of miscellaneous special functions, e.g. Bessel functions, Error function, Beta function etc and also are neither the limit of a function, nor the limit of a sequence of functions with point – wise or uniform convergence.Hence, this formula may be much more appropriate and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

scholarly journals An Analytic Exact form of Heaviside Function

2019 ◽  
Author(s):  
John Venetis
Keyword(s):  
Special Functions ◽  
Error Function ◽  
Operational Calculus ◽  
Delta Function ◽  
Step Function ◽  
Heaviside Function ◽  
Exact Form ◽  
Dirac Delta ◽  
Computational Procedures ◽  
Engineering Practices

In this paper, the author obtains an analytic exact form of Heaviside function, which is also known as Unit Step function and constitutes a fundamental concept of the Operational Calculus.In particulat, this function is explicitly expressed in a very simple manner by the aid of purely algebraic representations. The novelty of this work is that the proposed explicit formula is not performed in terms of non – elementary special functions, e.g. Dirac delta function or Error function and also is neither the limit of a function, nor the limit of a sequence of functions with point wise or uniform convergence. Hence, it may be much more appropriate and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

A Meshless Local Petrov-Garlerkin Method for Solving the Biharmonic Equation

2014 ◽  
Vol 931-932 ◽  
pp. 1488-1494
Author(s):  
Supanut Kaewumpai ◽  
Suwon Tangmanee ◽  
Anirut Luadsong
Keyword(s):  
Relative Error ◽  
Biharmonic Equation ◽  
Interpolation Method ◽  
Step Function ◽  
Special Treatment ◽  
Maximum Relative Error ◽  
Heaviside Step Function ◽  
Treatment Techniques ◽  
Point Interpolation Method ◽  
Point Interpolation

A meshless local Petrov-Galerkin method (MLPG) using Heaviside step function as a test function for solving the biharmonic equation with subjected to boundary of the second kind is presented in this paper. Nodal shape function is constructed by the radial point interpolation method (RPIM) which holds the Kroneckers delta property. Two-field variables local weak forms are used in order to decompose the biharmonic equation into a couple of Poisson equations as well as impose straightforward boundary of the second kind, and no special treatment techniques are required. Selected engineering numerical examples using conventional nodal arrangement as well as polynomial basis choices are considered to demonstrate the applicability, the easiness, and the accuracy of the proposed method. This robust method gives quite accurate numerical results, implementing by maximum relative error and root mean square relative error.

scholarly journals Originals of operating images for generalized problems of unsteady heat conductivity

2019 ◽  
Vol 14 (4) ◽  
pp. 77-86 ◽  
Author(s):  
E. M. Kartashov
Keyword(s):  
Mathematical Models ◽  
Bessel Functions ◽  
Applied Mathematics ◽  
Operational Calculus ◽  
Polymeric Materials ◽  
Thermal Reaction ◽  
Heaviside Step Function ◽  
External Influences ◽  
Wide Range ◽  
Variable Function

A series of operating (Laplace) non-standard images, the originals of which are absent in well-known reference books on operational calculus, are considered. By reducing one of the basic images to the Riemann-Mellin contour integral for the modified Bessel functions and analyzing the corresponding inversion formula using the approaches of the complex variable function theory, an analytical form of the original original is found, which is abrupt in nature with a break point. It is shown that analytical solutions of the corresponding mathematical models using the found originals have a wave character, which is expressed by the presence of the Heaviside step function in the solutions. The latter means that at any time there is a region of physical disturbance to the point of discontinuity and an unperturbed area after the point of discontinuity. The images studied are included in the operational solutions of mathematical models in many areas of applied mathematics. physics, thermomechanics, thermal physics, in particular in the theory of thermal shock of viscoelastic bodies, in the study of the thermal reaction of solids based on the classical Maxwell-Cattaneo-Lykov-Vernott phenomenology, taking into account the final rate of heat propagation. These models are needed to study the thermal reaction of relatively new consolidated structurally sensitive polymeric materials in structures exposed to high-intensity external influences. The analytical relations obtained for the originals and the original improper integrals resulting from them, containing combinations of Bessel functions, can be used in the general methodology of constructing and applying various mathematical models in a wide range of external influences on materials in many fields of science and technology.

scholarly journals Definite Integral of Algebraic, Exponential and Hyperbolic Functions Expressed in Terms of Special Functions

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1425
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Function ◽  
Special Functions ◽  
Complex Numbers ◽  
Definite Integral ◽  
Hyperbolic Functions ◽  
Closed Form Solutions ◽  
Definite Integrals ◽  
Arbitrary Complex ◽  
Famous Book

While browsing through the famous book of Bierens de Haan, we came across a table with some very interesting integrals. These integrals also appeared in the book of Gradshteyn and Ryzhik. Derivation of these integrals are not listed in the current literature to best of our knowledge. The derivation of such integrals in the book of Gradshteyn and Ryzhik in terms of closed form solutions is pertinent. We evaluate several of these definite integrals of the form ∫0∞(a+y)k−(a−y)keby−1dy, ∫0∞(a+y)k−(a−y)keby+1dy, ∫0∞(a+y)k−(a−y)ksinh(by)dy and ∫0∞(a+y)k+(a−y)kcosh(by)dy in terms of a special function where k, a and b are arbitrary complex numbers.

The Mesoscopic Investigation on Moduli Prediction in Visco-Elastic Particle Reinforced Composites

2008 ◽  
Vol 385-387 ◽  
pp. 329-332
Author(s):  
Xue Zhong Ding ◽  
Li Qiang Tang
Keyword(s):  
Volume Fraction ◽  
Step Function ◽  
Elastic Behavior ◽  
Loading Path ◽  
Elastic Model ◽  
Reinforced Composites ◽  
Heaviside Step Function ◽  
Equivalent Inclusion ◽  
Inclusion Theory ◽  
Elastic Particle

The visco-elastic mechanism of particles reinforced composites has been investigated through revised Eshelby equivalent inclusion theory. A visco-elastic model is applied. Furthermore, by introducing Heaviside step function and Laplace transform, the creep constitutional equation related to strain rate effect is achieved. Finally, by equivalent inclusion theory, introducing secant modulus, the material moduli with time and volume fraction concerning Glass/ED6 particles reinforced materials have been given. The results show that the visco-elastic property of composite material is mainly determined by the visco-elastic behavior of the matrix, which meet experiment results well. It can be concluded from the results that there exits close relationship between the inclusion shape, volume fraction and loading path.

scholarly journals An Asymptotic Approach for the Elastodynamic Problem of a Plate under Impact Loading

2010 ◽  
Vol 2010 ◽  
pp. 1-10
Author(s):  
Penelope Michalopoulou ◽  
George A. Papadopoulos
Keyword(s):  
Laplace Transform ◽  
Impact Loading ◽  
Step Function ◽  
Infinite Strip ◽  
Asymptotic Approach ◽  
Heaviside Step Function ◽  
Inertia Effects ◽  
Elastodynamic Problem ◽  
Line Force ◽  
The One

An approach is presented for analyzing the transient elastodynamic problem of a plate under an impact loading. The plate is considered to be in the form of a long strip under plane strain conditions. The loading is taken as a concentrated line force applied normal to the plate surface. It is assumed that this line force is suddenly applied and maintained thereafter (i.e., it is a Heaviside step function of time). Inertia effects are taken into consideration and the problem is treated exactly within the framework of elastodynamic theory. The approach is based on multiple Laplace transforms and on certain asymptotic arguments. In particular, the one-sided Laplace transform is applied to suppress time dependence and the two-sided Laplace transform to suppress the dependence upon a spatial variable (along the extent of the infinite strip). Exact inversions are then followed by invoking the asymptotic Tauber theorem and the Cagniard-deHoop technique. Various extensions of this basic analysis are also discussed.

scholarly journals Characterization and Mitigation of Model Bias in Parametric Mapping of Dopamine Response to Behavioral Challenge

2021 ◽  
Author(s):  
Michael A Levine ◽  
Joseph B Mandeville ◽  
Finnegan Calabro ◽  
David Izquierdo-Garcia ◽  
Julie C Price ◽  
...  
Keyword(s):  
Nucleus Accumbens ◽  
Step Function ◽  
Binding Potential ◽  
Reference Tissue ◽  
Heaviside Step Function ◽  
Average Bias ◽  
Adult Participants ◽  
Single Target ◽  
First Pass ◽  
Initial Uptake

Compartmental modeling of 11C-raclopride (RAC) is commonly used to measure dopamine response to intra-scan behavioral tasks. Bias in estimates of binding potential (BPND) and its dynamic changes (ΔBPND) can arise when the selected compartmental model deviates from the underlying biology. In this work, we characterize the bias associated with assuming a single target compartment and propose a model for reducing this bias by selectively discounting the contribution of the initial uptake period. Methods: 69 healthy young adult participants were scanned using RAC PET/MR while simultaneously performing a rewarded behavioral task. BPND and ΔBPND were estimated using an extension of the Multilinear Reference Tissue Model (MRTM2) with the task challenge encoded as a Heaviside step function. Bias was estimated using simulations designed to match the acquired data and was reduced by introducing a new model (DE-MRTM2) that reduces the biasing influence of the initial uptake period in the modeled estimation of BPND for both simulations and participant data. Results: Bias in ΔBPND was observed to vary both spatially with BPND and with the assumed value of k4. At the most likely value of k4 (0.13 min-1), the average bias and the maximum voxel bias magnitude in the nucleus accumbens were estimated to be 1.2% and 3.9% respectively. Simulations estimated that debiasing the contribution of the first 27 minutes of acquired data reduced average bias and maximum voxel bias in the nucleus accumbens ΔBPND to -0.3% and 2.4% respectively. In the acquired participant data, DE-MRTM2 produced modest changes in the experimental estimates of striatal ΔBPND, while extrastriatal bias patterns were greatly reduced. DE-MRTM2 also considerably reduced the dependence of ΔBPND upon the first-pass selection of k2'. Conclusion: Selectively discounting the contribution of the initial uptake period can help mitigate BPND- and k4-dependent bias in single compartment models of ΔBPND, while also reducing the dependence of ΔBPND on the first-pass estimation of k2'.

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