scholarly journals The use of fractional calculus for the optimal placement of electronic components on a linear array

2015 ◽  
Vol 28 (1) ◽  
pp. 77-84
Author(s):  
Mey de ◽  
Mariusz Felczak ◽  
Bogusław Więcek
Keyword(s):  
Integral Equation ◽  
Fractional Calculus ◽  
Forced Convection ◽  
Linear Array ◽  
The Other ◽  
Simple Solution ◽  
Optimal Placement ◽  
Critical Component ◽  
Electronic Components ◽  
One Dimensional

Cooling of heat dissipating components has become an important topic in the last decades. Sometimes a simple solution is possible, such as placing the critical component closer to the fan outlet. On the other hand this component will heat the air which has to cool the other components further away from the fan outlet. If a substrate bearing a one dimensional array of heat dissipating components, is cooled by forced convection only, an integral equation relating temperature and power is obtained. The forced convection will be modelled by a simple analytical wake function. It will be demonstrated that the integral equation can be solved analytically using fractional calculus.

scholarly journals Solution of Abel’s Integral Equation Using Tikhonov Regularization

2005 ◽  
Author(s):  
K. J. Daun ◽  
K. A. Thomson ◽  
F. Liu ◽  
G. J. Smallwood
Keyword(s):  
Integral Equation ◽  
Field Distribution ◽  
Tikhonov Regularization ◽  
Measurement Errors ◽  
Field Variable ◽  
The Other ◽  
One Dimensional ◽  
Variable Distribution ◽  
Data Points

This paper presents a method based on Tikhonov regularization for solving one-dimensional inverse tomography problems that arise in combustion applications. In this technique, Tikhonov regularization transforms the ill-conditioned set of equations generated by onion-peeling deconvolution into a well-conditioned set that is more stable to measurement errors that arise in experimental settings. The performance of this method is compared to that of onion-peeling and Abel three-point deconvolution by solving for a known field variable distribution from projected data contaminated with artificially-generated error. The results show that Tikhonov deconvolution provides a more accurate field distribution than onion-peeling and Abel three-point deconvolution, and is more stable than the other two methods as the distance between projected data points decreases.

The transport equation with plane symmetry and isotropic scattering

1964 ◽  
Vol 60 (4) ◽  
pp. 909-921 ◽  
Author(s):  
M. G. Smith
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Point Sources ◽  
The Other ◽  
Isotropic Scattering ◽  
Dimensional Problem ◽  
Plane Symmetry ◽  
Double Integral ◽  
One Dimensional ◽  
The One

AbstractThe double integral equation, which takes the place of the Milne equation in the one-dimensional problem, is derived from the governing partial differentio-integral equations. An analytical solution of the problem of a distribution of point sources on a plane, when the other boundaries are at infinity, is then found. The possibility of more complicated boundary conditions is discussed.

scholarly journals Two-scale mathematics and fractional calculus for thermodynamics

2019 ◽  
Vol 23 (4) ◽  
pp. 2131-2133 ◽  
Author(s):  
Ji-Huan He ◽  
Fei-Yu Ji
Keyword(s):  
Fractional Calculus ◽  
Three Dimensional ◽  
The Other ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Dimensional Approach ◽  
One Dimensional ◽  
Fractal Calculus ◽  
Lower Dimensional ◽  
The Continuum

A three dimensional problem can be approximated by either a two-dimensional or one-dimensional case, but some information will be lost. To reveal the lost information due to the lower dimensional approach, two-scale mathematics is needed. Generally one scale is established by usage where traditional calculus works, and the other scale is for revealing the lost information where the continuum assumption might be forbidden, and fractional calculus or fractal calculus has to be used. The two-scale transform can approximately convert the fractional calculus into its traditional partner, making the two-scale thermodynamics much promising.

scholarly journals Fredholm type integral equation with special functions

2018 ◽  
Vol 10 (1) ◽  
pp. 5-17 ◽  
Author(s):  
Frdric Ayant ◽  
Dinesh Kumar
Keyword(s):  
Integral Equation ◽  
Fractional Calculus ◽  
Special Functions ◽  
Fredholm Type ◽  
One Dimensional ◽  
Type Integral ◽  
The One ◽  
Dimensional Integral

Abstract Recently Chaurasia and Gill [7], Chaurasia and Kumar [8] have solved the one-dimensional integral equation of Fredholm type involving the product of special functions. We solve an integral equation involving the product of a class of multivariable polynomials, the multivariable H-function defined by Srivastava and Panda [29, 30] and the multivariable I-function defined by Prasad [21] by the application of fractional calculus theory. The results obtained here are general in nature and capable of yielding a large number of results (known and new) scattered in the literature.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

scholarly journals Assessing a Linear Nanosystem's Limiting Reliability from its Components

2008 ◽  
Vol 45 (03) ◽  
pp. 879-887 ◽  
Author(s):  
Nader Ebrahimi
Keyword(s):  
Present State ◽  
Size Range ◽  
Two Dimensions ◽  
The Other ◽  
One Dimension ◽  
Present Moment ◽  
One Dimensional ◽  
The One ◽  
Fixed Moment ◽  
Nanosized Systems

Nanosystems are devices that are in the size range of a billionth of a meter (1 x 10-9) and therefore are built necessarily from individual atoms. The one-dimensional nanosystems or linear nanosystems cover all the nanosized systems which possess one dimension that exceeds the other two dimensions, i.e. extension over one dimension is predominant over the other two dimensions. Here only two of the dimensions have to be on the nanoscale (less than 100 nanometers). In this paper we consider the structural relationship between a linear nanosystem and its atoms acting as components of the nanosystem. Using such information, we then assess the nanosystem's limiting reliability which is, of course, probabilistic in nature. We consider the linear nanosystem at a fixed moment of time, say the present moment, and we assume that the present state of the linear nanosystem depends only on the present states of its atoms.

scholarly journals Hybrid Bernstein Block-Pulse Functions Method for Second Kind Integral Equations with Convergence Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu ◽  
Fereshteh Babaei
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Convergence Analysis ◽  
Bernstein Polynomials ◽  
Linear Equations ◽  
The Other ◽  
New Combination ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Block Pulse Functions

We introduce a new combination of Bernstein polynomials (BPs) and Block-Pulse functions (BPFs) on the interval [0, 1]. These functions are suitable for finding an approximate solution of the second kind integral equation. We call this method Hybrid Bernstein Block-Pulse Functions Method (HBBPFM). This method is very simple such that an integral equation is reduced to a system of linear equations. On the other hand, convergence analysis for this method is discussed. The method is computationally very simple and attractive so that numerical examples illustrate the efficiency and accuracy of this method.

Lower Bounds to the Eigenvalues in One-Dimensional Problems by a Shift in the Weight Function

1970 ◽  
Vol 37 (2) ◽  
pp. 267-270 ◽  
Author(s):  
D. Pnueli
Keyword(s):  
Lower Bound ◽  
Weight Function ◽  
Lower Bounds ◽  
Variational Formulation ◽  
Ritz Method ◽  
The Other ◽  
One Dimensional ◽  
Systematic Shift ◽  
The One ◽  
Detailed Computation

A method is presented to obtain both upper and lower bound to eigenvalues when a variational formulation of the problem exists. The method consists of a systematic shift in the weight function. A detailed procedure is offered for one-dimensional problems, which makes improvement of the bounds possible, and which involves the same order of detailed computation as the Rayleigh-Ritz method. The main contribution of this method is that it yields the “other bound;” i.e., the one which cannot be obtained by the Rayleigh-Ritz method.

Sign in / Sign up

Export Citation Format

Share Document