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Research Papers

# An Accurate Numerical Method for Solving Unsteady Isothermal Flow of a Gas Through a Semi-Infinite Porous Medium

[+] Author and Article Information
Kourosh Parand

Department of Computer Sciences,
Shahid Beheshti University,
Tehran 19697-64166, Iran
e-mail: k_parand@sbu.ac.ir

Mehdi Delkhosh

Department of Computer Sciences,
Shahid Beheshti University,
Tehran 19697-64166, Iran
e-mail: mehdidelkhosh@yahoo.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 8, 2017; final manuscript received July 1, 2017; published online October 9, 2017. Assoc. Editor: Dan Negrut.

J. Comput. Nonlinear Dynam 13(1), 011007 (Oct 09, 2017) (9 pages) Paper No: CND-17-1021; doi: 10.1115/1.4037225 History: Received January 08, 2017; Revised July 01, 2017

## Abstract

The Kidder equation, $y″(x)+2xy′(x)/1−βy(x)=0, x∈[0,∞), β∈[0,1]$ with $y(0)=1$, and $y(∞)=0$, is a second-order nonlinear two-point boundary value ordinary differential equation (ODE) on the semi-infinite domain, with a boundary condition in the infinite that describes the unsteady isothermal flow of a gas through a semi-infinite micro–nano porous medium and has widely used in the chemical industries. In this paper, a hybrid numerical method is introduced for solving this equation. First, by using the method of quasi-linearization, the equation is converted to a sequence of linear ODEs. Then these linear ODEs are solved by using the rational Legendre functions (RLFs) collocation method. By using 200 collocation points, we obtain a very good approximation solution and the value of the initial slope $y′(0)=−1.19179064971942173412282860380015936403$ for $β=0.50$, highly accurate to 38 decimal places. The convergence of numerical results is shown by decreasing the residual errors when the number of collocation points increases.

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## References

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Parand, K. , Yousefi, H. , Delkhosh, M. , and Ghaderi, A. , 2016, “ A Novel Numerical Technique to Obtain an Accurate Solution to the Thomas-Fermi Equation,” Eur. Phys. J. Plus, 131(7), p. 228.
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## Figures

Fig. 1

Graphs of the logarithm of coefficients |ai| for various values of L, m = 200 and sixth iteration to calculate an approximation of the optimal value of L

Fig. 3

Graph of the logarithm of coefficients |ai| with m = 200 and n = 6 for showing the convergence of the method

Fig. 2

Graph of residual errors RESnm with m=100,125,150,175,200, n = 6 and β=0.50 for showing the convergence of the method

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1. Sep 7, 2016

### skyhj105

In some books, there are some terms which are ambiguous to me.
The parts which I cant understand is that "less mass axis(There are two axis in nanopositioning) frequency response for nomial load is plotted" and "with the maximum payload, the response frequency reduced to 415Hz."
What is the meaning of nominal load and payload in these sentences?

2. Sep 9, 2016

### Nidum

All those terms could make sense in the right context but as stated above they are just a jumble .

Conrol engineering applies to all sorts of systems large and small .

You mention nanopositioning but what is the actual hardware that you are dealing with ?

3. Sep 9, 2016

### skyhj105

Actual hardware what Im dealing with is commercial serial-kinematic nanopositioning stage which is composed with Noliac NAC2003 piezoelectric stack actuator, Micro 6810 capacitive sensor and 6504-01prove.
I will use this apparatus for physics research. Because It is a two-axis nanopositioning stage, it has two resonance frequencies for each axis.
Resonance frequency of the x-axis is 513Hz and the other axis has 727Hz.
Because of larger resonance frequency of x-axis than the other axis, x-axis will impose greater limitation on performance.(I don`t konw why but it is just written like this. If you know, could you let me know why it is?)
So, in what i have read, x-axis frequency response for nominal load and payload are plotted. The format of the plotted graph is a bode plot

4. Sep 10, 2016

### Tom.G

"Nominal" in this context would be either a "very light" load or the load that the device is designed to work with.
Without looking at it, the data sheet "maximum" load is probably the load beyond which no reliable position change takes place.

Something seems to be reversed in that description. The 513Hz resonant frequency of the x-axis is lower than "the other axis" at 727Hz.
In any case, the frequency response tells you how rapidly the device can respond to an input. 513Hz would be 1.9mS and 727Hz would be around 1.4mS. Depending on how they are defining it, the minimum pulse repetition time may be two to four times these values.

From the numbers you quoted, apparently the X axis stage carries the actuator for "the other axis", thus adding to the mass it must move, and it consequently has a lower mechanical resonant frequency. Think of it as a mechanical spring-mass system. For a given spring, a higher mass will yield a lower resonant frequency.

5. Sep 10, 2016

### skyhj105

First, You said that "maximum"load. Is it same with "maximum payload" that I mentioned before?

Second, frequency response is same as resonance frequency?
I said "Resonance frequency of the x-axis is 513Hz and the other axis has 727Hz" and ,In your reply, you said "the frequency response tells you how rapidly the device can respond to an input. 513Hz would be 1.9mS and 727Hz would be around 1.4mS."
The meaning of resonance frequency, in physically, is the frequency of force which can be absorbed effectively. So energy of system can be larger easily.
Why this is related with response?

6. Sep 12, 2016

### Tom.G

Yes.

I used "frequency response" as a general term, it is often displayed as a graph of response versus frequency. A resonance shows up on such a graph as either a peak or a dip in the response curve. Just as in a usual mechanical system, you usually need to stay below the resonant frequency to have effective control of the action. For instance with a spring and mass system, if you apply energy pulses above the resonant frequency, some of the pulses will occur while the mass is rebounding and energy is wasted fighting the kinetic energy (inertia). Also, some of the energy pulses may occur when the mass is away from the input point and be wasted.

In your nanopositioning stage, it takes a finite time for the actuator to actually move the stage to a new position (F=MA or A=F/M). If the input pulse stops before the movement is complete, you will not get the expected amount of movement. It also takes time for the actuator to return to its rest position after an input pulse. If the next pulse occurs too soon, you will again get either a smaller than expected movement of no movement at all.


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# Problems and Theorems in Classical Set Theory

'Problem Books in Mathematics'. Auflage 2006. Sprache: Englisch.
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This is the first comprehensive collection of problems in set theory. It contains well chosen sequences of exercises. Most of the problems are challenging and require work, wit, and inspiration. The book is destined to become a classic.
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Titel: Problems and Theorems in Classical Set Theory
Autor/en: Peter Komjath, Vilmos Totik

ISBN: 038730293X
EAN: 9780387302935
'Problem Books in Mathematics'.
Auflage 2006.
Sprache: Englisch.
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2. Mai 2006 - gebunden - 528 Seiten

## Beschreibung

This volume contains a variety of problems from classical set theory and represents the first comprehensive collection of such problems. Many of these problems are also related to other fields of mathematics, including algebra, combinatorics, topology and real analysis. Rather than using drill exercises, most problems are challenging and require work, wit, and inspiration. They vary in difficulty, and are organized in such a way that earlier problems help in the solution of later ones. For many of the problems, the authors also trace the history of the problems and then provide proper reference at the end of the solution.

## Inhaltsverzeichnis

Problems.
Operations on sets.
Countability.
Equivalence.
Continuum.
Sets of reals and real functions.
Ordered sets.
Order types.
Ordinals.
Ordinal arithmetic.
Cardinals.
Partially ordered sets.
Transfinite enumeration.
Euclidean spaces.
Zorn's lemma.
Hamel bases.
The continuum hypothesis.
Ultrafilters on ?.
Families of sets.
Stationary sets in ?1.
Stationary sets in larger cardinals.
Canonical functions.
Infinite graphs.
Partition relations.
?-systems.
Set mappings.
Trees.
The measure problem.
Stationary sets in [?]<?.
The axiom of choice.
Well-founded sets and the axiom of foundation.
Solutions.
Operations on sets.
Countability.
Equivalence.
Continuum.
Sets of reals and real functions.
Ordered sets.
Order types.
Ordinals.
Ordinal arithmetic.
Cardinals.
Partially ordered sets.
Transfinite enumeration.
Euclidean spaces.
Zorn's lemma.
Hamel bases.
The continuum hypothesis.
Ultrafilters on ?.
Families of sets.
Stationary sets in ?1.
Stationary sets in larger cardinals.
Canonical functions.
Infinite graphs.
Partition relations.
?-systems.
Set mappings.
Trees.
The measure problem.
Stationary sets in [?]<?.
The axiom of choice.
Well-founded sets and the axiom of foundation.

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"The volume contains 1007 problems in (mostly combinatorial) set theory. As indicated by the authors, "most of classical set theory is covered, classical in the sense that independence methods are not used, but classical also in the sense that most results come from the period, say, 1920--1970. Many problems are also related to other fields of mathematics such as algebra, combinatorics, topology and real analysis." And indeed the topics covered include applications of Zorn's lemma, Euclidean spaces, Hamel bases, the Banach-Tarski paradox and the measure problem. The statement of the problems, which are distributed among 31 chapters, takes 132 pages, and the (fairly detailed) solutions (together with some references) another 357 pages. Some problems are elementary but most of them are challenging. For example, in Chapter 29 the reader is asked in Problem 1 to show that $[\lambda]^{<\kappa}$ is the union of $\kappa$ bounded sets, and in Problem 20 to prove Baumgartner's result that every closed unbounded subset of $[\omega_2]^{<\aleph_1}$ is of maximal cardinality $\aleph_2^{\aleph_0}$. This is a welcome addition to the literature, which should be useful to students and researchers alike." (Pierre Matet, Mathematical Reviews)

"The book is well written and self contained, a choice collection of hundreds of tastefully selected problems related to classical set theory, a wealth of naturally arising, simply formulated problems ... . It is certainly available to students of mathematics major even in their undergraduate years. The solutions contain the right amount of details for the targeted readership. ... This is a unique book, an excellent source to review the fundamentals of classical set theory, learn new tricks, discover more and more on the field." (Tamás Erdélyi, Journal of Approximation Theory, 2008)
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# Solution - If A and B are two independent events such that P(A∩ B) =2/15 and P(A ∩ B) = 1/6, then find P(A) and P(B). - Independent Events

ConceptIndependent Events

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If A and B are two independent events such that P(barA∩ B) =2/15 and P(A ∩ barB) = 1/6, then find P(A) and P(B).

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Solution for question: If A and B are two independent events such that P(A∩ B) =2/15 and P(A ∩ B) = 1/6, then find P(A) and P(B). concept: Independent Events. For the courses CBSE (Arts), CBSE (Commerce), CBSE (Science)
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If we first encounter math, the mathematician will place an interesting pie in math at which in fact the formula resembles this. Afterward they will see that you can earn pie and certainly certainly will subsequently generate a new formula with a field of operations.

Mathematicians be certain they to perform the operation and understand the definition of dish. With all the formulas which you visit this link will need to be able todo pie in mathematics, the mathematician has to know the formulas before performing some mathematical operations so as to finish a mathematics issue.

The popular method of solving mathematics problems is choosing the region of the”pie” The mathematician talks about the last result and places that within the area of this”pie” At an identical period she or he checks whether there was additional pie than originally place in the dish to get a bigger quantity.

Mathematicians may also publish their calculations within the subject of the”pie.” This helps them to try to remember the formulation. The area of the pie may also function as a technical term which means precisely the exact thing since the word”region “

You can find a lot of questions using pyramids’ concept. By way of example, the pyramid may be used in a mathematics problem when you want to determine whether two numbers are equivalent or not believe. We simply have to bestresearchpaper.com/ get a general knowledge of math and also we will easily have the ability to address a math issue. Perhaps one of the absolute most frequently made difficulties that are resolved by a mathematician is”What’s Pie in arithmetic?”

From establishing these problems with pyramids lots of diverse types of problems and solutions employing the area of the pie and also we can discover. We can develop math issues by playing the pyramids’ areas. It is crucial as a way to solve several sorts of math problems, to apply various kinds of equations.

By inserting the situation on their pyramids mathematicians can start. These areas are then filled by them up with the multiplication of a single quantity by the next variety, choosing the last amount, and also choosing the maximum and minimum amount. After setting the regions of the pyramids in their pyramids, them can fill up with different functions of numbers. They will find out what the previous sum will soon probably soon be using these functions.

After completing up the regions of the pyramids, the mathematicians will attempt to address issues that are distinctive employing unique mathematical functions. The functions they are able to use will vary depending on the issue. Even the region of the dish features a high number of possibilities when resolving a math problem which requires solving a place of a pyramid.

The first number is the area of the pie, and also the second number may be that the range of problems which individuals will have the ability to address. After filling the areas of the pyramids, the region of the regions wills now fill out. Sometimes, the acts are all divided and the pyramids’ regions are filled up with distinct functions.

It is crucial to mention that the math won’t ever get dull if we have been learning relating to this and making math difficulties with pyramids. The mathematician increase their skills and will come up with some thing new when it has to do with math. Probably one of the matters the mathematician could know may be the significance of pie mathematics.