finite difference Method, For Numerical analysis. working matlab code. numeric analysis finite difference method. MATLAB provides tools to solve math. Using linear programing techniques we can easily solve system of equations. This file provides a running code of Finite difference matlab code
finite difference Method, For Numerical analysis. working matlab code. numeric analysis finite difference method. MATLAB provides tools to solve math. Using linear programing techniques we can easily solve system of equations. This file provides a running code of Finite difference matlab code
In this slide fourier series of Engineering Mathematics has been described. one Example is also added for you. Hope this will help you understand fourier series.
Divided Difference Method, For Numerical analysis. working matlab code. numeric analysis Divided Difference method. MATLAB provides tools to solve math. Using linear programing techniques we can easily solve system of equations. This file provides a running code of Divided Difference
Gaussian Quadrature Formulas, which are simple and will help learners learn about Gauss's One, Two and Three Point Formulas, I have also included sums so that learning can be easy and the method can be understood.
In this slide fourier series of Engineering Mathematics has been described. one Example is also added for you. Hope this will help you understand fourier series.
Divided Difference Method, For Numerical analysis. working matlab code. numeric analysis Divided Difference method. MATLAB provides tools to solve math. Using linear programing techniques we can easily solve system of equations. This file provides a running code of Divided Difference
Gaussian Quadrature Formulas, which are simple and will help learners learn about Gauss's One, Two and Three Point Formulas, I have also included sums so that learning can be easy and the method can be understood.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
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Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
First principle, power rule, derivative of constant term, product rule, quotient rule, chain rule, derivatives of trigonometric functions and their inverses, derivatives of exponential functions and natural logarithmic functions, implicit differentiation, parametric differentiation, L'Hopital's rule
The purpose of this research work is to study the hydrodynamic characteristics of a new type of artificial reef structure, in order to provide a structure with low flow resistance, which will be a more suitable shelter for fishes and marine organisms. The idea of the new artificial reef is based on the streamlined bicycle helmet design concept. The hydrodynamic characteristics of the helmet and hollow cube artificial reefs (ARs) of the same volume have been studied at different water depths and wave frequencies of Malaysia seas using Computational Fluid Dynamics (CFD) method. The finite volume RANSE code Ansys CFX was used for calculating the reefs drag force (FD) and flow characteristics, while the potential flow code Ansys Aqwa was used for calculating the reefs inertia force (FI). The Shear Stress Transport (SST) turbulence model was used in the RANSE code. The results of the two ARs were then compared for studying the hydrodynamic improvement due to the use of streamlined helmet artificial reef on the flow pattern around it. The streamlined body of the helmet artificial reef enhances the flow pattern at the aft region of the reef and provides flow zones with moderate flow speed at this area, which can help fishes and marine organisms from finding good shelter. The special shape of the different openings in the body of the helmet artificial reef improves the condition of the flow velocity distribution inside the unit than that of the hollow cube unit, which can increase the amount of the nutrient to the living fishes and organisms inside the reef.
Novel approach of bidirectional diffuser-augmented channels system for enhanc...Yasser Ahmed
Hydrokinetic is a recently introduced type of hydropower energy, having been proven as the most
effective and predictable renewable energy source available around the world, especially in the rural and
electrification areas. Most of these sites are dependent on small and micro scale stations to produce
cheap but abundantly available and effective electrical energy. Hydrokinetic energy that can be harnessed
from the flow of water in the irrigation and rainy channels is a promising technology in countries
with vast current energy. Micro hydrokinetic energy scheme presents an attractive, environmentally
friendly and efficient electric generation in rural, remote and hilly areas, as effort to reduce the everincreasing
greenhouse gas emissions and fuel prices in these sites. Though potential, this scheme is
yet to be fully discovered to the considerable extent, as researchers are still searching for solution for the
main problem of low velocity of current in the open flow channels. Deploying acceleration nozzle in the
channel is a unique solution for increasing the channels current flow systems' efficiency. Acceleration
nozzle channel method has numerous advantages especially on the environmental impact, yet has not
been given much attention in the renewable energy field. This paper proposes a novel system configuration
to capture as much as kinetic energy from in stream current water. This system, known as
bidirectional diffuser-augmented channel functions by utilizing dual directed nozzles in the flow, surrounded
by dual cross flow turbines. This type of turbine is commonly used for hydropower applications;
and this study proposes the employment of this turbine for hydrokinetic power generation. Numerical
investigations had been performed using finite volume Reynolds-Averaged NaviereStokes Equations
(RANSE) code Ansys CFX to investigate the flow field characteristics of the new system approach with
and without the turbines. The performance of the twin (lower and upper) cross flow turbines had also
been studied. It was found that the highest efficiency of 0.52 was recorded for lower turbine at tip speed
ratio (TSR) of 0.5.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
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The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
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Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
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• Compatible with MAFI CCR system.
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Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
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Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
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Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
5. 5
The Trapezoidal Rule
( ) ( )dxxfdxxf
b
a
b
a ∫∫ ≅= 1I
( ) ( ) ( ) ( )( )a-x
a-b
afbf
afxf
−
+=1
( ) ( ) ( )( ) ( ) ( ) ( )
2
bfaf
abdxa-x
a-b
afbf
afI
b
a
+
−=
−
+= ∫
All the Newton-Cotes formulas can be expressed in this general average height form
heightaveragewidthI ×≅
6. Example 1 Single Application of the Trapezoidal Rule
f(x) = 0.2 +25x – 200x2 + 675x3 – 900x4 + 400x5
Integrate f(x) from a=0 to b=0.8
%....E
.
..
.
)()(
)(
tt
:oferroranrepresentswhich
:RulelTrapezoida
58946773117280640531
17280
2
232020
80
2
==−=
=
+
=
+
−=
ε
bfaf
abI
∫
=
=
==
80
0
640531
.
.)(:valueintegralTrue
b
a
dxxfI
Solution: f(a)=f(0) = 0.2 and f(b)=f(0.8) = 0.232
7. 7
• The accuracy can be improved by dividing the
interval from a to b into a number of segments
and applying the method to each segment.
• The areas of individual segments are added to
yield the integral for the entire interval.
Using the trapezoidal rule, we get:
∫∫∫
−
+++=
===
−
=
n
n
x
x
x
x
x
x
n
dxxfdxxfdxxfI
xbxa
n
ab
h
1
2
1
1
0
)()()(
seg.of#n 0
L
++
−
=
+
++
+
+
+
=
∑
−
=
−
1
1
0
12110
2
2
222
n
i
in
nn
xfxfxf
n
ab
I
xfxf
h
xfxf
h
xfxf
hI
)()()(
)()()()()()(
L
The Multiple-Application Trapezoidal Rule
13. Example 4
The vertical distance covered by a rocket from to seconds is
given by:
∫
−
−
=
30
8
89
2100140000
140000
2000 dtt.
t
lnx
a) Use two-segment Trapezoidal rule to find the distance covered.
b) Find the true error, for part (a).
c) Find the absolute relative true error, for part (a).a∈
tE
14. Solution
a) The solution using 2-segment Trapezoidal rule is
+
++
−
= ∑
−
=
)b(f)iha(f)a(f
n
ab
I
n
i
1
1
2
2
2=n 8=a 30=b
2
830 −
=
n
ab
h
−
= 11=
24. )()(
)(
)(
2
2
1
212
2
2
1
2
1
≅⇒≅
h
h
hEhE
h
h
hE
hE
Romberg Integration
Successive application of the trapezoidal rule to attain efficient numerical integrals of functions.
Richardson’s Extrapolation:
Uses two estimates of an integral to compute a third and more accurate approximation.
sizes)stepdifferentforconstantis(assume
)()()()(
/)(/)()()(
)
2
(
2
12
2211
fhOfh
ab
E
hEhIhEhI
habnnabhhEhII
′′=′′
−
≅
+=+
−=−=+=
I = exact value of integral E(h) = the truncation error
I(h): trapezoidal rule (n segments, step size h)
Improved estimate of the integral.
It is shown that the error of this
estimate is O(h4). Trapezoidal rule
had an error estimate of O(h2).( )
[ ])()(
/
)(
)()(
122
21
2
22
1
1
hIhI
hh
hII
hEhII
−
−
+≅
+=
( )
( ) 1/
)()(
)()()(/)()( 2
21
12
222
2
2121
−
−
≅⇒+≅+
hh
hIhI
hEhEhIhhhEhI
25. 25
( )
[ ])()(
1/
1
)( 122
21
2 hIhI
hh
hII −
−
+≅
Example 7
Evaluate the integral of f(x) = 0.2 +25x – 200x2 + 675x3 – 900x4 + 400x5
from a=0 to b=0.8. I (True Integral value) = 1.6405
Segments h Integral εtr%
1 0.8 0.1728 89.5
2 0.4 1.0688 34.9
4 0.2 1.4848 9.5
%)6.16(0.2733675.16405.1E
3675.1)1728.0(
3
1
)0688.1(
3
4
:givetocombined2&1Segments
tt ==−=
=−≅
ε
I
In each case, two estimates with
error O(h2) are combined to give a
third estimate with error O(h4)
%)1(0.01716234.16405.1E
6234.1)0688.1(
3
1
)4848.1(
3
4
:givetocombined4&2Segments
tt ==−=
=−≅
ε
I
)2/(If 12 ⇒= hh [ ] )(
3
1
)(
3
4
)()(
12
1
)( 121222 hIhIhIhIhII −=−
−
+≅
26. 26
Simpson’s Rules
General idea:
Use high order polynomials other than trapezoids to achieve more
accurate integration result but using the similar amount of computation
Parabola Cubic polynomial
27. • More accurate estimate of an integral is obtained if
a high-order polynomial is used to connect the
points. These formulas are called Simpson’s rules.
Simpson’s 1/3 Rule: results when a 2nd order
Lagrange interpolating polynomial is used for f(x)
dxxf
xxxx
xxxx
xf
xxxx
xxxx
xf
xxxx
xxxx
I
dxxfdxxfI
x
x
b
a
b
a
xbxa
xf
∫
∫ ∫
−−
−−
+
−−
−−
+
−−
−−
=
≅=
==
2
0
)(
))((
))((
)(
))((
))((
)(
))((
))((
)()(
2
1202
10
1
2101
20
0
2010
21
20
2
Using
.polynomialorder-secondais)(2where
[ ] RULE1/3SSIMPSON'
2
)()(4)(
3
210
:resultsformulafollowingtheon,manipulatialgebraicandnintegratioafter
⇐
−
=++≅
ab
hxfxfxf
h
I
a=x0 x1 b=x2
Simpson’s Rules
43. Gauss Quadrature
• Gauss quadrature implements a strategy of
positioning any two points on a curve to define a
straight line that would balance the positive and
negative errors.
• Hence the area evaluated under this straight line
provides an improved estimate of the integral.
46. Method of Undetermined Coefficients
• The trapezoidal rule yields exact results when the function
being integrated is a constant or a straight line, such as y=1
and y=x:
)()( 10 bfcafcI +≅
)(
2
)(
2
2
0
22
22
110
10
10
10
2/)(
2/)(
10
2/)(
2/)(
bf
ab
af
ab
I
ab
cc
ab
c
ab
c
abcc
dxx
ab
c
ab
c
dxcc
ab
ab
ab
ab
−
+
−
=
−
==
=
−
+
−
−
−=+
=
−
+
−
−
=+
∫
∫
−
−−
−
−−
Solve simultaneously
Trapezoidal rule
47. 47
Derivation of the Two-Point Gauss-Legendre Formula/
• The object of Gauss quadrature is to determine the equations of
the form
• However, in contrast to trapezoidal rule that uses fixed end
points a and b, the function arguments x0 and x1 are not fixed
end points but unknowns.
• Thus, four unknowns to be evaluated require four conditions.
• First two conditions are obtained by assuming that the above
eqn. for I fits the integral of a constant and a linear function
exactly.
• The other two conditions are obtained by extending this
reasoning to a parabolic and a cubic functions.
)()( 1100 xfcxfcI +≅