Solving boundary value problems using the Galerkin's method. This is a weighted residual method, studied as an introduction to the Finite Element Method.
This is a part of a series on Advanced Numerical Methods.
Solving second order ordinary differential equations (boundary value problems) using the Least Squares Technique. Contains one numerical examples from Shah, Eldho, Desai
Solving second order ordinary differential equations (boundary value problems) using the Least Squares Technique. Contains one numerical examples from Shah, Eldho, Desai
Engineering Mathematics
First order differential equations
Basic Concepts
Separable Differential Equations
substitution Methods
Exact Differential Equations
Integrating Factors
Linear Differential Equations
Bernoulli Equations
Dr. Summiya Parveen
Department of Mathematics
COLLEGE OF ENGINEERING ROORKE (COER)
ROORKEE 247667, INDIA
Email Id : summiyaparveen82@gmail.com
Spline interpolation is a problem of "Numerical Methods".
This slide covers the basics of spline interpolation mostly the linear spline and cubic spline interpolation.
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
Finite Element Analysis of the Beams Under Thermal LoadingMohammad Tawfik
A report on the finite element analysis of a beam under thermal loading. Nonlinear deflections and solution procedures covered.
#WikiCourses
https://wikicourses.wikispaces.com/TopicX+Nonlinear+Solid+Mechanics
https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics
The Gaussian or normal distribution is one of the most widely used in statistics. Estimating its parameters using
Bayesian inference and conjugate priors is also widely used. The use of conjugate priors allows all the results to be
derived in closed form. Unfortunately, different books use different conventions on how to parameterize the various
distributions (e.g., put the prior on the precision or the variance, use an inverse gamma or inverse chi-squared, etc),
which can be very confusing for the student. In this report, we summarize all of the most commonly used forms. We
provide detailed derivations for some of these results; the rest can be obtained by simple reparameterization. See the
appendix for the definition the distributions that are used.
Engineering Mathematics
First order differential equations
Basic Concepts
Separable Differential Equations
substitution Methods
Exact Differential Equations
Integrating Factors
Linear Differential Equations
Bernoulli Equations
Dr. Summiya Parveen
Department of Mathematics
COLLEGE OF ENGINEERING ROORKE (COER)
ROORKEE 247667, INDIA
Email Id : summiyaparveen82@gmail.com
Spline interpolation is a problem of "Numerical Methods".
This slide covers the basics of spline interpolation mostly the linear spline and cubic spline interpolation.
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
Finite Element Analysis of the Beams Under Thermal LoadingMohammad Tawfik
A report on the finite element analysis of a beam under thermal loading. Nonlinear deflections and solution procedures covered.
#WikiCourses
https://wikicourses.wikispaces.com/TopicX+Nonlinear+Solid+Mechanics
https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics
The Gaussian or normal distribution is one of the most widely used in statistics. Estimating its parameters using
Bayesian inference and conjugate priors is also widely used. The use of conjugate priors allows all the results to be
derived in closed form. Unfortunately, different books use different conventions on how to parameterize the various
distributions (e.g., put the prior on the precision or the variance, use an inverse gamma or inverse chi-squared, etc),
which can be very confusing for the student. In this report, we summarize all of the most commonly used forms. We
provide detailed derivations for some of these results; the rest can be obtained by simple reparameterization. See the
appendix for the definition the distributions that are used.
01. Differentiation-Theory & solved example Module-3.pdfRajuSingh806014
Total No. of questions in Differentiation are-
In Chapter Examples 31
Solved Examples 32
The rate of change of one quantity with respect to some another quantity has a great importance. For example the rate of change of displacement of a particle with respect to time is called its velocity and the rate of change of velocity is
called its acceleration.
The following results can easily be established using the above definition of the derivative–
d
(i) dx (constant) = 0
The rate of change of a quantity 'y' with respect to another quantity 'x' is called the derivative or differential coefficient of y with respect to x.
Let y = f(x) be a continuous function of a variable quantity x, where x is independent and y is
(ii)
(iii)
(iv)
(v)
d
dx (ax) = a
d (xn) = nxn–1
dx
d ex =ex
dx
d (ax) = ax log a
dependent variable quantity. Let x be an arbitrary small change in the value of x and y be the
dx
d
(vi) dx
e
(logex) = 1/x
corresponding change in y then lim
y
if it exists, d 1
x0 x
is called the derivative or differential coefficient of y with respect to x and it is denoted by
(vii) dx
(logax) =
x log a
dy . y', y
dx 1
or Dy.
d
(viii) dx (sin x) = cos x
So, dy dx
dy
dx
lim
x0
lim
x0
y
x
f (x x) f (x)
x
(ix) (ix)
(x) (x)
d
dx (cos x) = – sin x
d (tan x) = sec2x
dx
The process of finding derivative of a function is called differentiation.
If we again differentiate (dy/dx) with respect to x
(xi)
d (cot x) = – cosec2x
dx
d
then the new derivative so obtained is called second derivative of y with respect to x and it is
Fd2 y
(xii) dx
d
(xiii) dx
(secx)= secx tan x
(cosec x) = – cosec x cot x
denoted by
HGdx2 Jor y" or y2 or D2y. Similarly,
d 1
we can find successive derivatives of y which
(xiv) dx
(sin–1 x) = , –1< x < 1
1 x2
may be denoted by
d –1 1
d3 y d4 y
dn y
(xv) dx (cos x) = –
,–1 < x < 1
dx3 ,
dx4 , ........, dxn , ......
d
(xvi) dx
(tan–1 x) = 1
1 x2
Note : (i)
y is a ratio of two quantities y and
x
(xvii) (xvii)
d (cot–1 x) = – 1
where as dy
dx
dy
is not a ratio, it is a single
dx
d
(xviii) (xviii)
(sec–1 x) =
1 x2
1
|x| > 1
quantity i.e.
dx dy÷ dx
dx x x2 1
(ii)
dy is
dx
d (y) in which d/dx is simply a symbol
dx
(xix)
d (cosec–1 x) = – 1
dx
of operation and not 'd' divided by dx.
d
(xx) dx
(sinh x) = cosh x
d
(xxi) dx
d
(cosh x) = sinh x
Theorem V Derivative of the function of the function. If 'y' is a function of 't' and t' is a function of 'x' then
(xxii) dx
d
(tanh x) = sech2 x
dy =
dx
dy . dt
dt dx
(xxiii) dx
d
(xxiv) dx
d
(coth x) = – cosec h2 x (sech x) = – sech x tanh x
Theorem VI Derivative of parametric equations If x = (t) , y = (t) then
dy dy / dt
=
(xxv) dx
(cosech x) = – cosec hx coth x
dx dx / dt
(xxvi) (xxvi)
(xxvii) (xxvii)
d (sin h–1 x) =
Level Slicing and Map Calculation studied here. Done using GRASS GIS software, an open source resource. This is the second part of the laboratory series on GRASS GIS.
This is a practical demostration of Watershed Analysis with GRASS. This follows a click-this and click-that approach, followed by questions and exercises.
These slides are the first amongst the series of documents made by me as a part of a LaTeX workshop. Provided for free as a help for researchers and document makers.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
A Survey of Techniques for Maximizing LLM Performance.pptx
FEM Introduction: Solving ODE-BVP using the Galerkin's Method
1. Solving ODE-BVP through Galerkin’s Method
FEM: Introduction
Suddhasheel Ghosh, PhD
Department of Civil Engineering
Jawaharlal Nehru Engineering College
N-6 CIDCO, 431003
Series on Advanced Numerical Methods
shudh (JNEC) Concepts MEStru2k1617 1 / 14
2. DiffEq1
Introduction to terminology
Given a differential equation
Ψ
d2
y
dx2
,
dy
dx
,y,x = 0, (1)
and the initial conditions,
F1
dy
dx
,y,x = a = 0 F2
dy
dx
,y,x = b = 0
So, given the points a and b, it is desired to find the solution of the
differential equation using the Galerkin’s Method.
shudh (JNEC) Concepts MEStru2k1617 2 / 14
3. DiffEq1
A second-order Boundary Value Problem
A boundary value problem is given as follows:
d2
y
dx2
+ P(x)
dx
dy
+ Q(x)y = R(x)
along with the conditions
y(x = a) = A, y(x = b) = B
shudh (JNEC) Concepts MEStru2k1617 3 / 14
4. GM
Concept of Linear Independence
In Vector Algebra, n vectors, namely v1,v2,...,vn are linearly independent,
when
n
i=1
aivi = 0 ⇐⇒ ai = 0,∀i = 1,...,n
Linear independence means that no vector can be expressed as a linear
combination of other vectors.
This concept of linear independence is not only limited to vectors, but has
also been extended to the area of functions and various algebraic
polynomials.
shudh (JNEC) Concepts MEStru2k1617 4 / 14
5. GM
Galerkin’s method I
Formulation
The Galerkin’s Method is a “weighted-residual” method. We will try to
solve the following differential equation:
d2
y
dx2
+ P(x)
dy
dx
+ Q(x)y = R(x)
with the following boundary conditions y(x = a) = A and y(x = b) = B.
Let us assume that the solution is in the form
y = α0 + α1x + α2x2
+ ··· + αnxn
=
n
i=0
αixi
shudh (JNEC) Concepts MEStru2k1617 5 / 14
6. GM
Galerkin’s method II
Formulation
Differentiating the above form, with respect to x, we have:
dy
dx
=
n
i=0
iαixi−1
(2)
d2
y
dx2
=
n
i=0
i(i − 1)αixi−2
(3)
Substituting, these into the differential equation we have:
n
i=0
αi i(i − 1)xi−2
+ iP(x)xi−1
+ Q(x)xi
= R(x) (4)
shudh (JNEC) Concepts MEStru2k1617 6 / 14
7. GM
Galerkin’s method III
Formulation
From the boundary conditions, we have
n
i=0
αiai
= A, (5)
n
i=0
αibi
= B (6)
We work out the residual function as follows:
(x) =
n
i=0
αi i(i − 1)xi−2
+ iP(x)xi−1
+ Q(x)xi
− R(x) (7)
If there are n − 1 unknowns, then there should be n − 1 linearly
independent polynomials chosen to be multiplied as weights to the
shudh (JNEC) Concepts MEStru2k1617 7 / 14
8. GM
Galerkin’s method IV
Formulation
residual function. Therefore, for each j = 1,...,n − 1, we should have
n − 1 equations
b
a
Nj(x) (x)dx = 0 (8)
where Nj(x) denotes the jth polynomial.
These equations are solved using linear algebra to obtain the values of
αi, i = 0,...,n
shudh (JNEC) Concepts MEStru2k1617 8 / 14
9. GM
Example I
Galerkin’s Method
Problem: Use the Galerkin’s method to solve the following differential
equation:
d2
y
dx2
− y = x
Use the boundary conditions y(x = 0) = 0 and y(x = 1) = 0. (Desai, Eldho,
Shah)
Solution: Let us assume that the solution to the given differential
equation is in the following form, where there are four unknowns:
y = α0 + α1x + α2x2
+ α3x3
shudh (JNEC) Concepts MEStru2k1617 9 / 14
10. GM
Example II
Galerkin’s Method
From the boundary conditions given, we have
α0 + α1(0) + α2(02
) + α3(03
) = 0 =⇒ α0 = 0
α0 + α1(1) + α2(12
) + α3(13
) = 0 =⇒ α1 + α2 + α3 = 0(or)α3 = −(α1 + α2
We calculate the derivatives as follows:
dy
dx
= α1 + 2α2x + 3α3x2
d2
y
dx2
= 2α2 + 6α3x
Substituting these into the differential equation, we have the following:
α1x + α2(2 − x2
) + α3(6x − x3
) = x
shudh (JNEC) Concepts MEStru2k1617 10 / 14
11. GM
Example III
Galerkin’s Method
Since α3 = −(α1 + α2), we will have
−α1x + α2(2 − x2
) + (α1 + α2)(x3
− 6x) = x
=⇒ α1(x3
− 7x) + α2(x3
− x2
− 6x + 2) = x (9)
We can therefore formulate
(x) = α1(x3
− 7x) + α2(x3
− x2
− 6x + 2) − x (10)
shudh (JNEC) Concepts MEStru2k1617 11 / 14
12. GM
Example IV
Galerkin’s Method
Since there are two unknown parameters here, we will consider two
functions N1(x) = x − x2
, and N2(x) = x2
− x3
, as weighting functions.
Therefore,
1
0
N1(x) (x)dx = 0 =⇒ −0.5500α1 − 0.1833α2 = 0.0833
1
0
N2(x) (x)dx = 0 =⇒ −0.3262α1 − 0.1429α2 = 0.0500
We will have this system
−0.5500 −0.1833
−03262 −0.1429
α1
α2
=
0.0833
0.0500
(11)
shudh (JNEC) Concepts MEStru2k1617 12 / 14
13. GM
Example V
Galerkin’s Method
Using this, the relations α3 = −(α1 + α2), and α0 = 0, we have
α0 = 0,α1 = −0.1456,α2 = −0.01743,α3 = 0.1631
Therefore, we can say that for our differential equation, we have the
following solution:
y = −0.1456x − 0.01743x2
+ 0.1631x3
shudh (JNEC) Concepts MEStru2k1617 13 / 14