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On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
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On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...BRNSS Publication Hub
We know that a large number of problems in differential equations can be reduced to finding the solution x to an equation of the form Tx=y. The operator T maps a subset of a Banach space X into another Banach space Y and y is a known element of Y. If y=0 and Tx=Ux−x, for another operator U, the equation Tx=y is equivalent to the equation Ux=x. Naturally, to solve Ux=x, we must assume that the range R (U) and the domain D (U) have points in common. Points x for which Ux=x are called fixed points of the operator U. In this work, we state the main fixed-point theorems that are most widely used in the field of differential equations. These are the Banach contraction principle, the Schauder–Tychonoff theorem, and the Leray–Schauder theorem. We will only prove the first theorem and then proceed.
We research behavior and sharp bounds for the zeros of infinite sequences of polynomials orthogonal with respect to a Geronimus perturbation of a positive Borel measure on the real line.
A. JBILOU, Convexity of the Set of k-Admissible Functions on a Compact Kähler Manifold (2014), International Journal of Innovation and Scientific Research, vol. 4, no. 1, pp. 42–46.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
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On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...BRNSS Publication Hub
We know that a large number of problems in differential equations can be reduced to finding the solution x to an equation of the form Tx=y. The operator T maps a subset of a Banach space X into another Banach space Y and y is a known element of Y. If y=0 and Tx=Ux−x, for another operator U, the equation Tx=y is equivalent to the equation Ux=x. Naturally, to solve Ux=x, we must assume that the range R (U) and the domain D (U) have points in common. Points x for which Ux=x are called fixed points of the operator U. In this work, we state the main fixed-point theorems that are most widely used in the field of differential equations. These are the Banach contraction principle, the Schauder–Tychonoff theorem, and the Leray–Schauder theorem. We will only prove the first theorem and then proceed.
We research behavior and sharp bounds for the zeros of infinite sequences of polynomials orthogonal with respect to a Geronimus perturbation of a positive Borel measure on the real line.
A. JBILOU, Convexity of the Set of k-Admissible Functions on a Compact Kähler Manifold (2014), International Journal of Innovation and Scientific Research, vol. 4, no. 1, pp. 42–46.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
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My paper for Domain Decomposition Conference in Strobl, Austria, 2005Alexander Litvinenko
We did a first step in solving, so-called, skin problem. We developed an efficient H-matrix preconditioner to solve diffusion problem with jumping coefficients
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Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
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!
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
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3. 2. Anharmonic Oscillator
The potential energy curves for most stretching vibrations have a form similar to a Morse potential
VM (x) = D[1 −e− βx
]2
= D[1 −2e− βx
+ e− 2βx
].
Expand in a power series
In contrast, most bending vibrations have an approximately quartic form
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For large values of x, the b x˜ operator should give a large, positive value. The derivative of a
positive gaussian in this region should be negative and very small (and will contribute positively
due to the negative sign in the equation). And the ψ0 itself is defined as a positive gaussian and
is therefore positive over all space. Therefore, because all terms are positive for very large x
values, the far right lobe should be positive.
4. Solution: We can interpret a Morse potential as a perturbation of a perfect harmonic
oscillator. These perturbations are the higher order terms in the power expansion of the
potential. Considering
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Here is some useful information:
The power series expansion of the vibrational energy levels is
Hint: The goal of this problem is to relate information about the potential surface (i.e. D and
β) to information about the energy level pattern we can obtain experimentally (i.e. ω˜,
ω˜x˜, etc.). We make these connections via perturbation theory.
A. For a Morse potential, use perturbation theory to obtain the relationships between (D, β)
and (ωe, ωex˜, ωey˜). Treat the (aˆ + aˆ† )3 term through second–order perturbation
theory and the (aˆ + aˆ† )4 term only through first order perturbation theory. [HINT: you
will find that ωey˜ = 0.]
5. onlythetermsoforder4orlessanddefiningω0=2Dβ2/m,wehave
The zero-order energies are those for a harmonic oscillator with frequency ω0
To calculate the first-order corrections it’s necessary to rewrite the perturbation term H(1)
in terms of raising and lowering operators.
The first-order corrections are the integrals Hv,v . In the equation above, the cubic term
has selection rules Δv = ±3, ±1, and so it will not contribute to the first-order corrections.
Expanding the quartic term and keeping only those parts that have a selection rule Δv =
0 (i.e. those terms which have two aˆ’s and two aˆ† ’s), we arrive at
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The second-order term is a bit more complicated. We will make the simplification of only
considering the second-order contributions from the cubic term in the potential (this is
reasonable because in real systems quartic terms are generally an order of magnitude smaller
than cubic terms). The second order correction to the energies is
7. eduassignmenthelp.com
Okay! Now we’ve done the hard work, let’s make the final connections. First let’s add up all of the
perturbation corrections to the energy levels:
Compare this to the “dumb” (i.e. a priori un-insightful) power series expansion of the energy
levels (which we would measure experimentally),
Matching powers of (v + 1/2), we can determine the following relations between the
experimentally determined molecular constants (˜ω, ω˜x . . . ˜ ) and the potential curve
parameters (the information we actually care about).
8. Solution: We approach the quartic bending potential in the same manner as above.
First we find the perturbation theory expressions for the energy levels and
manipulate them to be a power series in (v + 1/2), and then compare this to the
experimental power series expansion of the vibrational energy levels.
B. Optional Problem For a quartic potential, find the relationship between (ωe, ωex˜,
ωey˜) and (k, b) by treating (aˆ + aˆ† )4 through second–order perturbation theory.
series in (v + 1/2), and then compare this to the experimental power series expansion of
the vibrational energy levels.
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10. 3. Perturbation Theory for Harmonic Oscillator Tunneling Through a δ–function
Barrie
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are related to the potential parameters by matching powers:
Note that when we treat the quartic term up through second-order perturbation theory it
contributes to the linear, quadratic, and cubic terms in the (v + 1/2) power series. The
firstorder correction only contributes to the quadratic term (and all contribute to the
constant offset).
where C > 0 for a barrier. δ(x) is a special, infinitely narrow, infinitely tall function centered at
x = 0. It has the convenient property that
where ψv(0) is the value at x = 0 of the vth eigenfunction for the harmonic oscillator. Note that,
for all v = odd,
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A.
(i) The {ψv} are normalized in the sense
What are the units of ψ(x)?
Solution: The normalization for the zero-order wavefunctions is given as
(ii) From Eq. (3.2), what are the units of δ(x)?
Solution:
The δ function is defined by
(iii) V (x) has units of energy. From Eq. (3.1), what are the units of the constant, C?
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We can now estimate the value of the integrals of the perturbation term
D. Make the assumption that all terms in the sum over v0 (Eq. 3.12) except the v, v + 2 (0) (1)
(2) and v, v � 2 terms are negligibly small. Determine Ev = Ev + Ev + Ev and comment on the
qualitative form of the vibrational energy level diagram. Are the odd–v lev- (0) els shifted at all
from their Ev values? Are the even–v levels shifted up or down rela- (0) tive to Ev ? How does
the size of the shift depend on the vibrational quantum number?
Solution: Making the assumption that only the v0 = v � 2, v, v + 2 terms are non-negligible,
we can calculate the perturbed energy levels of the v–even states.
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Solution: The time-dependence of this two-level superposition state’s probability distribution
Ψ?(x,t)Ψ(x,t) will exhibit oscillations at the frequency corresponding to the two-level energy
difference ω � (E1 � E0)/}. As this motion causes the wavepacket to move back and forth
through the barrier, this oscillation frequency is called the tunneling frequency
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Solution:
To compute hxit, we need expressions for the perturbed wavefunctions in terms of the zero-order
harmonic oscillator wavefunctions
Odd-v wavefunctions are unperturbed, so ψ1 = ψ1 (0), but we do need to calculate the energy
of the perturbed ground state
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The value of ω10 will be ≈ ω plus additional perturbation corrections as determined in part 3D.
Note that the motion remains sinusoidal (why is this always true for a two-level superposition
state?).
18. Solution: The tunneling rate for a superposition of two adjacent states will be proportional to the
energy difference between them. As we saw in parts 3.D and 3.E, the difference in energy of the v =
0 and v = 1 level is smaller than the difference in energy between the v = 2 and v = 3 states
(because lower–v states are pushed up more). Therefore the v = 0, 1 superposition will have a
slower tunneling rate. Increasing the barrier height by increasing C will decrease the tunneling
rates. It pushes up each even-v state closer to the unperturbed odd-v state above it. For example,
the v = 0, 1 states will be closer together in energy and a superposition state of them will tunnel at
a slower rate.
4. Perturbation Theory for a Particle in a modified infinite box
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20. Solution: We will now explicitly calculate the perturbation integrals in order to evaluate the first
and second-order corrections to the energy levels.
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C. What is the selection rule for non-zero integrals
Solution:
The perturbation integrals are
Without evaluating any further, we can determine the selection rules for these integrals. We
require that the integrand is symmetric (or has a symmetric part). Since V 0 (x) is a (0) symmetric
function relative to the center of the box, this requires that the product of ψn (0) and ψm must
be symmetric. This requires that n and m are either both even or both od
22. Solution: The effect of switching the sign of V0 will change the perturbation from being a “well
in the box” to a “hill in the box”. We expect the energies to be perturbed upward. This can
easily be seen by the form of the first-order corrections in Eqs. (4.11)-(4.13). Switching the sign
of the perturbation will simply switch the sign of the first order corrections.
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The second-order corrections are equal to the sum
Limiting the infinite sum to m ≤ 5 and remembering our selection rule for the perturbation
integrals, we obtain