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This book contains an extensive illustration of use of finite
difference method in solving the boundary value problem
numerically. A wide class of differential equations has been
numerically solved in this book. Starting with differential
equations of elementary functions like hyperbolic, sine and cosine,
we have solved those of special functions like Hermite, Laguerre
and Legendre. Those of Airy function, of stationary localised
wavepacket, of the quantum mechanical problem of a particle in a 1D
box, and the polar equation of motion under gravitational
interaction have also been solved. Mathematica 6.0 has been used to
solve the system of linear equations that we encountered and to
plot the numerical data. Comparison with known analytic solutions
showed nearly perfect agreement in every case. On reading this
book, readers will become adept in using the method.
The book contains a detailed account of numerical solutions of
differential equations of elementary problems of Physics using
Euler and 2nd order Runge-Kutta methods and Mathematica 6.0. The
problems are motion under constant force (free fall), motion under
Hooke's law force (simple harmonic motion), motion under
combination of Hooke's law force and a velocity dependent damping
force (damped harmonic motion) and radioactive decay law. Also
included are uses of Mathematica in dealing with complex numbers,
in solving system of linear equations, in carrying out
differentiation and integration, and in dealing with matrices.
The book presents in comprehensive detail numerical solutions to
boundary value problems of a number of non-linear differential
equations. Replacing derivatives by finite difference
approximations in these differential equations leads to a system of
non-linear algebraic equations which we have solved using Newton's
iterative method. In each case, we have also obtained Euler
solutions and ascertained that the iterations converge to Euler
solutions. We find that, except for the boundary values, initial
values of the 1st iteration need not be anything close to the final
convergent values of the numerical solution. Programs in
Mathematica 6.0 were written to obtain the numerical solutions.
This book is intended for undergraduate students of Mathematics,
Statistics, and Physics who know nothing about Monte Carlo Methods
but wish to know how they work. All treatments have been done as
much manually as is practicable. The treatments are deliberately
manual to let the readers get the real feel of how Monte Carlo
Methods work. Definite integrals of a total of five functions ( ),
namely Sin( ), Cos( ), e , loge( ), and 1/(1+ 2), have been
evaluated using constant, linear, Gaussian, and exponential
probability density functions ( ). It is shown that results agree
with known exact values better if ( ) is proportional to ( ).
Deviation from the proportionality results in worse agreement. This
book is on Monte Carlo Methods which are numerical methods for
Computational Physics. These are parts of a syllabus for
undergraduate students of Mathematics and Physics for the course
titled "Computational Physics." Need for the book: Besides the
three referenced books, this is the only book that teaches how
basic Monte Carlo methods work. This book is much more explicit and
easier to follow than the three referenced books. The two chapters
on the Variational Quantum Monte Carlo method are additional
contributions of the book. Pedagogical features: After a thorough
acquaintance with background knowledge in Chapter 1, five
thoroughly worked out examples on how to carry out Monte Carlo
integration is included in Chapter 2. Moreover, the book contains
two chapters on the Variational Quantum Monte Carlo method applied
to a simple harmonic oscillator and a hydrogen atom. The book is a
good read; it is intended to make readers adept at using the
method. The book is intended to aid in hands-on learning of the
Monte Carlo methods.
This book contains an extensive illustration of use of finite
difference method in solving the boundary value problem
numerically. A wide class of differential equations has been
numerically solved in this book. Starting with differential
equations of elementary functions like hyperbolic, sine and cosine,
we have solved those of special functions like Hermite, Laguerre
and Legendre. Those of Airy function, of stationary localised
wavepacket, of the quantum mechanical problem of a particle in a 1D
box, and the polar equation of motion under gravitational
interaction have also been solved. Mathematica 6.0 has been used to
solve the system of linear equations that we encountered and to
plot the numerical data. Comparison with known analytic solutions
showed nearly perfect agreement in every case. On reading this
book, readers will become adept in using the method.
The book contains a detailed account of numerical solutions of
differential equations of a number of elementary problems of
physics using Euler and second order Runge-Kutta methods using
Mathematica 6.0. The problems are motion under constant force (free
fall), motion under Hooke's law force (simple harmonic motion),
motion under combination of Hooke's law force and a velocity
dependent damping force (damped harmonic motion) and radioactive
decay law. Also included are uses of Mathematica in dealing with
complex numbers, in solving system of linear equations, in carrying
out differentiation and integration, and in dealing with matrices.
Containing an extensive illustration of the use of finite
difference method in solving boundary value problem numerically, a
wide class of differential equations have been numerically solved
in this book.
The book contains a detailed account of numerical solutions of
differential equations of elementary problems of Physics using
Euler and 2nd order Runge-Kutta methods and Mathematica 6.0. The
problems are motion under constant force (free fall), motion under
Hooke's law force (simple harmonic motion), motion under
combination of Hooke's law force and a velocity dependent damping
force (damped harmonic motion) and radioactive decay law. Also
included are uses of Mathematica in dealing with complex numbers,
in solving system of linear equations, in carrying out
differentiation and integration, and in dealing with matrices.
Master's Thesis from the year 2013 in the subject Physics - Quantum
Physics, grade: -, Shahjalal University of Science and Technology
(Department of Physics), course: Nanostructure Physics, language:
English, abstract: We have numerically investigated parametric
variations of transmission peaks of symmetric rectangular double
barrier in non-tunneling regime. We have compared the variations
with those for tunneling regime. One of the three variations in
non-tunneling regime is completely different from that for
tunneling regime warranting rapid dissemination. The book contains
background on Quantum Mechanics, Microelectronics and Nanostructure
Physics to enable readers assimilate the book completely.
Master's Thesis from the year 2011 in the subject Physics - Quantum
Physics, grade: -, Shahjalal University of Science and Technology
(Department of Physics), course: Nanostructure Physics, language:
English, abstract: This book contains a comprehensive account of
application of WKB method to pure Physics of nanostructures
containing single or symmetric double barrier V(x) in their band
model in presence of longitudinal magnetic field applied along x
direction. It concentrates on effects on transmission coefficient
of single and symmetric double barriers by three dimensional
electron gas (3DEG). Analytical expressions for longitudinal
magnetic field dependent transmission coefficient of single and
symmetric double barrier of general shape are obtained first. These
general expressions are then used to obtain analytical expressions
of longitudinal magnetic field dependent transmission coefficient
of single and symmetric double barriers of many different shapes we
encounter in studying nanostructure Physics. This is followed by
thorough numerical investigation to bring out effects of
longitudinal magnetic field on transmission coefficient of all
these barriers. Comparisons with standard results where available
showed excellent agreements. Results of numerical investigation
have been explained completely. The book makes well documented,
with thorough calculation and discussion, pure Physics of
semiconductor nanostructures.
This book presents in comprehensive detail numerical solutions to
boundary value problems of a number of differential equations using
the so-called Shooting Method. 4th order Runge-Kutta method,
Newton's forward difference interpolation method and bisection
method for root finding have been employed in this regard. Programs
in Mathematica 6.0 were written to obtain the numerical solutions.
This monograph on Shooting Method is the only available detailed
resource of the topic.
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