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Video Lecture Course
Numerical Simulations on Physical
Problems
Lecturer: Ramaz Khomeriki
Lecture #1
1. Overview of the course
2. Basic MATLAB operations
3. Ordinary Differential Equations (ODE-s)
4. Simulating ODE-s with MATLAB
x
FDr  kx
Lecture #2
MATLAB Simulations on anharmonic oscillator
FDr  kx
FDr  kx  k3 x 3
d 2x
m 2  kx
dt
d 2x
m 2  kx  k3 x 3
dt
Lecture #3:
Mathematical pendulum Model;
Driven-Damped Oscillations
d
l 2   g sin 
dt
2
Double Pendulum Model:
MATLAB Simulations
Lecture #4:
Simulations on anharmonic oscillator chain
using MATLAB ODE toolbox
Fermi-Pasta-Ulam Lattice
d 2u n
3
3
m 2  k un1  un   k un1  un   k3 un1  un   k3 un1  un 
dt
Lecture #5:
Simulations on Pendula Chain: Sine-Gordon Equation
d 2 n
ml 2  k  n 1   n 1  2 n   mgsin  n ,
dt
Lecture #6:
MATLAB animations on Fermi-Pasta-Ulam
Model and Pendula Chain
Lecture #7:
Quantum-Mechanical Problems: Rabi
Oscillations in Two-Well potential
  2 
i
 2  V ( x, t )   0
t
x
Lecture #8:
Landau-Zener Tunneling in Two well potential
W0
  2 
i


0
2
t
x
cosh x  vt 
Lecture #9:
Bloch oscillations in Optical waveguide arrays
 
  
1 B(r , t )
  E (r , t )  
;
c t
 
  
1 D(r , t )
  H (r , t ) 
;
c t
  
  D(r , t )  0;
  2 
i
 2  a cos 2 x  bx  0
z x
  
  B(r , t )  0
Lecture #10:
Simulations on Magnetic spin systems:
Landau-Lifshitz Equation

H    JSi Si 1  Di  S
N
i 1
1
 J y S nz  S ny1  S ny1   J z S ny  S nz1  S nz1   2 DS ny S nz  ,

2
1
S ny   J z S nx  S nz1  S nz1   J x S nz  S nx1  S nx1   2 DS nx S nz  ,
2
1
S nz   J x S ny  S nx1  S nz1   J y S nx  S ny1  S ny1   .
2
S nx 

z 2
i

Lecture #11:
Simulations on the systems with two and more
dimensions
Lecture #12:
Bose-Hubbard model for two-boson system
and modelling of Hilbert space
j-1
N 1

j
j+1

N
Hˆ   g aˆ j 1aˆ j  aˆ j aˆ j 1  bˆ j 1bˆ j  bˆ j bˆ j 1  U  aˆ j bˆ j aˆ j bˆ j
j 1
j 1
Vectors in MATLAB
A  2, 3, 1.5, 7,  2
A  2 3 1.5 7 - 2
1 
 
3 
B  5 
 
  2
4 
 
B  1; 3; 5;  2; 4
B  1 3 5 - 2 4
5
Scalar Product
A * B   Ai Bi
i 1
Matrixes in MATLAB
1 3 2 


C  3 -1 5 
5 2 - 4


1 3 5 


C  3 -1 2 
 2 5 - 4


C  1, 3, 2; 3, - 1, 5; 5, 2, - 4
C * C 1
C^(-1)
1

 0
0

0 0

1 0
0 1
Multiplication of Matrixes in MATLAB
maSin
1 3 2 


C  3 -1 5 
5 2 - 4


 2 -1 0 


D  4 1 2 
3 0 2


1 * (-1)  3 *1  2 * 0
1 * 0  3 * 2  2 * 2   20 2 10 
 1* 2  3 * 4  2 * 3

 

C * D   3 * 2  (1) * 4  5 * 3 3 * (-1)  (-1) *1  5 * 0 3 * 0  (1) * 2  5 * 2   17 - 4 8 
 5 * 2  2 * 4  (4) * 3 5 * (-1)  2 *1  (-4) * 0 5 * 0  2 * 2  (-4) * 2   6 - 3 - 4 

 

A  2 3 1.5 7 - 2
1 
 
3 
B  5 
 
  2
4 
 
C*B
A*C
C*A’
Solving Algebraic Equations in MATLAB
 x  y  5 z  1

2 x  3 y  z  3
 x  6 y  z  2

 1 -1 5


S   2 3 - 1 ,
  1 6 1


S*X  N
x
 
X   y ,
z 
 
X  S ^ (1) * N
 - 1
 
N   3
 2
 
Plotting curves and surfaces in MATLAB
A  2 3 1.5 7 - 2
1 
 
3 
B  5 
 
  2
4 
 
plot ( A, B)
A  2 7 - 2
B   2 10 5
surf ( A, B, C )
1 3 2 


C  3 -1 5 
5 2 - 4


Ordinary Differential Equations (ODE-s) in MATLAB
ODE of the first order
dy
  ay  b
dt
ODE of the second order
d 2x
 kx
2
dt
Reduction to two
first order ODE-s
y  [x, v]
y (1)  x; y(2)  v
dx
v
dt
dy (1)
 y (2);
dt
dv
  kx
dt
dy (2)
 ky(1)
dt
First MATLAB file “osc.m” for simplest ODE
clear
tspan=0:0.1:400;
y0=1;
[T,y]=ode45('osc1',tspan,y0);
Second MATLAB file “osc1.m” for simplest ODE
function dydt=osc1(t,y)
a=0.01; b=0.0;
dydt=-a*y+b;