numerical approaches via matlab

8/14/2019 Numerical Approaches via MATLAB

1/23

Numerical Approaches Via MATLABNumerical Approaches Via MATLABTutorial on Computational Physics I (Phys.605)

Tsegaye Kassa

Department of Physics, Graduate Studies

Bahir Dar University, Bahir Dar, Ethiopia

E-mail:[email protected]

May 2008

Bahir Dar


2/23

OutlineOutline

Numerical Derivatives

Numerical Integrations

Ordinary Differential Equations


3/23

Numerical DerivativesNumerical Derivatives

% This program calculates the derivatives of 10+78x^2 at

any point you choose.

x=input('Enter the value of x that you want to calculate derivatives')

h=input('choose the step you want to use')

res=(neja_fun(x+h)-neja_fun(x))/h

% This function is developed to calculate the first derivative of a functionfunction[y]=neja_fun(x)

y=10+78*x^2;

Forward Derivative Formula


4/23

Comparison of Numerical and AnalyticalComparison of Numerical and AnalyticalSolutionsSolutions

%tsegaye-program to compute the Numerical Derivatives of the function

%and compare with the exact valuesclear; help tsegaye % clear memory and print header

x=input('Enter the location(x)-');

h=input('choose the step that want to use')

for i=1:3

hplot(i)=h; %record the value of h for plotting

%Right and Centered first derivativesresR(i)=(neja_fund(x+h)-neja_fund(x))/h;

resC(i)=(neja_fund(x+h)-(neja_fund(x-h)))/2*h;

% The Exact first derivativeresT(i)=neja_fund(x);

h=h/2; % halve the grid spacing at each iteration

end

%Plot numerical and true derivation on semilog scale

semilogx(hplot,resR,'+', hplot,resC,'g:', hplot,resT,'r-')

xlabel('interval size(h)')ylabel('First derivative')

legend('Right derivative', 'Centered derivative','Exact derivative')

title('Comparison of forward, centered and Exact derivatives')

%Displaying Outputs

fprintf('The value of right derivative=%g\n',resR)

printf('The value of centered derivative=%g\n',resC)

fprintf('The value ofexact derivative=%g\n',resR)

function[y]=neja_fund(x)

y=78*x^2+10;


5/23

ContC

ontHigher Order Derivative

% This program is developed to calculate the Second Derivative of e^x% at x=15.

x=input('Enter the value of x that you want to use')h=input('choose the step you want to use')

res=(neja_fun2(x+h)+neja_fun2(x-h)-(2)*neja_fun2(x))/h^2

% fprintf('The second derivative of the function is %f\n',res)

% This is a function developed to estimate the second derivative of the function.

function[y]= neja_fun2(x)

y=exp(x);


6/23

Numerical IntegrationNumerical Integration

RectangularRule

% This program is developed to calculate the integral of x^2 at any

% point you choose.

a=input('Enter the value of a')b=input('Enter the value of b')

dx=input('choose the step dx')

n=b-a/dx;

sum=0; % Initialization

for i=1:nsum=sum+neja_fun3(a+(i-1)*dx)*dx % Left rectangular rule.

end

% This function is develped to calcualate the integral of y=x^2

function[y]= neja_fun3(x)

y=x^2;


7/23

ContCont

Trapezoidal

Rule

help Integration ;%clear memory and print the headerfor x=0:0.5:2

f='78*x^2+10';

h=0.5;

fo=subs(f,'0','x');

eval(fo);

f1=subs(f,'0.5','x');

eval(f1);

f2=subs(f,'1','x');eval(f2);

f3=subs(f,'1.5','x');

eval(f3);

f4=subs(f,'2','x');

eval(f4);

f='78*x^2+10' ; %creates symbollic function

Iexact=int(f,0,2); %computes the exact value of the integration with limit of integration 0 and 2.

d=eval(Iexact);Iestim=h/2*(fo+2*f1+2*f2+2*f3+f4); %estimates the integration with in the given interval

e=eval(Iestim);

end

fprintf('The Analytical Soluion of the Integral is=%f\n',d)

fprintf('The Trapizoidal Estimate of the Integral is=%f\n',e)


8/23


9/23

O

rdinary Differential EquationsO

rdinary Differential EquationsEuler forward Method

%A program that computes second order ode given by

%using four alternative numerical methods.

clear global; help Numerical_methods;y(1)=1; x(1)=2;

h=0.1;

t=0:h:5;

for i=1:length(t)-1

y(i+1) = y(i)+h*x(i);x(i+1) = x(i)+4*h*y(i);

end

yexact=exp(2*t);

plot(t,y,'-b',t,yexact,'-g')

y"-4y=0


10/23

ContC

onthold on

for i=1:length(t)-1

xm(i+1) = x(i);

y(i+1) = y(i)+h/2*(x(i)+xm(i+1));

x(i+1) = x(i)+2*h*((y(i)+y(i+1)));

while abs(x(i+1)-xm(i+1))>0.001

x(i+1) = x(i)+2*h*((y(i)+y(i+1)));

xm(i+1) = x(i+1);

end

end

plot(t,y,'-r')

Euler Modified Method


11/23

ContC

onthold on

for i=1:length(t)-1

k1 = h*x(i);m1 = 4*h*y(i);

k2 = h*(x(i)+m1);

m2 = 4*h*(y(i)+k1);

y(i+1) = y(i)+0.5*(k1+k2);

x(i+1) = x(i)+0.5*(m1+m2);end

plot(t,y,':y')

Second - Order Runge Kutta Method


12/23

ContC

ontThird - Order Runge Kutta Methodhold on

for i=1:length(t)-1

k1 = h*x(i);

m1 = 4*h*y(i);

k2 = h*(x(i)+m1/2);

m2 = 4*h*(y(i)+k1/2);

k3 = h*(x(i)-m1+2*m2);

m3= 4*h*(y(i)-k1+2*k2);y(i+1) = y(i)+1/6*(k1+4*k2+k3);

x(i+1) = x(i)+1/6*(m1+4*m2+m3);

end

plot(t,y,':r')


13/23

ContC

ontFourth - Order Runge Kutta Methodhold on

for i=1:length(t)-1

k1 = h*x(i);

m1 = 4*h*y(i);k2 = h*(x(i)+m1/2);

m2 = 4*h*(y(i)+k1/2);

k3 = h*(x(i)+m2/2);

m3= 4*h*(y(i)+k2/2);

k4 = h*(x(i)+m3);

m4= 4*h*(y(i)+k3);

y(i+1)=y(i)+1/6*(k1+2*(k2+k3)+k4);

x(i+1)=x(i)+1/6*(m1+2*(m2+m3)+m4);

end

plot(t,y,':b')


14/23

ContC

ontxlabel('Time(sec)')

ylabel('y(m)')

title('Comparison of Different Numerical Methods')

legend('Euler forward','Analytical','Euler modified','Second-order R-K','Third-order R-K','Fourth order R-K')


15/23

Simple PendulumSimple Pendulumclear all; clc; clf;L=10; g=9.8; T=2*pi*sqrt(L/g);

h=0.04; theta(1)=0.5*pi;

w(1)=0;t=0:100*T;

for i=1:length(t)-1

k1=h*w(i);m1=-h*(g/L)*sin(theta(i));

k2=h*(w(i)+(m1)/2);

m2=-h*(g/L)*(sin(theta(i))+(k1)/2);

k3=h*(w(i)+(m2)/2);

m3=-h*(g/L)*(sin(theta(i))+(k2)/2);

k4=h*(w(i)+m3);

m4=-h*(g/L)*(sin(theta(i))+k3);

theta(i+1)=theta(i)+1/6*(k1+2*(k2+k3)+k4);

w(i+1)=w(i)+1/6*(m1+2*(m2+m3)+m4);

end

U

+(L/g)sin(U

)=0

Equation of Motion


16/23

Spectral AnalysisSpectral Analysis% Calculate data

f1=input

f2=input

t=

x = sin(2*pi*f1*t) +sin(2*pi*f2*t)+ 2*randn(size(t));

y = fft(x); %%Taking the 512-point fast Fourier transform (FFT): y = fft(x,512);

%m = y.*conj(y)/512; %%The power spectrum, a measurement of the power

at various frequencies, is m = y.* conj(y) / 512;

%f = 1000*(0:256)/512;; %%The data sampled at 1000 Hz.

f = (0:length(y)-1)'*100/length(y);;

K=abs(y);

p=unwrap(angle(y));


17/23

ContCont

% Create time plot

plot(t,x)

xlabel('Time (Seconds)')

legend('Time Series (Unfiltered)')

title('sin(2*pi*f1*t) +sin(2*pi*f2*t)+ 2*randn(size(t))','fontsize',12.2)

%title('Time Plot of sin(2*pi*f1*t) + sin(2*pi*f2*t)')

grid on

% Create frequency plot

plot(f,K)


18/23

ContC

ontxlabel('Frequency (Hz)')

legend('Frequency Series (Unfiltered)')

%title('Frequency Plot of sin(2*pi*f1*t) + sin(2*pi*f2*t)')

grid on

%Creat frequency plot of noise free wavex = sin(2*pi*f1*t) +sin(2*pi*f2*t);

y = fft(x);

m = y.*conj(y)/512;

K=abs(y);

%f = 1000*(0:256)/512;; %

f = (0:length(y)-1)'*100/length(y);;

plot(f,K)

xlabel('Frequency (Hz)')

%title('Frequency Plot of sin(2*pi*f1*t) + sin(2*pi*f2*t)')

legend('Frequency Series (Filtered)')


19/23

ContC

ontx = sin(2*pi*f1*t) +sin(2*pi*f2*t);

y = fft(x,512); %%Taking the 512-point fast Fourier transform (FFT): y = fft(x,512);

m = y.*conj(y)/512; %%The power spectrum, a measurement of the power at

various frequencies, is m = y.* conj(y) / 512;

f = (0:length(y)-1)'*100/length(y);; %%The data sampled at 1000 Hz.%f = 1000*(0:256)/512;; %

p=unwrap(angle(y));

% Create time plot

plot(f,p*180/pi)

legend('Phase Angle (Filtered)')


20/23

InterpolationInterpolationLinear Interpolation 1%This program is developed to determine the missed data point in a given

%data points by using linear interpolation.

clear; help linear; %clear memory and print the header

x=input('Enter the value of x')

datax=[1 3 4 6];

datay=[3 9 12 18];

if x>=1 & x


21/23

ContC

ontLinear Interpolation 2

%This program is developed to determine the missed data point in a given

%data points by using linear interpolation.

x=input('Enter the value of x')

datax=[0,20];datay=[10,50];

if x>=0 & x


22/23

Interpolation and Extrapolation of ln2Interpolation and Extrapolation of ln2% Interpolation and Extrapolation-This program is developed to %determine the missed

data point by using interpolation and %extrapolation between the given functionclear;help Interpolation and extrapolation;

datax=[1 4];

datay=[0 1.386294];

x=input('Enter the value of x that you choose')

if x>=1 & x


23/23

ContCont

% Extrapolate

datax=[1 4]

datay=[0 1.386294]

x=input('Enter the value of x that you choose')

if x=4z=datay(1)+(x-datax(1))*(datay(2)-datay(1))/(datax(2)-datax(1))

end

datay=exp(datax);

subplot(212)

plot(datax,datay,'o-', x,z,'*r')xlabel('Dependant variabel')

ylabel('Independent variabel')

title('The first order Lagrange extrapolation')

numerical approaches via matlab

Documents