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Plots

Problem 2

The desired convergence was reached at N=20

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Problem 3

Plot for finding initial guesses for solving eigenvalues:

Plotting function expansion for increasing number of terms N = 1, 2, 3, 4, and 5.

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Problem 4

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MATALB Source code

PROBLEM 2

clear all

clc

close all

N=20;

%Half-period

L=1;

%Definition of the function to be simulated

syms x

f = x -x.^2;

%Fourier sine coefficients

n=0;

forj=1:N

B(j)=2*double(int(f*sin(((2*n+1)/2)*pi*x),0, L));

n=n+1;

end

%X-vector for plottingxx=(0:0.01*L:L);

%Reconstruction of the function

fsim=0;

n=0;

forj=1:N

fsim=fsim + B(j)*sin(((2*n+1)/2)*pi*xx);

n=n+1;

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end

%Plot of original and simulated function

close

figure('Position',[20,150,700,300],'Color','w')

subplot('Position', [0.10,0.18,0.85,0.73])

ezplot(f, [0,L])

hold on

plot(xx, fsim, 'linewidth', 1.5, 'color', 'r')

hold off

axis([0,L, -0.5, 0.5])

grid on

set(gca,'fontsize',11)

xlabel('{\it x/L}','FontName', 'Times','FontAngle', 'normal','fontsize',16);

ylabel('{\it f}','FontName', 'Times','FontAngle', 'normal','fontsize',16);

legend('Original function', ['Simulated function (', num2str(N), ' terms)'], -1)

PROBLEM 3

clc

clear all

close all

syms x

ezplot(tan(sqrt(x)),[0 300 -15 15]) % ezplot(fun2,[xmin,xmax,ymin,ymax])

hold on

ezplot(-sqrt(x), [0, 300 -15 15])

title( 'tan(x^1/2) = -x^1/2')

% Numerical evaluation of eignevalues

% vpasolve returns the closest, depending on the initial guess,

% The initial guesses were changed based on the intersection points on the

% plot

vpasolve(tan(sqrt(x)) == -sqrt(x),x,5)

vpasolve(tan(sqrt(x)) == -sqrt(x),x,30)

vpasolve(tan(sqrt(x)) == -sqrt(x),x,63)

vpasolve(tan(sqrt(x)) == -sqrt(x),x,123)

vpasolve(tan(sqrt(x)) == -sqrt(x),x,200)

Plotting Eigenfunctions

figure

ezplot(sin(sqrt(4.1.*x)),[0,1])

x1 = 0.15

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y1 = sin(sqrt(4.1.*x1));

str1 = '\leftarrow X_1 = sin \lambda_1^{1/2}';

text(x1,y1,str1)

hold on

ezplot(sin(sqrt(24.1.*x)),[0,1])x1 = 0.08

y1 = sin(sqrt(24.1.*x1));

str1 = '\leftarrow X_2 = sin \lambda_2^{1/2}';

text(x1,y1,str1)

hold on

ezplot(sin(sqrt(63.7.*x)),[0,1])

x1 = 0.1

y1 = sin(sqrt(63.7.*x1));

str1 = '\leftarrow X_3 = sin \lambda_3^{1/2}';

text(x1,y1,str1)

hold on

ezplot(sin(sqrt(122.8.*x)),[0,1])

x1 = 0.1y1 = sin(sqrt(122.8.*x1));

str1 = '\leftarrow X_4 = sin \lambda_4^{1/2}';

text(x1,y1,str1)

hold on

ezplot(sin(sqrt(201.9.*x)),[0,1])

x1 = 0.1

y1 = sin(sqrt(201.9.*x1));

str1 = '\leftarrow X_5 = sin \lambda_5^{1/2}';

text(x1,y1,str1)

Plotting Fourier Sine Seriesclear all

clc

close all

N = 5;

eig = [4.1159 24.1393 63.6591 122.8892 201.8513];

%Half-period

L=1;

%Definition of the function to be simulatedsyms x

f = x-x.^2;

%Fourier sine coefficients

forj=1:N

d = 1/2 - (1/(4*sqrt(eig(j))))*sin(2*sqrt(eig(j)));

B(j)=double(int(f*sin(sqrt(eig(j))*x),0,L))/d;

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end

%X-vector for plotting

xx=(0:0.01*L:L);

%Reconstruction of the function

fsim=0;

forj=1:N

fsim=fsim + B(j).*sin(sqrt(eig(j))*xx);

end

%Plot of original and simulated function

close

figure('Position',[20,150,700,300],'Color','w')

subplot('Position', [0.10,0.18,0.85,0.73])

ezplot(f, [0,L])

hold on

plot(xx, fsim, 'linewidth', 1.5, 'color', 'r')

hold offaxis([0,L, -0.5, 0.5])

grid on

set(gca,'fontsize',11)

xlabel('{\it x/L}','FontName', 'Times','FontAngle', 'normal','fontsize',16);

ylabel('{\it f}','FontName', 'Times','FontAngle', 'normal','fontsize',16);

legend('Original function', ['Simulated function (', num2str(N), ' terms)'], -1)

PROBLEM 4

clear all

clc

close all

%number of eigenfunctions considered

N=12;

%dx increment

dx=0.00001;

%Evaluating Legendre polinomials

%x interval for Legendre polinomial

x = (-1:dx:1);

leg0 = legendre(0,x);

leg1 = legendre(1,x);

leg2 = legendre(2,x);

leg3 = legendre(3,x);

leg4 = legendre(4,x);

leg5 = legendre(5,x);

leg6 = legendre(6,x);

leg7 = legendre(7,x);

leg8 = legendre(8,x);

leg9 = legendre(9,x);

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leg10 = legendre(10,x);

leg11 = legendre(11,x);

%Final vector, each row contains the LEgendre polinomial evaluated at x

%interval

leg = [leg0(1,:) ; leg1(1,:) ; leg2(1,:) ; leg3(1,:) ; leg4(1,:) ; leg5(1,:) ;

leg6(1,:) ; leg7(1,:) ; leg8(1,:); leg9(1,:) ; leg10(1,:); leg11(1,:)];

figure

plot(x,leg(1,:))

hold on

plot(x,leg(2,:))

hold on

plot(x,leg(3,:))

hold on

plot(x,leg(4,:))

hold on

plot(x,leg(5,:))

hold onplot(x,leg(6,:))

legend('P0','P1','P2','P3', 'P4', 'P5')

%Numerical integration X limits

X=(-1:dx:1);

%Function to be simulated

fori=1:length(X)

ifX(i)< 0

f(i) = -1;

else

f(i)= 1;

endend

%Series expansion coefficient numerical integration versus dx

n=0;

forj=1:N

c(j)=0;

fork=1:length(X)

fun = f(k);

legpol = leg(j,k);

c(j)=c(j)+dx.*fun.*legpol;

end

%Scaling cn by 2n+1/2

c(j)=((2*n+1)/2)*c(j);

n=n+1;

end

%Removing odd terms of c because they are supposed to be zero

c = c(abs(c)>0.01)

%Reconstruction of the function

fsim=0;

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n=0;

forj=1:6

fsim=fsim + c(j).*leg(2*j,:);

n=n+1;

end

%Plot of original and simulated function

close

figure('Position',[20,150,700,300],'Color','w')

subplot('Position', [0.10,0.18,0.85,0.73])

plot(X,f)

hold on

plot(X, fsim, 'linewidth', 1.5, 'color', 'r')

hold off

axis ([-1 1 -3 3])

grid on

set(gca,'fontsize',11)

xlabel('{\it x/L}','FontName', 'Times','FontAngle', 'normal','fontsize',16);

ylabel('{\it f}','FontName', 'Times','FontAngle', 'normal','fontsize',16);legend('Original function', ['Simulated function (', num2str(j), ' terms)'], -1)

Published with MATLAB R2015a

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