Chapter 1Chapter 1Chapter 1Chapter 1

>> r=2.5; >> I=(pi/8-8/(9*pi))*r^4

>> c=5; >> k=(4*c-1)/(4*c-4)+0.615/c

>> B=0.6; >> K=3/(1-B)^3*(0.5-2*B+B*(1.5-log(B)))

Homework Problems not solved.

>> A=1.7;B=1.2; >> D=1.265*((A*B)^3/(A+B))^(1/5)

>> n=6;M=1/sin(pi/n);h=(1+M^2)/(4*M); >> alpha=acos(sqrt(h^2+2)-h); >> answer=M*(1-M^2)*sin(alpha)/(1+M^2-2*M*cos(alpha))^2

>> L=3000;d=45;V=1600; >> p=0.03*L/d^1.24*(V/1000)^1.84

>> v=0.3;E=3e+7;d1=1.5;d2=2.75;F=100;z =0.01 >> a=(3*F/8*2*((1-v^2)/E)/(1/d1+1/d2))^(1/3); >> pmax=3*F/(2*pi*a^2); >> Qz=-pmax/(1+z^2/a^2) >> Qx=-pmax*((1-z/a*atan(a/z))*(1-v)-0.5*(1+z^2/a^2)^-1)

>> v=0.3;E=3e+7;d1=1.5;d2=2.75;F=100;L=2;z=0.001; >> b=sqrt(2*F/(pi*L)*2*((1-v^2)/E)/(1/d1+1/d2));h=(1+z^2/b^2); >> pmax=2*F/(pi*b*L); >> Qz=-pmax/sqrt(h); >> Qy=-pmax*((2-1/h)*sqrt(h)-2*z/b); >> Qx=-2*v*pmax*(sqrt(h)-z/b); >> Tyz=0.5*(Qy-Qz);

>> e=0.8;a=(1-e^2); >> NL=pi*e*sqrt(pi^2*(a)+16*e^2)/a^2

>> h=1.25; d0=0.25; d1=0.625; E=3e+7; d2=d1+h*tan(pi/6); >> y=log((d2-d0)*(d1+d0)/((d2+d0)*(d1-d0))); >> k=pi*E*d0*tan(pi/6)/y

>> alpha=1.2e-5;E=3e+7;v=0.3;Ta=500;Tb=300;a=0.25;b=0.5;r=0.375; >> Tc=Ta-Tb; c=log(b/a); d=log(b/r); k=a^2/(b^2-a^2); >> T=Tb+Tc*d/c; h=alpha*E*Tc/(2*(1-v)*c); >> Qt=h*(1-k*(b^2/r^2+1)*c-d); >> Qr=h*(k*(b^2/r^2-1)*c-d);

>> p=0.3;k=1.4; % pe/po=p >> Y=sqrt(k/(k-1)*(p^(2/k)-p^((k+1)/k)))

>> x=0.45;y=sqrt(16*x^2+1); >> K=1.2/x*(y+1/(4*x)*log(y+4*x))^(-2/3)

>> n1=0; n2=1; >>Pi1=(sqrt(8)/9801*gamma(4*n1+1)*(1103+26390*n1)/(gamma(n1+1)^4*396^(4*n1)))^(-1); >>Pi2=(1/Pi1+(sqrt(8)/9801*gamma(4*n2+1)*(1103+26390*n2)/(gamma(n2+1)^4*396^(4*n2))))^-1; >> % These two statement cannot be broken up as shown. They have been presented in two lines because of page width restrictions. >> pi-Pi1 %show the approximation of pi with a different less than 10^-7 ans = -7.64235e-008 >> pi-Pi2 %show the approximatoin of pi with a different less than 10^-15 ans = -4.44089e-016

>> k=1.4; r=10; rc=3; >> eta=1-1/r^(k-1)*(rc^k-1)/(k*(rc-1))

>> k=1.4 ; M=2 ; %A/A*=X >> X=1/M*(2/(k+1)*(1+(k-1)/2*M^2))^((k+1)/(2*(k-1)))

Chapter 2Chapter 2Chapter 2Chapter 2

>> a=-1:2:13;b=1:2:15; >>a+b; %part a) >>a-b; %part b) >>a'*b ; det(a'*b); %part c) >>a*b'; %part d)

>> x=[17 -3 -47 5 29 -37 51 -7 19]; a=sort(x); >> b=sort(a(1:4),'descend');c=sort(a(5:9),'descend'); y=[b c]; >> %we can use fliplr function too. >> b=fliplr(a(1:4));c=fliplr(a(5:9)); y=[b c];

>> y=[0 -0.2 0.4 -0.6 0.8 -1.0 -1.2 -1.4 1.6]; >> z=sin(y);h=sort(z) h = -0.985 -0.932 -0.841 -0.564 -0.198 0 0.389 0.717 0.999 >> a = min(h(1:5)); b = max(h(1:5)); %part a) >> c=sqrt(h(6:9)); %part b)

>> x=logspace( log10(6) , log10(106) , 8); >> y=x(5); %part a) >> h=[x(1) x(3) x(5) x(7)]; %part b)

>> z=magic(5); >> z(: , 2)=z(: , 2)/sqrt(3); % part a) I >> z(5 , :)=z(3 , :) + z(5 , :); % part a) II >> z(: , 1)=z(: , 1) .* z(: , 4); % part a)III >> q=z-diag(diag(z))+diag([2 2 2 2 2]); % part a) IV >> diag(q*q'); % part b) >> c=q.^2; % part c) >> max(max(c)); min(min(c)); % part d)

>> w=magic(2); >> a=repmat(w, 2, 2); % part a) >> b=repmat(w, 3, 1); % part b) >> c=repmat([w w'], 3 ,1); % part c) >> da=[w w ;w w];db=[w ;w ;w];dc=[w w';w w';w w']; % part d)

>> x=magic(3); >> new=[x(2,:) ; x(3,:) ; x(1,:)]; % part a >> new=[x(:,3) , x(:,1) , x(:,2)]; % part b

>> a=1;b=1.5;e=0.3;phi=0:360; >> s=a*cosd(phi)+sqrt(b^2-(a*sind(phi)-e).^2); >> plot(phi,s)

>> t=1/sqrt(19);Pt=4.3589;n=linspace(2,25,300); %to/T=t >> Po=1+2*sin(n*pi*t).^2./(n*pi*t).^2; >> plot(n,Po)

>> x=1:0.5:5; >> n=1:100; >> [xx,nn]=meshgrid(x,n); >> sn=prod(1-xx.^2./(nn.^2-2.8)); >> sj=a./(sin(pi*a)*sqrt(a^2+x.^2)).*sin(pi*sqrt(a2̂+x.^2)); >> en=100*(sn-sj)./sj;

>> x=[72 82 97 113 117 126 127 127139154 159 199 207]; >> B=3.644; >> Delta=(1/length(x))*sum(x.^B) Delta = 3.0946e+028

>> phi=linspace(0,90,10);theta=linspace(0,360,24);b=2; >> [theta2,phi2]=meshgrid(theta,phi); >> x=b*sind(phi2).*cosd(theta2); >> y=b*sind(phi2).*sind(theta2); >> z=b*cosd(phi2); >> mesh(x,y,z); >>% The result after using the rotate icon in the figure window.

>> x=linspace(0.1,1,5); y=pi*x*sqrt(2); n=0:25; >> t=2*pi^4./y.^3.*(sinh(y)+sin(y))./(cosh(y)-cos(y)); >> [xx,nn]=meshgrid(x,n); >> S= sum(1./(nn.^4+xx.^4)); >> compare=[t' S' (t-S)'];

>> x=[1 2 3 4 5 6]; k=0:25;n=2; >> [xx,kk]=meshgrid(x,k); >> p=sum((-1).^kk.*(xx/2).^(2.*kk+n)./(gamma(kk+1).*gamma(kk+n+1))); >> besselj(2,x); >> r=besselj(2,x); >> compare=[p' r']

>> k=1:13;n=7; >> s=sum(cos(k*pi/n))

>> w=0.5*[-1 -1; 1 -1;-1 1; 1 1]; q=0.5*[1 -1 -1 1; 1 1 -1 -1; 1 -1 1 -1; 1 1 1 1]; >> I1=w'*w; I2=q'*q;

>>e=pi/6;a1=1;a2=2; a3=3; >>A1=[cos(e) -sin(e) 0 a1*cos(e);sin(e) cos(e) 0 a1*sin(e);0 0 1 0;0 0 0 1]; >>A2=[cos(e) -sin(e) 0 a2*cos(e);sin(e) cos(e) 0 a2*sin(e);0 0 1 0;0 0 0 1]; >>A3=[cos(e) -sin(e) 0 a3*cos(e);sin(e) cos(e) 0 a3*sin(e);0 0 1 0;0 0 0 1]; >>T3=A1*A2*A3; >>qx=T3(1,4);qy=T3(2,4); >>%For finding orientation with respect to Fig2.8 orientation of X3 is %arctan(uy/ux) and orientation of Y3 is arctan(vy/vx) >>X3=atand(t(2,1)/t(1,1)) X3 = 90.0 >>Y3=atand(t(2,2)/t(1,2)) Y3 = 0.00 >>%They are parallel to Y0 and X0 axes respectively.

>> X= [17 31 5; 6 5 4; 19 28 9; 12 11 10];

>> H=X*inv(X'*X)*X'; >> diag(H)

Part a) >> n=1:2:399; t=linspace(-0.5,0.5,200); >> u= sin(2*pi*n'*t); >> fa=4/pi*(1./n*u); >> plot(t,fa)

part b) >> n= 1:200; t=linspace(-1,1,200); >> u= sin(2*pi*n'*t); >> fb=1/2+1/pi*(1./n*u); >> plot(t,fb) >> % figure of part a & b

part c) >> n= 1:200; t=linspace(-1,1,200); >> u= sin(2*pi*n'*t); >> fb=1/2-1/pi*(1./n*u); >> plot(t,fb)

part d) >> n= 1:200; t=linspace(-1,1,200); >> u= cos((2*n-1)'*pi*t); >> fd=pi/2-4/pi*(1./(2*n-1).^2*u); >> plot(t,fd) >> % figure of part c & d

part e) >> n= 1:200; t=linspace(-1,1,200); >> u= cos(2*n'*pi*t); >> fe=2/pi+4/pi*(1./(1-4*n.^2)*u); >> plot(t,fe) part f) >> n=2:2:106;t=linspace(-2,2,200); >> ff=1/pi+1/2* sin(pi*t)-2/pi*(1./(n.^2-1)*cos(pi*n'*t)); >> plot(t,ff)

>> % Figure of part e & f

part g) >> n=1:250;t=linspace(0,4*pi,350); >> u1=1./(1+n.^2)*cos(n'*t); >> u2=n./(1+n.^2)*sin(n'*t); >> fg=(exp(2*pi)-1)/pi*(1/2+u1-u2); >> plot(t,fg) part h) >> a=0.25;n=1:2:105;t=linspace(2,-2,200); >> fh=4/a^2*(sin(n*pi*a)./(pi*n).^2*sin(n'*pi*t)); >> plot(t,fh) >> % Figure of part g & h

>> n=1:25;a=sqrt(3);theta=(10:10:80)*pi/180; >> S1n=1./(n.^2+a^2)*cos(n'*e); >> S1inf=pi*cosh(a*(pi-e))/(2*a*sinh(pi*a))-1/(2*a2̂); >> S2n=n./(n.^2+a^2)*sin(n'*e); >> S2inf=pi*sinh(a*(pi-e))/(2*sinh(pi*a)); >> e1=100*((S1n-S1inf)./S1inf) >> e2=100*((S2n-S2inf)./S2inf)

>> eta=0:1/14:1;E=0:1/14:1;n=1:2:length(E);a=2; >> u1=1./(n.*sinh(n*pi*2)); u2=sinh(n'*eta*pi*2); u3=sin(pi*n'*E); >> [eta1,u11]=meshgrid(eta,u1); >> T=(u11.*u2)'*u3; >> mesh(E,eta,T)

>> n=1:50;a=0.25;t=0:0.05:2;eta=0:0.05:1; >> u1=sin(n*pi*a)./n.^2;u2=sin(n'*pi*eta);u3=cos(n'*pi*t); >> [eta1,u11]=meshgrid(eta,u1); >> c=2/(a*pi*(1-a)); >> u=c*(u11.*u2)'*u3; >> mesh(t,eta,u) >>% The result after using the rotate icon in the figure window.

Homework problems not solved.

>> v=0.4;E1=3e+5;E2=3.5e+4;Uo=0.01; >> a=0.192;b=0.25;c=0.312;t=1+v;h=1-v; >> S=[1 a^2 0 0 1 b^2 -1 -b^2 -t h*b^2 t*E1/E2 -h*b^2*E1/E2 0 0 -t h*c^2] S = 1.0000 0.0369 0 0 1.0000 0.0625 -1.0000 -0.0625 -1.4000 0.0375 12.0000 -0.3214 0 0 -1.4000 0.0584 >> y=[0 0 0 -Uo*E2*c]; >> w=inv(S)*y';x=w'; >> A1=x(1);B1=x(2);A2=x(3);B2=x(4); >> Qee1=-A1/b^2+B1; >> Qee2=-A2/b^2+B2;

Chapter 3Chapter 3Chapter 3Chapter 3

>>a=(1+sqrt(5))/2;b=(1-sqrt(5))/2;c=1/sqrt(5); >>n=0:15; >>Fn=c*(a.^n-b.^n); >>disp([repmat('F',16,1) num2str(n') repmat(' = ',16,1) num2str(Fn') ]); >>fprintf(1,'F %2.0f = %3.0f\n ',[n ;Fn]);

% To go next line without execution use shift+Enter >>ft=input('Enter the value of length in feet :'); disp([num2str(ft) ' ft = ' num2str(ft*0.3048) ' m'])

Homework Problem not solved.

>>D=input('Enter a positive integer < 4.5x10^15 : '); % shift+Enter disp(['The binary representation of ' num2str(D) ' is ' dec2bin(D)] )

z=input('Enter the complex number:'); % shift+Enter disp(['The magnitude and phase of ' num2str(z) ' is']); % shift+Enter disp(['Magnitude = ' num2str(abs(z)) ' phase angle = ' num2str(angle(z)*180/pi) ' degrees']); >>%The last ‘disp’ statement cannot be broken up as shown. It has been presented in two lines because of page width restrictions.

Chapter 4Chapter 4Chapter 4Chapter 4

function y=myexp1(h,a,b) for n=1:length(h); if h(n)>a && h(n)<b; h(n)=1; else h(n)=0; end end disp([h]) end

function myexp2(x) k=0; for m=2:length(x) xn=1/m*sum(x(1:m)); Sn2=1/(m-1)*(sum(x(1:m).^2)-m*(xn).^2); k=k+1; y(k)=Sn2; end disp(['Sn2 = [' num2str(y) ' ]']) end

function myexp3(a,b) k=1; c=0.5*(b+a/b); %where c=b(n+1) while abs(b-c)>1e-6 k=k+1; b=0.5*(b+a/b); c=0.5*(b+a/b); end disp(['n-iteration=' num2str(k)])

end

part a)* function y=myexp4(x0) x0=0:200; r=x0.^2+0.25; x=0:5:200; for t=1:(length(x)-1); y(t)= r(5*t); end y=[r(1) y]; plot(x,y,'ks') end part b) function y=myexp5(x0) x0=0:200; r=x0.^2+0.25; x=0:5:200;y=[];k=0; while (length(x)-1)~=length(y) k=k+1; y(k)= r(5*k); end y=[r(1) y]; plot(x,y,'ks') end

function chisquare(x,e) k=0;e=[e 5];x=[x 0]; for n=1:length(e) if e(n)<5 e(n:n+1)=cumsum(e(n:n+1));

x(n:n+1)=cumsum(x(n:n+1)); else k=k+1; p(k)=n; end ;end h=x(p);u=e(p); %e modified= u and x modified=h if (u(end)-5)<5 u(end)=u(end)-5+u(length(u)-1); u(length(u)-1)=[]; h(end)= h(length(h)-1)+h(end); h(length(h)-1)=[]; end y=sum((h-u).^2./u); disp(['e modified = [' num2str(u) '], x modified = [' num2str(h) '], X2=' num2str(y) ]) end >>%Thist ‘disp’ statement cannot be broken up as shown. It has been presented in two lines because of page width restrictions.

function myexp6(p,s) a=length(p) ;b=length(s); if a>=b t=s+p(a-b+1:a); h=[p(1:(a-b)) t] elseif a<b t=p+s(b-a+1:b); h=[s(1:b-a) t] end end

Homework problems not solved.

ChapterChapterChapterChapter 5555

>>h=[6 3 2 1.5 1.2 1.1 1.07 1.05 1.03 1.01]; %h=D/d >>a=[0.33 0.31 0.29 0.26 0.22 0.24 0.21 0.2 0.18 0.17]; >>c=[0.88 0.89 0.91 0.94 0.97 0.95 0.98 0.98 0.98 0.92]; >>%Fifth order polynomial >>aa=polyfit(h,a,5);cc=polyfit(h,c,5); >>%Obtain Orginal Value Of Kt >>kt=c.*(h/2-0.5).^(-a); >>%Compare >>k1=polyval(cc,h).*(h/2-0.5).^(-polyval(aa,h)); >>k1-k ans = Columns 1 through 6 -1.0971e-008 -1.8084e-005 8.9483e-4 -0.011004 0.087181 -0.09516 Columns 7 through 10 -0.059276 -0.060901 0.016685 0.16796 >>%Fit data with a spline >> n=linspace(1.01,6,100); >> aa=spline(h,a,n); >> cc=spline(h,c,n); >> k=0; >> for h=[6 3 2 1.5 1.2 1.1 1.07 1.05 1.03 1.01] k=k+1; a2(k)=interp1(n,aa,h); c2(k)=interp1(n,cc,h); end >>k2=c2.*(h/2-0.5).^(-a2) >>%Compare >>k2-k >> -kt+kt2 ans = 0 0.0005 0.0009 -0.0062 0.0101 -0.0271 0.0321 -0.0637 -0.0408 0 >>%2nd method (spline) is better.

>> Qx=100;Qy=-60;Qz=80;Txy=-40;Tyz=50;Tzx=70; >> C0=Qx*Qy*Qz+2*Txy*Tyz*Tzx-Qx*Tyz^2-Qy*Tzx^2-Qz*Txy^2; >> C1=Txy^2+Tyz^2+Tzx^2-Qx*Qy-Qy*Qz-Qz*Qx; >> C2=Qx+Qy+Qz; >> disp([C0 C1 C2]) -844000 11800 120 >> F=inline('Q.^3-120*Q.^2-11800*Q-(-844000)','Q'); >> x=linspace(-100,200,500); >> plot(x,F(x)); >> Q1=fzero(F,170);Q2=fzero(F,50); Q3=fzero(F,-100); >> T12=(Q1-Q2)/2;T23=(Q2-Q3)/2;T13=(Q1-Q3)/2;

>> F=inline('tan(x)-x','x'); >> x=linspace(0,15,5000); >> plot(x,F(x),'k',[0 15],[0 0],'r') >> axis([0 15 -5 5]) >> r1=fzero(F,[-1 1]);r2=fzero(F,[4.2 4.6]);r3=fzero(F,[7.6 7.8]); >> r4=fzero(F,[10.85 10.95]);r5=fzero(F,[14 14.1]); >> disp([r1 r2 r3 r4 r5]);

>> %Part a >> H=inline('2*cot(x)-10*x+0.1./x','x'); >> x=linspace(0,15,5000); >> plot(x,H(x),'k',[0 15],[0 0],'r'); >> axis([0 15 -4 4]) >> r1=fzero(H,[0.4 0.6]);r2=fzero(H,[3.2 3.3]);r3=fzero(H,[6.3 6.4]); >> r4=fzero(H,[9.44 9.46]);r5=fzero(H,[12.57 12.6]); >> disp([r1 r2 r3 r4 r5]); >> %Part b >> H=inline('2*cot(x)-x+1./x','x'); >> x=linspace(0,15,5000);

>> plot(x,H(x),'k',[0 15],[0 0],'r'); >> axis([0 15 -4 4]) >> r1=fzero(H,[1 2]);r2=fzero(H,[3.5 4]);r3=fzero(H,[6.5 7]); >> r4=fzero(H,[9.5 10]);r5=fzero(H,[12.6 13]); >> disp([r1 r2 r3 r4 r5]);

>> G=inline('besselj(0,x).*bessely(0,2*x)-besselj(0,2*x).*bessely(0,x)','x'); >> x=linspace(0,20,5000); >> plot(x,G(x),'k',[0 20],[0 0],'r'); >> r1=fzero(G,[2 4]);r2=fzero(G,[6,8]);r3=fzero(G,[8 10]); >> r4=fzero(G,[12 14]);r5=fzero(G,[14 16]); >> disp([r1 r2 r3 r4 r5]);

>> %Mo/mo= m >> % Case 1 >> H=inline('0*x.*(cos(x).*sinh(x)-sin(x).*cosh(x))+cos(x).*cosh(x)+1','x'); >> x=linspace(0,20,5000); >> plot(x,H(x),'k',[0 20],[0 0],'r');; >> axis([0 20 -20 20]) >> r1=fzero(H,[1 4]);r2=fzero(H,[4,6]);r3=fzero(H,[6 8]); >> r4=fzero(H,[10 12]);r5=fzero(H,[13 16]); >> disp([r1 r2 r3 r4 r5]) >> % Case 2 >> H=inline('0.2*x.*(cos(x).*sinh(x)sin(x).*cosh(x))+cos(x).*cosh(x)+1','x'); >> plot(x,H(x),'k',[0 20],[0 0],'r'); >> axis([0 20 -20 20]) >> r1=fzero(H,[1 3]);r2=fzero(H,[3,5]);r3=fzero(H,[6 8]); >> r4=fzero(H,[10 12]);r5=fzero(H,[12 14]); >> disp([r1 r2 r3 r4 r5]) >> % Case 3 >> H=inline('x.*(cos(x).*sinh(x)-sin(x).*cosh(x))+cos(x).*cosh(x)+1','x'); >> plot(x,H(x),'k',[0 20],[0 0],'r'); >> axis([0 20 -20 20])

>> r1=fzero(H,[1 3]);r2=fzero(H,[3,5]);r3=fzero(H,[6 8]); >> r4=fzero(H,[10 12]);r5=fzero(H,[12 14]); >> disp([r1 r2 r3 r4 r5])

>> >>G=inline('tanh(x)-tan(x)','x'); >> x=linspace(0,15,5000); >> plot(x,G(x),'k',[0 15],[0 0],'r'); >> axis([0 15 -1 1]) >> r1=fzero(G,[-0.5 0.5]);r2=fzero(G,[3.5 4.5]);r3=fzero(G,[6.5 7.5]); >> r4=fzero(G,[9 10.5]);r5=fzero(G,[13 14]); >> disp([r1 r2 r3 r4 r5])

>> % Case 1 >> G=inline('besselj(0,x).*besseli(1,x)+besseli(0,x).*besselj(1,x)','x'); >> x=linspace(0,15,5000); >> plot(x,G(x),'k',[0 15],[0 0],'r'); >> axis([0 15 -22 22]) >> r1=fzero(G,[0 1]);r2=fzero(G,[2 4]);r3=fzero(G,[5 7]); >> disp([r1 r2 r3]) >> % Case 2 >> G=inline('besselj(1,x).*besseli(2,x)+besseli(1,x).*besselj(2,x)','x'); >> plot(x,G(x),'k',[0 15],[0 0],'r'); >> axis([0 15 -22 22]) >> r1=fzero(G,[0 1]);r2=fzero(G,[4 6]);r3=fzero(G,[7 10]); >> disp([r1 r2 r3]) >> % Case 3 >> G=inline('besselj(2,x).*besseli(3,x)+besseli(2,x).*besselj(3,x)','x'); >> plot(x,G(x),'k',[0 15],[0 0],'r'); >> axis([0 15 -22 22]) >> r1=fzero(G,[0 1]);r2=fzero(G,[5 7]);r3=fzero(G,[7 10]); >> disp([r1 r2 r3])

>> % Case 1 >> G=inline('tan(x)-x+4*x.^3/(2*pi^2)','x'); >> x=linspace(0,15,5000); >> plot(x,G(x),'k',[0 15],[0 0],'r'); >> axis([0 15 -10 20]) >> r1=fzero(G,[0 1]);r2=fzero(G,[2 4]);r3=fzero(G,[4.75 5]); >> r4=fzero(G,[7.86 7.87]);r5=fzero(G,[10.998 11]); >> disp([r1 r2 r3 r4 r5]) >> % Case 2 >> G=inline('tan(x)-x+4*x.^3/(4*pi^2)','x'); >> plot(x,G(x),'k',[0 15],[0 0],'r'); >> axis([0 15 -10 20]) >> r1=fzero(G,[0 1]);r2=fzero(G,[2 4]);r3=fzero(G,[4.75 5]); >> r4=fzero(G,[7.86 7.9]);r5=fzero(G,[11 11.1]); >> disp([r1 r2 r3 r4 r5]) >> % Case 3 >> G=inline('tan(x)-x+4*x.^3/(8*pi^2)','x'); >> plot(x,G(x),'k',[0 15],[0 0],'r'); >> axis([0 15 -10 20]) >> r1=fzero(G,[0 1]);r2=fzero(G,[2 4]);r3=fzero(G,[5 5.5]); >> r4=fzero(G,[7.86 7.96]);r5=fzero(G,[11 11.1]); >> disp([r1 r2 r3 r4 r5])

>> %case 1 >> G=inline('(1+0.4*x).^2.*(x.^2-x.^3)-0.2','x'); >> x=linspace(0,1,500); >> plot(x,G(x),'k',[0 1],[0 0],'r'); >> axis([0 1 -1 1]) >> r1=fzero(G,[0.4 0.7]);r2=fzero(G,[0.7 1]); >> disp([r1 r2]); >> %using "roots" function >> n=0.4; m=0.2; %co=n c1=m >> y=roots([-n^2 (n^2-2*n) (2*n-1) 1 0 -m]); >> disp(y(y>0)) >> %case 2

>> G=inline('(1+7*x).^2.*(x.^2-x.^3)-4','x'); >> plot(x,G(x),'k',[0 1],[0 0],'r'); >> axis([0 1 -1 1]) >> r1=fzero(G,[0.4 0.7]);r2=fzero(G,[0.7 1]); >> disp([r1 r2]); >> %using "roots" function >> n=7; m=4; %co=n c1=m >> y=roots([-n^2 (n^2-2*n) (2*n-1) 1 0 -m]); >> disp(y(y>0))

>> G=inline('2*5^2*sind(b).^2.*tand(b-35).^2-tand(b).*tand(b-35)-tand(b).^2','b'); >>%This ‘inline’ statement cannot be broken up as shown. It has been presented in two lines because of page width restrictions. >> x=linspace(0,90,5000); >> plot(x,G(x),'k',[0 90],[0 0],'r'); >> axis([0 90 -0.5 0.5]) >> r1=fzero(G,[0 10]);r2=fzero(G,[10 40]); >> r3=fzero(G,[40 60]);r4=fzero(G,[60 85]); >> disp([r1 r2 r3 r4])

>> H=inline('(sum(x.^B.*log(x))./(sum(x.^B)-1/14*sum(log(x)))).^-1-B','x','B'); >> B=linspace(-1,1,length(x)); >> x=[72 82 97 103 113 117 126 127 127 139 154 159 199 207]; >> plot(B,H(x,B),'k',[-1 1],[0 0],'r'); >> syms B >> y=H(x,B); >> G=inline(vectorize(y),'B'); >> r=fzero(G,[0.1 0.2]);

>> H=inline('x.*log(sqrt(x.^2-1)+x)-sqrt(x.^2-1)-0.5*x','x');

>> x=linspace(1.01,4,500); >> plot(x,H(x),'k',[0 4],[0 0],'r'); >> r=fzero(H,[2 3])

Homework problems not solved.

This problem is simple but has much long number as entering data. For first part, just make an inline function with variable r and tau. In the second part we have the values of p and tau. Multiply both side of equation by r and now you have just one variable and you can use fsolve to find the value of r then put this value and tau in equation of part one you will find z(r,tau). For part c do same procedure that done for part two, but this time put tau as variable.

>> %case 1 >> G=inline('(2*log10(2.51./(1e5*sqrt(y))+0.27/200)).^-2-y','y'); >> y=linspace(0,1,200); >> plot(y,G(y),'k',[0 1],[0 0],'r') >> r=fzero(G,[0.01,1]); >> r=fzero(G,[0.01,1]) >> %case 2 >> G=inline('(2*log10((1e5*sqrt(y))/2.51)).^-2-y','y'); >> plot(y,G(y),'k',[0 1],[0 0],'r') >> r=fzero(G,[0.01,1])

Homework problems not solved.

>> %As you see this equation has unexpected parenthesis or bracket. >> %I can't find the correct equation, but solve the problem. >> %the procedure is correct, but the answer is different. It’s a simple problem. >> G=inline( '(1-(1-cos(x))).*2.*0.6.^1.5.*cos(x)' ,'x'); >> I=1/(2*pi)*quadl(G,-a,a)

>> H=inline('c1./(y.^5.*(exp(c2./(y.*T1))-1))','y','c1','c2','T1'); >> c1=3.742e8;c2=1.439e4;sig=5.667e-8;T1=300; >> Int=quadl(H,1e-6,150,[],[],c1,c2,T1) >> sig1=Int/T1^4; >> %error >> (sig-sig1)/(sig1) *100; >>%you can do this procedure for other values of T.

>> H=inline('cos(x-y).*exp(-x.*y./(pi^2))','x','y'); >> I=dblquad(H,0,pi/2,pi/4,pi);

Ode45 differential equation solver (numerically)

function xp=myexp3(t,x) L=150;m=70;k=10;co=0.00324;g=9.8; xp=zeros(2,1);

xp(1)=x(2); xp(2)=-k/m*(x(1)-L).*(x(1)>=L)-co*sign(x(2)).*(x(2).^2)+g; end >> [t,x]=ode45('myexp3',[0 15],[0 0]); >> %the first column of x is distance and the second is velocity; >> tnew=linspace(0,15,5000); >> distance=spline(t,x(:,1),tnew); >> velocity=spline(t,x(:,2),tnew); >> L=150;m=70;k=10;co=0.00324;g=9.8; >> acceleration=-k/m*(distance-L).*(distance>L)-co*sign(velocity).*(velocity.^2)+g; >>%The last statement cannot be broken up as shown. It has been >> %presented in two lines because of page width restrictions. >>%the sign of answers depends on some other principles. >> %part a) >> %using interpolation to find a value that lie between two others, >> interp1(tnew,distance,11.47); >> %part b) >> interp1(tnew,distance,5.988); interp1(tnew,velocity,5.988); >> %part c) >> interp1(tnew,acceleration,11.18)

function xp=myexp5(t,x) M=0;B=10;a=0.1; xp=zeros(2,1); xp(1)=x(2); xp(2)=-a*x(2)+sin(x(1))-B*sin(x(1)).*(1-1./sqrt(5-4*cos(x(1)))); end >> [t,x]=ode45('myexp5',[0 50],[pi/4 0]); >> r=linspace(0,50,1000); >> y=spline(t,x(:,1),r); >> plot(r,y) %for plot theta as function of t >> plot(x(:,1),x(:,2)) % for plot theta versus it’s derivative

function xp=myexp4(t,x) p=0.375;q=7.4e-4; xp=zeros(2,1); xp(1)=x(2); xp(2)=-sign(x(2)).*p*(x(2).^2)-q*x(1); end >> [t,x]=ode45('myexp4',[0 150],[10 0]); >> interp1(x(:,1),t,0); >> [t,x]=ode45('myexp4',[0 300],[20 0]); >> interp1(x((60:80),1),t(60:80),0); >>%It seems that the answers of problem in book aren’t precise because using another method to solve problem. You can check the correct answer after plot it and zoom in.You can plot the curves with this script. >> ode45('myexp4',[0 300],[20 0]);

function xp = halfsin(t,x,z) if t<=5 h=sin(pi*t/5); elseif t>5 h=0; end xp=ones(2,1); xp(1)=x(2); xp(2)=-2*z*x(2)-x(1)+h; end >> k=0; >> for z=0.05:0.05:0.95 [t,x]=ode45(@halfsin,[0 35],[1 0],[],z); k=k+1; f(k)=sum((x(:,1)-1).^2); end >> [x,y]=min(f); >> z=0.05:0.05:0.95; >> z(y);

>>%part a) >> k=0; >> for B=[0.02,0.05 , 0.08 0.11 0.15 0.18 0.23 0.3] syms x H=B./sin(x)+1./cos(x); H=inline(vectorize(H),'x'); t=fminbnd(H,0,pi/4); k=k+1; y(k)=t; end >> disp('Corresponding min value of k with respect to B');disp([num2str(y')]); >> %part b) >> H=inline('0.16./sin(x)+1./cos(x)-1.5','x'); >> plot(x,H(x),'k',[0 pi/4],[0 0],'r'); >> axis([0 pi/4 -1 1]) >> r1=fzero(H,[0.2 0.5]);r2=fzero(H,[0.5 0.7]); >> disp([r1 r2]);

>> H=inline('sqrt(1.4/0.4)*sqrt(p.^(2/1.4)-p.^(2.4/1.4))','p'); >> p=linspace(0,1,1000); >> plot(p,H(p),'k',[0 1],[0 0],'r') >> [H,index]=max(H(p)); >> maxvalue=p(index) maxvalue = 0.5285 >> k=1.4; >> verify=(2/(k+1))^(k/(k-1)) verify = 0.5283

Note: We use fsolve to solve nonlinear system of equations. We have to define equations like a matrix in editor window as function.

For “fsolve” we have to indicate our variables as elements of a vector for example if we have 3 equations and variables ‘x’, ’y’, ’z’, we can’t use all of them in our equations. We select a variable like ‘h’ and use h(1) , h(2) , h(3) as our variables that ‘h’ refers to the vector after execution of fsolve function . %For part (a ) first define function in editor window as below: function w=myexp(e) w=[e(1)*(1-cos(e(2)))-3;e(1)*(e(2)-sin(e(2)))-1]; end >> option=optimset('display' , 'off'); >> z=fsolve(@myexp,[1 1],option); >> %part b) >> H=inline('3*(x-sin(x))-(1-cos(x))','x'); >> theta=fzero(H,1); >> k=3/(1-cos(theta))

function u=myexp2(x) sig=5.667e-8;T1=373;T2=293; u=[T1^4-x(1)^4-x(2)/sig; x(1)^4-x(3)^4-x(2)/sig;x(3)^4-T2^4-x(2)/sig]; end >> option=optimset('display' , 'off'); >> z=fsolve(@myexp2,[10 10 10],option); >> %part b) >> A=[1 0 1/sig;1 -1 -1/sig;0 1 -1/sig]; >> B=[T1^4 0 T2^4]; >> x=inv(A)*B'; >> %where x(1)=Ta^4 x(2)=Tb^4 >> Ta=(x(1))^(1/4);Tb=(x(2))^(1/4);Q=x(3);

>> %part a) >> syms x >> limit((1-sin(2*x))^(1/x),x,0);

>> %part b) >> limit(log(x^n)/(1-x^2),x,1)

>> syms a b w >> x=a+b*cos(w); >> f=(1-exp(-x))/(1+x^3); >> g=inline(vectorize(f),'a','b','w'); >> a=1.2;b=-0.45;w=pi/3; >> g(a,b,w)

>> k=0; >> for b=linspace(0,4*pi,10) syms x y=int((2*x+5)./(x.^2+4*x+5),x,0,b); k=k+1; h(k)=y; end >> disp(h)