ENGT5140 Numerical Techniques: Root Finding and ODE Solutions

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ENGT5140
Numerical Techniques in Engineering

Contents
1..................................................................................................................................................................3
(a)............................................................................................................................................................3
(b)............................................................................................................................................................4
6..................................................................................................................................................................7
(a)............................................................................................................................................................7
(b)............................................................................................................................................................7
8..................................................................................................................................................................9
(a)............................................................................................................................................................9
(b)............................................................................................................................................................9
9................................................................................................................................................................10
(a)..........................................................................................................................................................10
(b)..........................................................................................................................................................10
(c)...........................................................................................................................................................11

1.
Code:
function [value] = func(x)
value = 14*x.*exp(x-2) - 12*exp(x-2) - 7*x.^3 + 20*x.^2 - 26*x + 12;
end
% func_prime.m
function [value] = func_prime(x)
value = 14*x*exp(x-2) + 14*exp(x-2) -12*exp(x-2) -21*x^2 +40*x -26;
end
% Newton_Raphson.m
x = input('Starting guess :');
tolerance = 1e-8;
iterations = 0;
while (iterations<30) & (abs(func(x))>tolerance)
x = x-func(x)/func_prime(x);
iterations = iterations + 1;
end
if iterations==30
disp('No root found')
else
disp(['Root =',num2str(x,10) ,'(found in ', int2str(iterations),'
iterations.)'])
end
(a)
i.

ii.
Root 1
Root 2

iii.
Root 2 converges linearly but other doesn’t.
(b)
function [value] = func1(x)
value = nthroot((1-3/4*x),3);
end
% func_prime.m
function [value] = func_prime1(x)
value = 14*x*exp(x-2) + 14*exp(x-2) -12*exp(x-2) -21*x^2 +40*x -26;
end
% Newton_Raphson.m
x = input('Starting guess :');
tolerance = 1e-8;
iterations = 0;
while (iterations<50) & (abs(func(x))>tolerance)
x = x-func1(x)/func_prime1(x);
iterations = iterations + 1;
end
if iterations==50
disp('No root found')
else
disp(['Root =',num2str(x,10) ,'(found in ', int2str(iterations),'
iterations.)'])
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Plot

6
(a)
function [t,y] = ode_RK4(f,tspan,y0,N,varargin)
%Runge-Kutta method to solve vector differential eqn y’(t) = f(t,y(t))
% for tspan = [t0,tf] and with the initial value y0 and N time steps
if nargin < 4 || N <= 0, N = 100; end
if nargin < 3, y0 = 0; end
y(1,:) = y0(:)'; %make it a row vector
h = (tspan(2) - tspan(1))/N; t = tspan(1)+[0:N]'*h;
for k = 1:N
f1 = h*feval(f,t(k),y(k,:),varargin{:}); f1 = f1(:)';
f2 = h*feval(f,t(k) + h/2,y(k,:) + f1/2,varargin{:}); f2 = f2(:)';
f3 = h*feval(f,t(k) + h/2,y(k,:) + f2/2,varargin{:}); f3 = f3(:)';
f4 = h*feval(f,t(k) + h,y(k,:) + f3,varargin{:}); f4 = f4(:)';
y(k + 1,:) = y(k,:) + (f1 + 2*(f2 + f3) + f4)/6;
end
(b)
clear all
tspan = [1 10];
N=200;
f = inline('[y(2); (0.12 ./t)*(1+ y(1).^2)^1/2]','t','y');
[t1,yr] = ode_RK4(f,tspan,[100 25],N);

8
(a)
[t,n]=ode45(@equation,[1 2],[1/3 1/12]);
n(length(t))
plot(n(:,1))
hold on
c=size(n);
g=linspace(1/3,1/12,c(1));
g1=1/3*(t).^2;
plot(g1)
(b)
Values are very much deviating as
1/3*t(1)^2=0.33

9
(a)
A=[0 1 1 1; 1 0 1 1;1 1 0 1; 1 1 1 0];
h=5;
B=[4; 4; 4; 4].*(1/h^2)-[0;0;0;100]
T=A\B
(b)
H=2.5
H=0.25

H=0.025
H=0.0025
Decreasing the size of grid increase value of t.
(c)
x=1; h=2.5

y=1;
x=1; h=2.5;
y=1;