Related
Basically I would like to use the fsolve command in order to find the roots of an equation.
I think I should create a function handle that evaluates this equation in the form "right hand side - left hand side =0", but I've been struggling to make this work. Does anyone know how to do this?
The equation itself is 1/sqrt(f) = -1.74log((1.254/((1.27310^8)sqrt(f)))+((110^-3)/3.708)). So I would like to find the point of intersection of the left and right side by solving for 1/sqrt(f)+(1.74log((1.254/((1.27310^8)sqrt(f)))+((110^-3)/3.708))) = 0 using fsolve.
Thanks a lot!
The code so far (not working at all)
f = #(x) friction(x,rho,mu,e,D,Q, tol, maxIter) ;
xguess = [0, 1];
sol = fsolve(x, xguess ) ;
function y = friction(x,rho,mu,e,D,Q, tol, maxIter)
D = 0.1;
L = 100
rho = 1000;
mu = 0.001;
e = 0.0001;
Q = 0.01;
U = (4*Q)/(pi*D^2);
Re = (rho*U*D)/mu ;
y = (1/sqrt(x))-(-1.74*log((1.254/(Re*sqrt(x)))+((e/D)/3.708)))
end
Error message:
Error using lsqfcnchk (line 80)
FUN must be a function, a valid character vector expression, or an inline function object.
Error in fsolve (line 238)
funfcn = lsqfcnchk(FUN,'fsolve',length(varargin),funValCheck,gradflag);
Error in Untitled (line 6)
sol = fsolve(x, xguess ) ;
opt = optimset('Display', 'Iter');
sol = fsolve(#(x) friction(x), 1, opt);
function y = friction(x)
D = 0.1;
L = 100; % note -- unused
rho = 1000;
mu = 0.001;
e = 0.0001;
Q = 0.01;
U = (4*Q)/(pi*D^2);
Re = (rho*U*D)/mu ;
y = (1/sqrt(x))-(-1.74*log((1.254/(Re*sqrt(x)))+((e/D)/3.708)));
end
sol = 0.0054
the first argument of fsolve should be the function not variable. Replace:
sol = fsolve(x, xguess );
with
sol = fsolve(f, xguess );
And define Rho, mu, e etc before you define f (not inside the friction function).
I want to solve coupled partial differential equations of first order, which are of stiff nature. I have coded in MATLAB to solve this pde's, I have used Method of line to convert PDE into ODE, and i have used beam and warmings(second order upwind) method to discritize the spatial derivative. The discretization method is total variation diminishing(TVD) to eliminate the oscillation. But rather using TVD and ode15s solver to integrate resultant stiff ode's the resultant plot is oscillatory(not smooth). What should i do to eliminate this oscillation and get correct results.
I have attached my MATLAB code.. please see it and suggest some improvement.
∂y(1)/∂t=-0.1 ∂y(1)/∂x + (0.5*e^(15*(y(2)⁄(1+y(2))))*(1- y(1))
∂y(2)/∂t=-0.1 ∂y(2)/∂x - (0.4*e^(15*(y(2)⁄(1+y(2))))*(1- y(1))-0.4
Initial condition: at t = 0 y(1)= y(2)=0
Boundary condition: y(1)= y(2) = 0 at x=0
I have attached my MATLAB code.. please see it and suggest some improvement.
function brussode(N)
if nargin<1
N = 149;
end
tspan = [0 10];
m = 0.00035
t = (1:N)/(N+1)*m;
y0 = [repmat(0,1,N); repmat(0,1,N)];
p = 0.5
q = 0.4
options = odeset('Vectorized','on','JPattern',jpattern(N));
[t,y] = ode15s(#f,tspan,y0,options);
a = size(y,2)
u = y(:,1:2:end);
x = (1:N)/(N+1);
figure;
%surf(x,t(end,:),u);
plot(x,u(end,:))
xlabel('space');
ylabel('solution');
zlabel('solution u');
%--------------------------------------------------------------
%Nested function -- N is provided by the outer function.
%
function dydt = f(t,y)
%Derivative function
dydt = zeros(2*N,size(y,2)); %preallocate dy/dt
x = (1:N)/(N+1);
% Evaluate the 2 components of the function at one edge of the grid
% (with edge conditions).
i = 1;
%y(1,:) = 0;
%y(2,:) = 0;
dydt(i,:) = -0.1*(N+1)*(y(i+2,:)-0)+ (0.01/2)*m*((N+1).^3)*(y(i+2,:)-0) + p*exp(15*(0/(1+0)))*(1-0);
dydt(i+1,:) = -0.1*(N+1)*(y(i+3,:)-0)+ (0.01/2)*m*((N+1).^3)*(y(i+3,:)-0) - q*exp(15*(0/(1+0)))*(1-0)+0.25;
i = 3;
%y(1,:) = 0;
%y(2,:) = 0;
dydt(i,:) = -0.1*(N+1)*(y(i+2,:)-y(i,:)) + (0.01/2)*m*((N+1).^3)*(y(i+3,:)-y(i,:)) + p*exp(15*(y(i+1,:)/(1+y(i+1,:))))*(1-y(i,:));
dydt(i+1,:) = -0.1*(N+1)*(y(i+3,:)-y(i+1,:)) + (0.01/2)*m*((N+1).^3)*(y(i+3,:)-y(i,:)) - q*exp(15*(y(i+1,:)/(1+y(i+1,:))))*(1-y(i,:))+0.25;
%Evaluate the 2 components of the function at all interior grid
%points.
i = 5:2:2*N;
%y(1,:) = 0;
% y(2,:) = 0;
dydt(i,:) = (-0.1/2)*(N+1)*(3*y(i,:)-4*y(i-2,:)+y(i-4,:)) +(0.01/2)*m*((N+1).^3)*(y(i,:)-2*y(i-2,:)+y(i-4,:))+ p*exp(15*(y(i+1,:)/(1+y(i+1,:))))*(1-y(i,:));
dydt(i+1,:) = (-0.1/2)*(N+1)*(3*y(i+1,:)-4*y(i-1,:)+y(i-3,:))+(0.01/2)*m*((N+1).^3)*(y(i+1,:)-2*y(i-1,:)+y(i-3,:)) - q*exp(15*(y(i+1,:)/(1+y(i+1,:))))*(1-y(i,:))+0.25;
end
%-------------------------------------------------------------
end %brussode
%-------------------------------------------------------------
% Subfunction -- the sparsity pattern
%
function S = jpattern(N)
% Jacobian sparsity patter
B = ones(2*N,5);
B(2:2:2*N,2) = zeros(N,1);
B(1:2:2*N-1,4) = zeros(N,1);
S = spdiags(B,-2:2,2*N,2*N);
end
%-------------------------------------------------------------
For my studies I had to write a PDE solver for the Poisson equation on a disc shaped domain using the finite difference method.
I already passed the Lab exercise. There is one issue in my code I couldn't fix. Function fun1 with the boundary value problem gun2 is somehow oscillating at the boundary. When I use fun2 everything seems fine...
Both functions use at the boundary gun2. What is the problem?
function z = fun1(x,y)
r = sqrt(x.^2+y.^2);
z = zeros(size(x));
if( r < 0.25)
z = -10^8*exp(1./(r.^2-1/16));
end
end
function z = fun2(x,y)
z = 100*sin(2*pi*x).*sin(2*pi*y);
end
function z = gun2(x,y)
z = x.^2+y.^2;
end
function [u,A] = poisson2(funame,guname,M)
if nargin < 3
M = 50;
end
%Mesh Grid Generation
h = 2/(M + 1);
x = -1:h:1;
y = -1:h:1;
[X,Y] = meshgrid(x,y);
CI = ((X.^2 +Y.^2) < 1);
%Boundary Elements
Sum= zeros(size(CI));
%Sum over the neighbours
for i = -1:1
Sum = Sum + circshift(CI,[i,0]) + circshift(CI,[0,i]) ;
end
%if sum of neighbours larger 3 -> inner note!
CI = (Sum > 3);
%else boundary
CB = (Sum < 3 & Sum ~= 0);
Sum= zeros(size(CI));
%Sum over the boundary neighbour nodes....
for i = -1:1
Sum = Sum + circshift(CB,[i,0]) + circshift(CB,[0,i]);
end
%If the sum is equal 2 -> Diagonal boundary
CB = CB + (Sum == 2 & CB == 0 & CI == 0);
%Converting X Y to polar coordinates
Phi = atan(Y./X);
%Converting Phi R back to cartesian coordinates, only at the boundarys
for j = 1:M+2
for i = 1:M+2
if (CB(i,j)~=0)
if j > (M+2)/2
sig = 1;
else
sig = -1;
end
X(i,j) = sig*1*cos(Phi(i,j));
Y(i,j) = sig*1*sin(Phi(i,j));
end
end
end
%Numberize the internal notes u1,u2,......,un
CI = CI.*reshape(cumsum(CI(:)),size(CI));
%Number of internal notes
Ni = nnz(CI);
f = zeros(Ni,1);
k = 1;
A = spalloc(Ni,Ni,5*Ni);
%Create matix A!
for j=2:M+1
for i =2:M+1
if(CI(i,j) ~= 0)
hN = h;hS = h; hW = h; hE = h;
f(k) = fun(X(i,j),Y(i,j));
if(CB(i+1,j) ~= 0)
hN = abs(1-sqrt(X(i,j)^2+Y(i,j)^2));
f(k) = f(k) + gun(X(i,j),Y(i+1,j))*2/(hN^2+hN*h);
A(k,CI(i-1,j)) = -2/(h^2+h*hN);
else
if(CB(i-1,j) ~= 0) %in negative y is a boundry
hS = abs(1-sqrt(X(i,j)^2+Y(i,j)^2));
f(k) = f(k) + gun(X(i,j),Y(i-1,j))*2/(hS^2+h*hS);
A(k,CI(i+1,j)) = -2/(h^2+h*hS);
else
A(k,CI(i-1,j)) = -1/h^2;
A(k,CI(i+1,j)) = -1/h^2;
end
end
if(CB(i,j+1) ~= 0)
hE = abs(1-sqrt(X(i,j)^2+Y(i,j)^2));
f(k) = f(k) + gun(X(i,j+1),Y(i,j))*2/(hE^2+hE*h);
A(k,CI(i,j-1)) = -2/(h^2+h*hE);
else
if(CB(i,j-1) ~= 0)
hW = abs(1-sqrt(X(i,j)^2+Y(i,j)^2));
f(k) = f(k) + gun(X(i,j-1),Y(i,j))*2/(hW^2+h*hW);
A(k,CI(i,j+1)) = -2/(h^2+h*hW);
else
A(k,CI(i,j-1)) = -1/h^2;
A(k,CI(i,j+1)) = -1/h^2;
end
end
A(k,k) = (2/(hE*hW)+2/(hN*hS));
k = k + 1;
end
end
end
%Solve linear system
u = A\f;
U = zeros(M+2,M+2);
p = 1;
%re-arange u
for j = 1:M+2
for i = 1:M+2
if ( CI(i,j) ~= 0)
U(i,j) = u(p);
p = p+1;
else
if ( CB(i,j) ~= 0)
U(i,j) = gun(X(i,j),Y(i,j));
else
U(i,j) = NaN;
end
end
end
end
surf(X,Y,U);
end
I'm keeping this answer short for now, but may extend when the question contains more info.
My first guess is that what you are seeing is just numerical errors. Looking at the scales of the two graphs, the peaks in the first graph are relatively small compared to the signal in the second graph. Maybe there is a similar issue in the second that is just not visible because the signal is much bigger. You could try to increase the number of nodes and observe what happens with the result.
You should always expect to see numerical errors in such simulations. It's only a matter of trying to get their magnitude as small as possible (or as small as needed).
I am solving Burger equation with Newton-Raphson method in Mathlab.
For the description of the problem see 1.
My problem is the following this code finds the solution upto time
$t=1$, but at this time a discontinuity develops and then the wave
moves forward (like a step function), but this code does not produce
correct solutions after time $t=1$.
Any suggestions or comments to improve the code.
Here is the Matlab code that I am using
function BurgerFSolve2
clc; clear;
% define a 1D mesh
a = -1; b = 3; Nx = 100;
x = linspace(a,b,Nx);
dx = (b-a)/Nx;
J = length(x);
% Iinitial condition
p_init = zeros(size(x));
p_init(x<=0) = 1;
p_init(x>0 & x<1)= 1-x(x>0 & x<1);
% storing results
P = zeros(length(p_init),3001);
P(:,1) = p_init;
% Boundary condition
pL = 1; pR = 0;
% solver
dt = 0.001;
t = 0;
T = zeros(1,3001);
c = dt/dx;
for i = 1:3000
t = t+dt;
T(i+1) = t;
options=optimset('Display','iter'); % Option to display output
p = fsolve(#(p) myfun1(p, pL, pR, c, J, P(:,i)), p_init, ...
options);
% Call solver
P(:,i+1) = p;
p_init = p;
figure(1);
plot(x, p, '-o');
title(['t= ' num2str(t) ' s']);
drawnow;
end
end
function F = myfun1(p, pL, pR, c, J, p_Old)
% Rewrite the equation in the form F(x) = 0
F(1) = p(1) + c*(p(1)^2 - p(1)*pL) - p_Old(1);
for i=2:J-1
F(i) = p(i) + c*(p(i)^2 - p(i-1)*p(i)) - p_Old(i);
end
F(J) = p(J) + c*(p(J)^2 - p(J-1)*p(J)) - p_Old(J);
end
I'm trying to write a cubic spline interpolation program. I have written the program but, the graph is not coming out correctly. The spline uses natural boundary conditions(second dervative at start/end node are 0). The code is in Matlab and is shown below,
clear all
%Function to Interpolate
k = 10; %Number of Support Nodes-1
xs(1) = -1;
for j = 1:k
xs(j+1) = -1 +2*j/k; %Support Nodes(Equidistant)
end;
fs = 1./(25.*xs.^2+1); %Support Ordinates
x = [-0.99:2/(2*k):0.99]; %Places to Evaluate Function
fx = 1./(25.*x.^2+1); %Function Evaluated at x
%Cubic Spline Code(Coefficients to Calculate 2nd Derivatives)
f(1) = 2*(xs(3)-xs(1));
g(1) = xs(3)-xs(2);
r(1) = (6/(xs(3)-xs(2)))*(fs(3)-fs(2)) + (6/(xs(2)-xs(1)))*(fs(1)-fs(2));
e(1) = 0;
for i = 2:k-2
e(i) = xs(i+1)-xs(i);
f(i) = 2*(xs(i+2)-xs(i));
g(i) = xs(i+2)-xs(i+1);
r(i) = (6/(xs(i+2)-xs(i+1)))*(fs(i+2)-fs(i+1)) + ...
(6/(xs(i+1)-xs(i)))*(fs(i)-fs(i+1));
end
e(k-1) = xs(k)-xs(k-1);
f(k-1) = 2*(xs(k+1)-xs(k-1));
r(k-1) = (6/(xs(k+1)-xs(k)))*(fs(k+1)-fs(k)) + ...
(6/(xs(k)-xs(k-1)))*(fs(k-1)-fs(k));
%Tridiagonal System
i = 1;
A = zeros(k-1,k-1);
while i < size(A)+1;
A(i,i) = f(i);
if i < size(A);
A(i,i+1) = g(i);
A(i+1,i) = e(i);
end
i = i+1;
end
for i = 2:k-1 %Decomposition
e(i) = e(i)/f(i-1);
f(i) = f(i)-e(i)*g(i-1);
end
for i = 2:k-1 %Forward Substitution
r(i) = r(i)-e(i)*r(i-1);
end
xn(k-1)= r(k-1)/f(k-1);
for i = k-2:-1:1 %Back Substitution
xn(i) = (r(i)-g(i)*xn(i+1))/f(i);
end
%Interpolation
if (max(xs) <= max(x))
error('Outside Range');
end
if (min(xs) >= min(x))
error('Outside Range');
end
P = zeros(size(length(x),length(x)));
i = 1;
for Counter = 1:length(x)
for j = 1:k-1
a(j) = x(Counter)- xs(j);
end
i = find(a == min(a(a>=0)));
if i == 1
c1 = 0;
c2 = xn(1)/6/(xs(2)-xs(1));
c3 = fs(1)/(xs(2)-xs(1));
c4 = fs(2)/(xs(2)-xs(1))-xn(1)*(xs(2)-xs(1))/6;
t1 = c1*(xs(2)-x(Counter))^3;
t2 = c2*(x(Counter)-xs(1))^3;
t3 = c3*(xs(2)-x(Counter));
t4 = c4*(x(Counter)-xs(1));
P(Counter) = t1 +t2 +t3 +t4;
else
if i < k-1
c1 = xn(i-1+1)/6/(xs(i+1)-xs(i-1+1));
c2 = xn(i+1)/6/(xs(i+1)-xs(i-1+1));
c3 = fs(i-1+1)/(xs(i+1)-xs(i-1+1))-xn(i-1+1)*(xs(i+1)-xs(i-1+1))/6;
c4 = fs(i+1)/(xs(i+1)-xs(i-1+1))-xn(i+1)*(xs(i+1)-xs(i-1+1))/6;
t1 = c1*(xs(i+1)-x(Counter))^3;
t2 = c2*(x(Counter)-xs(i-1+1))^3;
t3 = c3*(xs(i+1)-x(Counter));
t4 = c4*(x(Counter)-xs(i-1+1));
P(Counter) = t1 +t2 +t3 +t4;
else
c1 = xn(i-1+1)/6/(xs(i+1)-xs(i-1+1));
c2 = 0;
c3 = fs(i-1+1)/(xs(i+1)-xs(i-1+1))-xn(i-1+1)*(xs(i+1)-xs(i-1+1))/6;
c4 = fs(i+1)/(xs(i+1)-xs(i-1+1));
t1 = c1*(xs(i+1)-x(Counter))^3;
t2 = c2*(x(Counter)-xs(i-1+1))^3;
t3 = c3*(xs(i+1)-x(Counter));
t4 = c4*(x(Counter)-xs(i-1+1));
P(Counter) = t1 +t2 +t3 +t4;
end
end
end
P = P';
P(length(x)) = NaN;
plot(x,P,x,fx)
When I run the code, the interpolation function is not symmetric and, it doesn't converge correctly. Can anyone offer any suggestions about problems in my code? Thanks.
I wrote a cubic spline package in Mathematica a long time ago. Here is my translation of that package into Matlab. Note I haven't looked at cubic splines in about 7 years, so I'm basing this off my own documentation. You should check everything I say.
The basic problem is we are given n data points (x(1), y(1)) , ... , (x(n), y(n)) and we wish to calculate a piecewise cubic interpolant. The interpolant is defined as
S(x) = { Sk(x) when x(k) <= x <= x(k+1)
{ 0 otherwise
Here Sk(x) is a cubic polynomial of the form
Sk(x) = sk0 + sk1*(x-x(k)) + sk2*(x-x(k))^2 + sk3*(x-x(k))^3
The properties of the spline are:
The spline pass through the data point Sk(x(k)) = y(k)
The spline is continuous at the end-points and thus continuous everywhere in the interpolation interval Sk(x(k+1)) = Sk+1(x(k+1))
The spline has continuous first derivative Sk'(x(k+1)) = Sk+1'(x(k+1))
The spline has continuous second derivative Sk''(x(k+1)) = Sk+1''(x(k+1))
To construct a cubic spline from a set of data point we need to solve for the coefficients
sk0, sk1, sk2 and sk3 for each of the n-1 cubic polynomials. That is a total of 4*(n-1) = 4*n - 4 unknowns. Property 1 supplies n constraints, and properties 2,3,4 each supply an additional n-2 constraints. Thus we have n + 3*(n-2) = 4*n - 6 constraints and 4*n - 4 unknowns. This leaves two degrees of freedom. We fix these degrees of freedom by setting the second derivative equal to zero at the start and end nodes.
Let m(k) = Sk''(x(k)) , h(k) = x(k+1) - x(k) and d(k) = (y(k+1) - y(k))/h(k). The following
three-term recurrence relation holds
h(k-1)*m(k-1) + 2*(h(k-1) + h(k))*m(k) + h(k)*m(k+1) = 6*(d(k) - d(k-1))
The m(k) are unknowns we wish to solve for. The h(k) and d(k) are defined by the input data.
This three-term recurrence relation defines a tridiagonal linear system. Once the m(k) are determined the coefficients for Sk are given by
sk0 = y(k)
sk1 = d(k) - h(k)*(2*m(k) + m(k-1))/6
sk2 = m(k)/2
sk3 = m(k+1) - m(k)/(6*h(k))
Okay that is all the math you need to know to completely define the algorithm to compute a cubic spline. Here it is in Matlab:
function [s0,s1,s2,s3]=cubic_spline(x,y)
if any(size(x) ~= size(y)) || size(x,2) ~= 1
error('inputs x and y must be column vectors of equal length');
end
n = length(x)
h = x(2:n) - x(1:n-1);
d = (y(2:n) - y(1:n-1))./h;
lower = h(1:end-1);
main = 2*(h(1:end-1) + h(2:end));
upper = h(2:end);
T = spdiags([lower main upper], [-1 0 1], n-2, n-2);
rhs = 6*(d(2:end)-d(1:end-1));
m = T\rhs;
% Use natural boundary conditions where second derivative
% is zero at the endpoints
m = [ 0; m; 0];
s0 = y;
s1 = d - h.*(2*m(1:end-1) + m(2:end))/6;
s2 = m/2;
s3 =(m(2:end)-m(1:end-1))./(6*h);
Here is some code to plot a cubic spline:
function plot_cubic_spline(x,s0,s1,s2,s3)
n = length(x);
inner_points = 20;
for i=1:n-1
xx = linspace(x(i),x(i+1),inner_points);
xi = repmat(x(i),1,inner_points);
yy = s0(i) + s1(i)*(xx-xi) + ...
s2(i)*(xx-xi).^2 + s3(i)*(xx - xi).^3;
plot(xx,yy,'b')
plot(x(i),0,'r');
end
Here is a function that constructs a cubic spline and plots in on the famous Runge function:
function cubic_driver(num_points)
runge = #(x) 1./(1+ 25*x.^2);
x = linspace(-1,1,num_points);
y = runge(x);
[s0,s1,s2,s3] = cubic_spline(x',y');
plot_points = 1000;
xx = linspace(-1,1,plot_points);
yy = runge(xx);
plot(xx,yy,'g');
hold on;
plot_cubic_spline(x,s0,s1,s2,s3);
You can see it in action by running the following at the Matlab prompt
>> cubic_driver(5)
>> clf
>> cubic_driver(10)
>> clf
>> cubic_driver(20)
By the time you have twenty nodes your interpolant is visually indistinguishable from the Runge function.
Some comments on the Matlab code: I don't use any for or while loops. I am able to vectorize all operations. I quickly form the sparse tridiagonal matrix with spdiags. I solve it using the backslash operator. I counting on Tim Davis's UMFPACK to handle the decomposition and forward and backward solves.
Hope that helps. The code is available as a gist on github https://gist.github.com/1269709
There was a bug in spline function, generated (n-2) by (n-2) should be symmetric:
lower = h(2:end);
main = 2*(h(1:end-1) + h(2:end));
upper = h(1:end-1);
http://www.mpi-hd.mpg.de/astrophysik/HEA/internal/Numerical_Recipes/f3-3.pdf