I have a column vector of linear equations. each equations is fun of series of variables (x1,x2,......x20,y1,y2,.....,y20),
all the equaions=0
how to solve these equations
from Matlab help
sol = solve([eqn1, eqn2, eqn3], [x, y, z]);
xSol = sol.x
ySol = sol.y
zSol = sol.z
how to call the matrix instead of each equations eqn1,eqn2,...especially for the very large number of equations
the same for the variables can call them as vectors instead of individuals/one by one.
Also, I keep getting the same warning
[Warning: Cannot solve symbolically. Returning a numeric
approximation instead. ]
Thanks
Related
I have a complex equation involving matrices:
R = expm(X)*A + (expm(X)-I)*inv(X)*B*U;
where R, B and U are known matrices.
I is an identity matrix.
I need to solve for X. Is there any way to solve this in MATLAB?
If your equation is nonlinear and you have access to MATLAB optimization toolbox you can use the fsolve function (You can still use it for a linear equation, but it may not be the most efficient approach). You just need to reformat your equation into the form F(x) = 0, where x is a vector or a matrix. For example, if X is a vector of length 2:
Define your function to solve:
function F = YourComplexEquation(X)
Fmatrix = expm(X)*A + (expm(X)-I)*inv(X)*B*U - R
% This last line is because I think fsolve requires F to be a vector, not a matrix
F = Fmatrix(:);
Then call fsolve with an initial guess:
X = fsolve(#YourComplexEquation,[0;0]);
I want to solve the differential equation. MATLAB shows warning:
clear all
syms x f(x) theta
eq = (-6*x+(-7+theta)*f*diff(f,x))*(1+diff(f,x)^2)+x*f*(diff(f,x,x))==0
cond = f(0)==1
dsolve(eq,cond)
Warning: Explicit solution could not be found.
> In dsolve (line 201)
In dsolvef (line 5)
ans =
[ empty sym ]
Is there a way to solve it?(analytically or numerically)
Thank you
There is a very high probability that no symbolic solution exists. In general the set of ODE that have symbolic solutions is "thin" in that slight variations of a symbolically solvable ODE make it unsolvable.
For a numerical solution use the ode45 solver or implicit solvers like ode15. The all need an ODE function that encodes the ODE as explicit order 1 system.
function dz = derivs(x,z)
y=z(1); dy = z(2)
ddy = -(-6*x+(-7+theta)*y*dy)*(1+dy^2) / (x*y)
dz = [ dy ddy ]
end
See also similar topics:
How do you plot nonlinear differential equations in matlab
How to draw the direction field of van der pol oscillator?
I have a problem with some differential equations of first-order.
I'm trying to solve them with ode23 and ode23s.
The differential equations are:
y'+3y+z=0
z'-y+z=0
with the initial values:
y(0)=1 and z(0)=1
I also want to compare it with the analytical solution:
y=exp(-2x)(1-2x)
z=exp(-2x)(1+2x)
I want to do it this way because I need to do the comparison in order to choose the better solver: ode23 or ode23s, whichever one is closer to the analytical solution.
My code is:
function dy=projectb1(t,y)
%y'=-4y
%z'= 0
%y(1)=y'
%y(2)=z'
dy = [-4*y(2); 0*y(1)];
and:
% Comparison of analytical solution
clear
options= odeset('RelTol',1e-4,'AbsTol', [1e-4 1e-4]);
%figure
%t1=cputime;
[t23,y23]= ode23('projectb1',[0 12],[1 1],options);
[t23s,y23s]= ode23s('project1',[0 20],[1 0],options);
%tobl = cputime -t1
figure
ya=exp(-2*t23).*(1-2*t23);
za=exp(-2*t23).*(1+2*t23);
plot(t23,ya,za,'r',t23,y23(:,1),'g-.',t23s,y23s(:,1),'b');
%legend('ya','ode23','ode23s',0)
text(3.4,-1.7,'ya')
title('\bf{Analytical and numerical solutions using} \it{ode23s, ode23}')
But it doesn't work. Could someone help me?
The error Matlab throws right away has to do with the line
plot(t23,ya,za,'r',t23,y23(:,1),'g-.',t23s,y23s(:,1),'b');
This should be
plot(t23,ya,t23,za,'r',t23,y23(:,1),'g-.',t23s,y23s(:,1),'b');
You missed that extra t23.
Another problem appears to be in the definition of the differential equation.
For systems of differential equations, the Matlab ODE suite passes a vector x whose components are the values of the functions you are attempting to approximate.
Therefore, as in the example below, the first component of x is the value of y at time t, and the second component of x is the value of z at time t:
function dx = projectb1(t,x)
y = x(1);
z = x(2);
dy = -3*y - z;
dz = y - z;
dx = [dy;dz];
end
I changed the input y to x to make it clear that what is input is a vector of values of y and z.
Also, note that while ode23 has the initial conditions [1,1], ode23s has [1,0], which means it is solving a different initial value problem.
I am trying to solve a second order differential using ODE45 in Matlab with matrix as inputs. I am struck with couple of errors that includes :
"In an assignment A(I) = B, the number of elements in B and
I must be the same."
Double order differential equations given below:
dy(1)= diag(ones(1,100) - 0.5*y(2))*Co;
dy(2)= -1 * Laplacian(y(1)) * y(2);
Main function call is:
[T,Y] = ode45(#rigid,[0.000 100.000],[Co Xo]);
Here, Co is Matrix of size 100x100 and Xo is a column matrix of size 100x1. Laplacian is a pre-defined function to compute matrix laplacian.
I will appreciate any help in this. Should I reshape input matrices and vectors to fall in same dimensions or something?
Your guess is correct. The MATLAB ode suite can solve only vector valued ode, i.e. an ode of the form y'=f(t,y). In your case you should convert y, and dy, back and forth between a matrix and an array by using reshape.
To be more precise, the initial condition will be transformed into the array
y0 = reshape([Co Xo], 100*101, 1);
while y will be obtained with
y_matrix = reshape(y, 100, 101);
y1 = y_matrix(:,1:100);
y2 = y_matrix(:,101);
After having computed the matrices dy1 and dy2 you will have to covert them in an array with
dy = reshape([dy1 dy2], 100*101, 1);
Aside from the limitations of ode45 your code gives that error because, in MATLAB, matrices are not indexed in that way. In fact, if you define A = magic(5), A(11) gives the eleventh element of A i.e. 1.
My overall goal is to use the MATLAB symbolic toolbox to simplify the process of formulating and solving for the sensitivities of solutions to ordinary differential equations with respect to the parameters in the equations. In my case I have an ODE with 2 states and 10 parameters. A smaller, but representative, example would look like
X = sym('X', [2 1]) % Vector representing state variables
p = sym('p', [3 1]) % Vector representing parameters
% Fitzhugh Nagumo Equations
rhs_1 = symfun(p(3)*(X(1) - X(1)^3/3 + X(2)), [X; p])
rhs_2 = symfun(-(X(1) - p(1) + p(2)*X(2))/p(3), [X; p])
I can then get the partial derivatives, which are used to solve for the sensitivities, of the RHS of the ODE wrt to the parameters using a command like 'gradient(rhs_1, p)'. But then I would like to convert this gradient to a matlab function that is a function of the vectors X and p, not a function of the elements of these vectors. I need these functions to be of this form because otherwise I cannot use the CVODES solver in the sundialsTB toolbox. Is this possible? Is there an easier way to accomplish what I am trying to do?
Recognizing that a comma-separated list of function inputs is really just a cell array, you can do this by converting your vector inputs to a cell arrays of scalar using mat2cell:
x=1:2;
p=1:3;
v = mat2cell([x(:);p(:)],ones(numel(x)+numel(p),1),1);
y1 = rhs_1(v{:})
y2 = rhs_2(v{:})