numerical-methods,ode,newtons-method,numerical-stability

Usually you chose the solution that is closest to y(i). The correct solution should satisfy y(i+1)=y(i)+h*f(x(i),y(i)) + O(h²) However, for stiff problems the constant in O(h²) can be very large so that this relation is not as helpful as it seems. If L is the Lipschitz constant of f and...

matlab,matlab-figure,ode,differential-equations,subplot

The error is thrown when you run the function and MATLAB attempts to evaluate D_t = 3*exp(-0.05*t);. Since no value of t was given, MATLAB throws an error saying that the up-to-that-point unused t variable must be specified. The main problem with the code is in the function's design. Namely,...

This should solve this issue: def resol(t, y, par): p1111, p1212, p2121, p1112, p1121, p1122, p1221, p1222, p2122 = y gamma, h, H = par You will get more problems such as that delta needs to be called delta_t (needs renaming) and ComplexWarning: Casting complex values to real discards the...

matlab,ode,differential-equations,numerical-integration

Summary: There are problems in integrating Hamiltonian systems with normal numerical integrators, and your special initial conditions aggravate this to the point where the numerical solution has no resemblance with the correct one. There's nothing wrong with your implementation per se. However, the initial conditions you use are not the...

Here we go... The ODE are linear, we know it, there is an algebraic solution and so. For using ode23 you need an explicit set, on the form -- ps. note the vectorial form usage on the dependent variable x and the independent variable t hereonforth: x'=f(t,x) Which on this...

You need to solve for both functions in the same time. Something like: clear all; clc; %// Constants - units look nice as comments :-D Ca_ER = 10e-6; Ca_cyto = 1.7e-6; Ir = 1; alpha = 0.5; p_open3 = 0.15; c0 = 4e-6; c1 = 0.185; v1 = 6; v2...

You are using {} to index, which is for cell arrays. You should be using (): dydpdt = 1*(-f(2)-f(1)*dydp); and also dydp0 = zeros(size(f(2))); [t2,JJ]=ode45(@(t,y)funsensitivity(t,y,f),0:tstep:tfinal,dydp0); EDIT based on comments: I have tried running your code in Octave (don't have MATLAB), and it looks like your problem is unstable by looking...

First issue is: n = range(0,t_f) r_n = r0*(n*dt) Here you define n as a list and try to multiply the list n with the integer dt. This will not work. Pure Python is NOT a vectorized language like NumPy or Matlab where you can do vector multiplication like this....

Yes, all of the solvers in the Apache Commons Math library are mainly for non-stiff problems. DormandPrince853Integrator is a good general choice, but even for mildly stiff problems it may not perform well. You might look for implementations of LSODE/ODEPACK or call a C or FORTRAN version of this directly....

Here's an example where I pass a variable coeff to the differential equation: function [T,Q] = main() t_span = [0 10]; q0 = [0.1; 0.2]; % initial state ode_options = odeset(); % currently no options... You could add some here coeff = 0.3; % The parameter we wish to pass...

Partial answer, compare accuracy-runtime estimates for Euler, improved Euler and RK4 Assume that the most time-consuming part is the evaluation of the system function, i.e., the vector arithmetic of the Runge-Kutta methods are a small overhead. Given a time interval of T=1, restrict the methods to 100 function evaluations. Then...

The error is exactly what it says... undefined variable x. You haven't defined it anywhere in your code, yet you start instantly using it at line 13. However, there is a function call just before line 13, and it looks like the ODE solving is performed there... but you're not...

You could use the inline package. From the beginning of the examples section for cfunction(): ## A simple Fortran example code <- " integer i do 1 i=1, n(1) 1 x(i) = x(i)**3 " cubefn <- cfunction(signature(n="integer", x="numeric"), code, convention=".Fortran") cubefn(n, x)$x ...

You're right in thinking that numba doesn't recognise fdot as a numba compiled function. I don't think you can make it recognise it as a function argument, but you can use this approach (using variable capture so fdot is known when the function is built) to build an ODE solver:...

You are integrating over a pole of the ODE function. At the pole, every solution ends. Trajectories to the left and right of the pole can not joined into one larger trajectory. The pole is the first positive solution t of 1 = mii*c_gama*cos(om*t). If mii*c_gama >= 1 there is...

It really is the order of arguments in y. In the cited source it was in the "natural" order, the reason for your mix is not really clear. Giving names to some sub-expressions, the ODE function might also be written as int func(double t, const double y[], double f[], void...

You need to pass the parameters to your function: function xprime = MyLogisticFunction(t,X,a,N) %% The ODE % Parameters (passed as function arguments) % N = 10 % Carrying Capacity % a = 0.5 % Growth Rate x1prime = 2*a*X(1)*(1 - X(1)/N ); xprime = [x1prime ]'; end and then when...

python,scipy,constraints,ode,numerical-integration

if algebraic manipulation fails, you can go for a numerical solution of your constraint, running for example fsolve at each timestep: import sys from numpy import linspace from scipy.integrate import odeint from scipy.optimize import fsolve y0 = [0, 5] time = linspace(0., 10., 1000) F_lon = 10. mass = 1000....

matlab,simulink,solver,ode,differential-equations

Can you solve it in closed form? Looks doable to me. I advise anyone to have the answer in hand if possible before you start a numerical solution. Here's what I get. Check me: y(x) = e^(-x)*(8e^3x + 3e^4x + 1)/12 Wolfram Alpha says this is correct. (Note: Trouble for...

matlab,plot,matlab-figure,ode,differential-equations

You're plotting theta and phi on V-theta and V_phi axes - too many initial conditions, which means the axes chosen by matlab to display theta_0 and phi_0 are zoomed out too far to display the line plot. Try simply commenting out the first plot command %plot(initial_phi,initial_velocity_phi,'ok','MarkerSize',5,'MarkerFaceColor','g'); to get a graph...

It really depends on the function of a, a=f(T). If you can take the derivative of f(T) I suggest you include a as another state, for instance X(3) and then you can access it from the X argument in xprime easily, plus it will help Matlab determine the proper step...

matlab,ode,differential-equations,runge-kutta

You need to create a file, for example, diffeq.m function xdot = diffeq(t, x) x4 = t; R1 = -0.000001*(1-cos(theta*x4))-theta*sin(theta*x4)+1; R2 = -(1-cos(theta*x4)); xdot(1) = x(2); xdot(2) = x(3); xdot(3) = R1*x(1)+R2*x(2)-alpha*x(3); xdot = xdot'; % ODE solver expects a column vector. Then call it using the ODE45 syntax (from...

python,numpy,ode,differential-equations

The comment made by @WarrenWeckesser is a very good one, you should start there. I'll merely try to highlight the differences between the implicit plot and the explicit plot. First, the setup: import numpy as np from scipy import integrate import matplotlib.pyplot as plt time=np.linspace(0,15,5*1024) def derivN(N, t): """Return the...

It is a simple copy-paste error. In the line x4=... there should be no f, as it is in x1=..., x2=..., x3=.... Then everything looks nice as it should. This is one reason to stay as close as possible to the cook-book implementation, use vector-valued functions and vector arithmetic. By...

When I ran your code in ipython I got: Integrator `vode` can be used to solve only a single problem at a time. If you want to integrate multiple problems, consider using a different integrator (see `ode.set_integrator`) Apparently you have to use: ode.set_integrator to integrate multiple problems at the same...

Vectorize, vectorize, then vectorize some more. And use data structures that facilitate vectorization. The function __connectionistModel uses a lot of the access pattern A[i*m+j], which is equivalent to an access to row i and column j in a 2D array with a total of m columns. This suggests that a...

I've solved this problem. It was because my F(t,y) is wrong. I checked the jacobian only by then I managed to see that I have some lines of zeros.

matlab,integration,simulink,ode,differential-equations

Short answer: yes it's possible, but nobody's going to do it for you. The steps would be: Write down on paper the differential equation of the system Put it in a form dy/dt = f(t,y) and write it as a MATLAB function Solve the differential equation with one of the...

If you have a predefined no. of ic you can use this for loop: for i=1:NoOfic test_mat(i,:)=test_ode_45(x(i),tspan); end You should predefine your matrix (test_mat) in a sufficient dimension e.g. test_mat = zeros(NoOfic, SizeOftspan) Obviously I have defined the variables NoOficand SizeOftspanto the specific numbers My whole test-code is: tspan =...

Yes, that's quite possible. Define x as [c;ce], then you have something like: function dx = my_ode(t,x) % parameters definition kf = ...; % whatever value you are using P0 = ...; % whatever value you are using Jer = ...; % whatever value you are using J_serca = ...;...

The simplest way is just to add a big force that kicks the particle out from the forbidden region. This requires an ODE solver that is able to handle "stiff" problems, but luckily odeint uses lsoda which automatically switches to a mode suitable for such problems. For example, F(x) =...

The declaration syms b(i) creates a symbolic function b of i. So, if a vector of doubles is passed to b(i), the output will be a vector of function values: >> syms b(i) >> b(1:6) ans = [ b(1), b(2), b(3), b(4), b(5), b(6)] >> b(i) = i^2; % Defining...

matlab,ode,differential-equations

I believe your question is really related to passing parameters to a function used by ode45. Your function that defines the right hand side of your ODE should take the angle as an input parameter and should be of the form f(t,y,angle) You should then wrap this function inside another...

One problem is that you have to call fsolve on prob5diff which takes a single vector input since your guess is a single vector: prob5diff(x) M = x(1); Fx = x(2); Fy = x(3); ...

matlab,ode,numerical-integration

The error is pretty clear: arrays of function handles aren't allowed in Matlab. You may be able to use cell arrays in some cases. See this question. I'm not sure that the ODE suite functions support the evaluation of separate events functions in a cell array (R2013b gives me an...

matlab,signals,solver,ode,square

A partial answer: If you put the line fprintf('%g\t%g\n', t, y) into your function test, you will see that the solver calls the function only at values of t where y is 24: 0 24 0 24 0 24 1.57072e-14 24 1 24 2 24 3 24 4 24 5...