# Questions tagged [ode]

Ordinary Differential Equations (ODEs) contain functions of only one independent variable, and one or more of their derivatives with respect to that variable. This tag is intended for questions on modeling phenomena with ODEs, solving ODEs, and other related aspects.

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### Runge Kutta solution blows up for a first order ODE with very large coefficients

I am solving a first-order ODE: $\frac{\partial \rho }{\partial t} = -a \rho^2 + b |A(t)|^2 \rho +c|A(t)|^{2m}$ This is the evolution of the plasma density in the presence of a laser pulse (complex ...
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### Numerically solving differential equations, the domain is very long, [0, +10^6), so the calculating time is very long

Is there any method to deal with this problem? I am using Mathematica to solve the differential equations, but the calculating time is so long because of the large domain $x\in[0,10^{6})$. In fact I ...
106 views

### The Schrodinger equation for time-dependent Hamiltonian after one timestep, taking exponential or use ode solver?

The Schrodinger equation for time-dependent Hamiltonian is $$i\hbar\frac{d}{dt}\psi(t) = H(t)\psi(t) \, .$$ Assuming I knew $\psi(t)$, I want to know $\psi(t+\Delta t)$. Should I take exponential ...
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### Will the numerical solving of the differential equation be wrong if I take the step too small? [closed]

If I take the step too large I will get error, while if I take the step too small I also get an error. In my case, instead of seeing the function decreasing, i have it increasing if I take the step ...
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### Implementing odespy for system of PDEs

After trying to use RK4 to solve the below system of equations, it appears the output had large and fast oscillations even with an adaptive time step I incorporated using the Cash-Karp method. I am ...
729 views

### Why does LSODA fail to integrate the logistic function?

I'm comparing some of the different ODE integrators in scipy.integrate.ode on solving the logistic function: $$x(t) = \frac{1}{1+e^{-rt}}$$ $$\dot{x} = rx(1-x)$$ ...
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### Object falling with air resistance using Runge-Kutta

I am not very familiar with differential equations, nor physics in general. I am trying to program an object falling with air resistance with the use of a numerical algorithm called Runge-Kutta. The ...
234 views

### Solving a differential equation using numerical methods and matlab

(a) Consider the following differential Equation $$Y'(t)=\frac{1}{1+t^{2}}-2[Y(t)]^2$$ $$Y(0)=0$$ The exact solution is $$Y(t)=\frac{t}{1+t^2}$$ Using the Euler method to solve the following ...
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### Multiple Coupled Differential Equation solution in Python

I have 4 ordinary differential equations that are coupled. The variables in the 4 equations are functions of time and space and one of them is second order in space. \begin{equation} \frac{ \partial ...
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### Solving ODEs of Switched Systems using MATLAB ODE suite

I have the following code to execute: $X$ = [0;0;0;0]; $sw$ = 0; ...
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### Solving ODE with multiple equilibriums

Consider an ODE of the form: $$u'(t)=-\frac{1}{\varepsilon}u(u-\frac{1}{2})(u-1)$$ with the initial value $$u(0)=u_0.$$ Here $\varepsilon>0$ is a constant. It is easy to verify that $u\equiv0$ ...
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### How to find conditional Lyapunov exponents

For a synchronized ODE system,I wanted to know if there is a program code in MATLAB available for plotting the conditional Lyapunov exponent. For example, synchronization of identical Rossler system ...
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### Comparison of velocity Verlet and leapfrog algorithms

Many sources present the Euler, Verlet, velocity Verlet, and leapfrog algorithms for integrating Newton's equations. Based on the order of accuracy, it is agreed that velocity Verlet, Verlet, and ...
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### What is the best option in terms of library or software to solve this system of hyperbolic PDEs?

I want to solve a system of coupled nonlinear 1-D PDE $(\partial_{tt} + \alpha\partial_t)u_i(x,t)=\partial_{xx}(\sum_{j=1}^{j<i}ju_j(x,t)+i\sum_{j=i}^{n}u_j(x,t))-\sin(u_i(x,t))+f$, using method of ...
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### Spectral Collocation (or Weighted Residual) Methods to solve Stiff ODEs?

I have a system of ODEs which is (at least moderately) stiff. Consider the class of spectral collocation methods https://en.wikipedia.org/wiki/Spectral_method or the related class of weighted ...
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### How can I use ODE events in MATLAB? [closed]

I need to have a better understanding about how to define ODE events. What I know is that if I have my ODE defined as ...
256 views

### Algorithm to Compute Separatrix of Nonlinear ODE

The solution space of a nonlinear ordinary differential equation (ODE) often includes a separatrix that is unstable in the sense that nearby solutions depart exponentially from it. The nonlinear ...
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### Adaptive ODE algorithm in Python

I want to integrate a particle path in 2D using the integrate.ode module. Things that are a bit different in my case are that, I only want to integrate up to a ...
1k views

### How can one describe the accuracy of a Runge-Kutta method?

I am solving a nonlinear ODE with a regular singularity using MATLAB ODE45 or ODE113. I am wondering what precision and accuracy they have and what one can say about the numerical error. The idea ...
112 views

### PDEs appropriate for adaptive time stepping algorithms

I'm looking for some physical phenomena for which an adaptive time stepping algorithm would be ideal. A PDE or ODE that showed very large gradients in time at a small period of time and smoother ...