# 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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### Adam Bashforth 4 method: how to determine starting values and stil keep the the order of accuracy

I am using an Adam Bashforth 4 method to solve an IVP problem so I need other numerical method to estimate the first 3 values. I am very much interested in finding a way to estimate the first 3 values ...
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### Step-size selection for an Trapezoidal Method ODE solver (ode23t)

I was reading the documentation of the MatLab ODE solver ode23t, and I've seen that the trapezoidal rule is used. Moreover, the error is estimated by ...
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### ODE: should Euler implicit be more accurate than Euler explicit for a given computational step?

I am aware than Euler explicit is conditionally stable, and Euler implicit is unconditionally stable. And I am aware that it is probably pointless to use Euler implicit with a small computational step ...
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### How to solve an implicit ODE with forward Euler?

Consider the implicit ODE $$M(y)\dot{y} = F(t,y)$$ If $M$ is non-singular for all $y$ How to use the forward-Euler method to numerically solve for $y$ without inverting $M(y)$? I only came out ...
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### Can a second-order ODE be “inconsistent” with its boundary conditions?

I am trying to solve a set of coupled, nonlinear ODEs. The only dependent variable is a 1-dimensional spatial coordinate, let's call it $x$. For now, I've managed to approximate away some of the ...
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### How well do explicit Runge-Kutta “tableau” methods compare to the state of the art ODE solvers and when do they fail?

How well do explicit Runge-Kutta "tableau" methods compare to the state of the art ODE solvers and when do they fail? I've been reading Butcher's ODE book and he does a good job at introducing ...
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### Euler Method Instability. Why?

I am currently enjoying writing computational codes as a hobby. Right now I've worked out an Euler method and results are pretty good with up to $x=1$. Over $x=1$, instability starts to kick in. May I ...
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### Nonlinear ODE to solve Duffing's equation

I am trying to solve the Duffing's equation in MATLAB. $m\ddot{y}+c\dot{y}+ky+k_{3}y^{3} = f(t)$ where $f(t) = A \sin{\omega t}$ To do that I wrote a function to be given to the ode45. ...
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### Ordinary differenial equation with numerical right hand side

I'm interested in solving a differential equation in which I don't know the analytical form of the right hand side, I only know its numerical value for a finite set of values of the independent ...
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### How do I find the default ODE solver tolerances in Matlab?

You can set the Absolute or Relative ODE solver tolerances in Matlab with the options structure from an odeset command. But how do I find the default values for the ...
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### Estimating error at mid timesteps for Runge-Kutta methods

When a timestep $h$ is rejected using a Runge-Kutta pair, such as Dormand–Prince, the algorithm resumes from the same initial point $t$ with a smaller timestep. A different idea is to resume at an ...
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### Which Runge-Kutta method is more accurate: Dormand-Prince or Cash-Karp?

I simply want to know whether the Dormand-Prince Numerical Method or the Cash-Karp Numerical Method is more accurate.
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### Solving an ODE from many starting points

I have a situation where I'm interested in solving the same ODE many times, from different initial locations. More precisely, I'm interested in solving an ODE of the form \begin{align} \frac{dx}{dt} &...