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.
4
votes
2answers
31 views
Optimal ODE method for fixed number of RHS evaluations
In practice, the runtime of numerically solving an IVP
$$
\dot{x}(t) = f(t, x(t)) \quad \text{ for } t \in [t_0, t_1]
$$
$$
x(t_0) = x_0
$$
is often dominated by the duration of evaluating the ...
0
votes
1answer
38 views
Spring damper model does not work very well
I'm trying to model a spring damper system from a tutorial that I've found on this site.
If I use the exact same parameters as the ones in the tutorial the system is not stable. I've downloaded the ...
1
vote
0answers
8 views
Combining trend estimation and constrained Marquart fit
This title certainly needs some clarification: I need to compute parameters $a_i$ for a helper function $f(\vec{a};k)$ (for grid interpolation) which is fitted to a number of values $y_k$ which are ...
4
votes
1answer
49 views
Solving Coupled ODE eigenvalue problem
I've been trying to find some resources that would help me figure out how to numerically solve a coupled system of ODEs which is also an eigenvalue problem.
The system is something like:
$ \tag{1} ...
19
votes
3answers
158 views
How does one test a numerical ODE solver implementation?
I'm about to start working on a software library of numerical ODE solvers, and I'm struggling with how to formulate tests for the solver implementations. My ambition is that the library, eventually, ...
2
votes
2answers
58 views
Best form for a system of ODEs to solve with Runge_kutta
Recently when I was solving a system of ODEs using runge-Kutta method , I got much different results when I transformed the variables from spherical coordinates ($r$ and $\theta$ ) to cylindrical ...
1
vote
1answer
76 views
4th order Padé scheme formula derivation
I am trying to derive the formula of the 4th order Padé scheme that passes through the points $x_i$, $x_{i-1}$ and $x_{i+1}$
$$\Big(\frac{\partial\phi}{\partial x} \Big)_i = ...
1
vote
2answers
104 views
Solver error in SciPy/LSODA with a very specific parameter set
I'm implementing a very simple Susceptible-Infected-Recovered model with a steady population for an idle side project - normally a pretty trivial task. But I'm running into solver errors using either ...
6
votes
4answers
170 views
Reference request: Rigorous analysis of algorithms for PDE and ODE
I'm interested in suggestions for book references on the subject of numerical PDE and ODE, in particular, a rigorous analysis of such methods in a manner written for professional mathematicians. It ...
6
votes
3answers
128 views
Numerically stable explicit solution of small linear system
I have an inhomogeneous linear system
$$
Ax=b
$$
where $A$ is a real $n\times n$ matrix with $n\leq 4$. The nullspace of $A$ is guaranteed to be of zero dimension so the equation has a unique ...
0
votes
0answers
44 views
Memory errors with GSL ODE solver
I am trying to solve a (large) system of ODEs with GSL solvers. When I use driver method I get an error message of could not allocate space for gsl_interp_accel, ...
2
votes
0answers
43 views
Dissipation and symplectic manifolds
I'm working on an API for simulation of port-Hamiltonian systems. As far as I understand it, a Hamiltonian system is symplectic if it is power conserving, and so including resistive elements would ...
3
votes
1answer
50 views
Modern alternatives to DRESOL Riccati solver
I am looking for a modern version or an alternative to the DRESOL package for differential matrix Riccati equations. The main issue that the original package uses single-precision type ...
4
votes
2answers
67 views
numerical investigation of stability of motion (confinement)
I am trying to find the required specifications of a RF trap, in which a proton can be confined.(trap dimensions,voltage frequency and amplitude used, etc). I have to solve the equations of motion ...
5
votes
3answers
165 views
Is there a way to reduce aberration in computations of planets' trajectories?
I don't think the title is very accurate , sorry for that.
I simulate bodies in space using two timestep:
the TIMESTEP is the Δt wich I use to make the calculation
and XTIME is the number of times ...
5
votes
3answers
103 views
which numerical method for ode with mixed BCs
I've got a second order nonlinear ODE (nothing fancy), but the BC are a little odd to me:
$y'(0) = 0$
$y \rightarrow y_a$ as $x \rightarrow \infty$
What's a good numerical method for solving ...
0
votes
0answers
136 views
solving nonlinear differential equations by finite differences+matlab code [closed]
Here is the equation that I don't know how to solve by finite differences. I will appreciate when someone can help me.
$$
\frac{\partial{^2T}}{\partial{x^2}} = 0.01 \cdot (T-20)^4 \\
T(0) = 200 \\
...
3
votes
2answers
157 views
Backward Euler method
Can you explain me how does the backward Euler method works?
I have seen the formula and try to understand the method, but what I can't understand is why and how to use the Newton-Rapson method.
Do ...
2
votes
0answers
43 views
System of non-linear ODEs and estimating unspecified initial conditions on Maple 12
I have the following 1st order equations and need to solve them using Maple 12.
There are unspecified initial conditions and can only be estimated through the Newton raphson method. My problem is how ...
4
votes
1answer
101 views
How do I solve an ODE Two-Point Boundary Value Problem?
I have a feeling my question is a very basic one, but I am not at all well versed in computational sciences.
My equations are of the form:
$$
y \in \mathbb{R}^3 \\
\dot{y}(t) = f(y(t)) \\
y_1(0) = a ...
2
votes
1answer
152 views
How to apply a Galerkin finite element method to a linear, one-dimensional boundary value problem
I have the following boundary value problem:
$$-(\alpha u')' + \gamma u = f $$
in $\Omega = (a,b)$ with b.c. $u(a) = u(b) = 0$
and $\alpha > 0, \gamma ≥ 0$ and $f:(a,b) \to \Re $
The weak ...
5
votes
5answers
292 views
What other method can be used to solve differential equations except ode functions in Matlab?
Recently I am taking advantage of the ode45, as well as ode23s, ode15s solvers in ...
0
votes
0answers
60 views
How to solve the following differential equations using ode functions of Matlab_part2? [closed]
This post is the ccontinuation of this pose enter link description here
...
0
votes
0answers
81 views
How to solve the following differential equations using ode functions of Matlab_part1? [closed]
As shown in the following matlab program which is used to find numerical solutions of a dynamic equations (differential equations) which are developed by Lagrange method, the ...
12
votes
3answers
250 views
Numerical methods for discontinuous r.s. ODEs
what are state of art methods for numerical solution of ODEs with discontinuous right side? I'm mostly interested piecewise-smooth right side functions, e.g. sign.
I'm trying to solve the equation of ...
9
votes
1answer
132 views
Algorithms for linear system of ODEs
I wonder: what is the best algorithm to solve
\begin{equation}
\frac{du}{dt} = Au
\end{equation}
Where $A$ is a real $n\times n$ matrix. A is not explicitly time dependent, usually sparse but not ...
2
votes
0answers
220 views
How to find Lyapunov exponent for coupled system
Answer gives a software for calculating conditional lyapunov exponent (CLE) for coupled oscillators in chaos synchronization. However, its hard to follow and there is no graphical output of the plot ...
6
votes
5answers
208 views
What algorithm for solving a set of stiff ODEs would be easiest to port to high precision floating point arithmetic?
I want to solve a relatively small system of stiff ODEs (< 10 first-order equations) using high precision floating point arithmetic (using MPFR or alike). What would be the easiest algorithm to ...
5
votes
2answers
907 views
Solving non-linear singular ODE with SciPy odeint / ODEPACK
I want to solve the Lane-Emden isothermal equation [PDF, eq. 15.2.9]
$$\frac{d^2 \!\psi}{d \xi^2} + \frac{2}{\xi} \frac{d \psi}{d \xi} = e^{-\psi}$$
with the initial conditions
$$\psi(\xi = 0) = 0 ...
9
votes
2answers
135 views
How do you improve the accuracy of a finite difference method for finding the eigensystem of a singular linear ODE
I am attempting to solve an equation of the type:
$ \left( -\tfrac{\partial^2}{\partial x^2} - f\left(x\right) \right) \psi(x) = \lambda \psi(x) $
Where $f(x)$ has a simple pole at $0$, for the ...
4
votes
1answer
132 views
Convert ODE into discrete probabilistic model
how can I turn an ODE equation into a discrete probabilistic model?
I take for example the Verhulst equation for the growth of a population.
$$\frac{dP}{dt} = rP(1-P/K)$$
I was thinking to simulate ...
8
votes
4answers
919 views
solving coupled ODEs with initial-value and final-value constraints
The essence of my question is the following: I have a system of two ODEs. One has an initial-value constraint and the other has a final-value constraint. This can be thought of as a single system with ...
7
votes
5answers
433 views
Symbolic solution of a system of 7 nonlinear equations
I've got a system of ordinary differential equations - 7 equations, and ~30 parameters governing their behavior as part of a mathematical model of disease transmission. I'd like to find the steady ...
7
votes
3answers
292 views
Can I use an explicit time stepping scheme to determine numerically whether an ODE is stiff?
I have an ODE:
$u'=-1000u+sin(t)$
$u(0)=-\frac{1}{1000001}$
I know that this particular ODE is stiff, analytically. I also know that if we use an explicit (forward) time stepping method ...
9
votes
4answers
280 views
Runge-Kutta and Reusing Datapoints
I am trying to implement the fourth order Runge-Kutta method for solving a first order ODE in Python i.e. $\frac{dy}{dx} = f(x,y)$. I understand how the method works, but am trying to write an ...
11
votes
4answers
1k views
The definition of stiff ODE system
Consider an IVP for ODE system $y'=f(x,y)$, $y(x_0)=y_0$. Most commonly this problem is considered stiff when Jacobi matrix $\frac{\partial f}{\partial y}(x_0,y_0)$ has both eigenvalues with very ...
3
votes
1answer
356 views
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 ...
3
votes
3answers
361 views
Implementing Euler's method for initial value ODEs
In my physics class, I had to calculate the trajectory of a projectile that was fired (very fast) with $v_0$ in an angle off a planet (radius $R$, mass $M$) from the surface. The projectile would ...
4
votes
0answers
101 views
Richardson extrapolation for strong rate of convergence of SDE
Is it possible to apply Richardson extrapolation with Euler-Maruyama scheme to improve strong rate of convergence of stochastic differential equations?
12
votes
2answers
146 views
Best practices for describing agent-based models
I work fairly heavily in mathematical biology/epidemiology, where most of the modeling/computational science work is still dominated by sets of ODEs, admittedly sometimes fairly elaborate sets of ...
15
votes
5answers
661 views
How can the gravitational n-body problem be solved in parallel?
How can the gravitational n-body problem be solved numerically in parallel?
Is precision-complexity tradeoff possible?
How does precision influence the quality of the model?
6
votes
3answers
87 views
Numerics: How do I renormalize the following ODE
This question is more about how to tackle a problem numerically.
In a small project I wanted to simulate the coorbital motion of Janus and Epimetheus. This is basically a three body problem. I ...
7
votes
2answers
394 views
Is there an open source set of ODE solvers for C that use the native C99 complex type?
I've been using GSL as the foundation of many of my simulations, but it's a little bit overkill for my purposes and it defines its own complex type for legacy reasons. Rather than code my own ...
