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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9
votes
4answers
353 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 ...
11
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3answers
1k 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 ...
1
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0answers
244 views

Memory allocation error with GSL ODE solver applied to system of 4 ODEs

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, ...
3
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0answers
90 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 ...
5
votes
1answer
167 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 ...
7
votes
2answers
328 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
213 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
239 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 this? ...
5
votes
2answers
6k views

How to implement Newton's method for solving the algebraic equations in the 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 ...
8
votes
2answers
741 views

How to solve the stiff equation in this Restricted Three Body Problem numerically?

I've come across a stiff equation in solving the Circular Restricted Three Body Problem. [An object is moving considering the effect of the gravitational forces caused by two gravitational sources ...
2
votes
0answers
93 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
210 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 ...
1
vote
1answer
483 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 ...
6
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5answers
2k 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 ...
15
votes
3answers
1k 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 ...
12
votes
1answer
213 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 ...
9
votes
1answer
1k 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, it is hard to follow and there is no graphical output of the ...
7
votes
5answers
562 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 ...
8
votes
2answers
5k 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 \...
11
votes
2answers
838 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
315 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 ...
11
votes
4answers
6k 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 ...
9
votes
5answers
2k 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 ...
10
votes
3answers
834 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 (...
11
votes
4answers
477 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 ...
17
votes
4answers
4k 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 ...
4
votes
1answer
993 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 ...
5
votes
3answers
982 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 ...
7
votes
1answer
304 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?
14
votes
3answers
227 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 them....
24
votes
6answers
4k 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?
9
votes
3answers
149 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 ...
12
votes
2answers
2k 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 Runge-...