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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Solvers for stiff initial value ODEs with sparse Jacobian

What ODE solvers are optimized for solving stiff systems with sparse Jacobian? Such systems appear, for instance, when a parabolic PDE is discretized in space using typical finite difference or finite ...
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1answer
40 views

Libraries with the method of lines for parabolic PDEs [on hold]

Could you please advise some programs or libraries for solving parabolic PDEs (or its systems) in 1D, 2D and 3D, for example, with the method of lines? The system of parabolic PDEs can be nonlinear in ...
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1answer
31 views

Matlab equivalent of scipy's 'vode' and 'zvode' ode routines

In python I have used the ode method from scipy.integrate. There I used the vodeintegrator ...
2
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1answer
52 views

scipy odeint - Excess work done on this call

I'm newbie both in calculus and Python/Scipy so I apologize if this question is too dumb. I'm trying to model flow between two pressure vessels. Let's say we have two points and a link between them ...
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1answer
55 views

Runge-Kutta 4th order for 4 coupled first order differential equation [closed]

I have to solve 4 coupled first order differential equations for $f(t)$ ,$g(t)$, $h(t)$ and $w(t)$ witch are only functions of $t$ , but for every reference link a function of 3 variables is assumed ...
2
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1answer
41 views

Solving large, non-linear systems of ODEs numerically: what do I need to consider in order to figure out which solver to use?

I would prefer recommendations that don't require the use of proprietary tools (such as Matlab). I know of two ODE solving options for the Python ecosystem: PyDSTool (Dopri, Radau, other Runge-Kutta ...
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1answer
24 views

Why might the time taken to compute the solution of an ODE system over some interval increase non-linearly with increasing size of interval?

Currently, my problem requires me to solve a system a large system of non-linear ODEs (up to ~5000). So far, I have been using scipy.integrate.odeint as my ...
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1answer
17 views

Possible to reduce effort needed to solve non-linear ODEs by taking some coefficients/parameters as constant over small time intervals?

So far, I have been using scipy.integrate.odeint as my "workhorse" ODE solver. My current problem requires that I solve a large system (up to ~5000) ODEs. Here's ...
2
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1answer
57 views

How to know whether a boundary-value ODE problem is well defined?

I am using bvp4c from Matlab to solve a boundary values ODEs problem. Given the ODEs and boundary conditions, is there any way to have more information on the solutions? How many do I expect? 0, 1, ...
0
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1answer
65 views

scipy.integrate.odeint: how can odeint access a parameter set that is evolving independently of it?

I might have some non-linear ODEs that are being solved by scipy.integrate.odeint. However, a parameter at each time step might have to be updated by using a non-DE ...
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1answer
40 views

How to impose an integral conservation in solving ODEs boundary value problems (BVP)?

I have a system of coupled ODEs that I want to solve. The functions are A(x), B(x), C(x). It is a boundary values problem. I am using Matlab bvp4c. So far I am not satisfied with my solutions. For ...
2
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2answers
117 views

Why does scipy's odeint function give a non-monotonic solution for a problem whose solution should be monotone?

The solution to the ode below looks like it is monotonically increasing: However on closer inspection we see that it is not: How can I ensure that the numerical solution is monotonically ...
6
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1answer
72 views

Numerical methods for boundary-value ODEs with a jump condition

I want to solve a non linear system of equations of a particular kind. I find it hard to formulate clearly so I directly give a simple example. $ f''=A(f,g)\\ g''=B(f,g) $ with the boundary ...
6
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2answers
394 views

Is the shooting method the only general numerical method for solving nonlinear boundary value ODEs?

During my wandering in Mathematica.se, I gradually noticed that a certain kind of differential equation solving problem is "troubling" us all the time, that is, the boundary value problem (BVP) of ...
3
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1answer
187 views

Non-conservative implementation implicit Euler

In Matlab R2013a I have implemented the Implicit Euler (time) integration scheme. To find the $x^{n+1}$ value I use fixed point iterations: $x^{n+1} = \Delta t f(x^{n+1}) + x^n$ To test this, I use ...
4
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1answer
83 views

Constructing explicit Runge Kutta methods of order 9 and higher

Some older books I've seen say that the minimum number of stages of an explicit Runge-Kutta method of a specified order is unknown for orders $\geq 9$. Is this still true? What libraries are there ...
3
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1answer
62 views

ODE boundary value problem relaxation method reference request

This is a somewhat basic question, I guess. Take the ODE boundary value problem $$ \frac1\lambda y''-y'=0, \qquad y(0) = 0, \quad y(1) = 1, $$ with the solution $$ y(x) = ...
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1answer
91 views

Solving a system of nonlinear equations with an ODE solver is faster than with the Newton method?

This is somehow unexpected, but my recent experience with solving a system of nonlinear equations is that treating them as the right hand side of a system of ordinary equations and then evolve the ...
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1answer
50 views

Appropriate algorithm for (non-linear) ODE with integral equilibrium constraint: collocation?

I have a problem of the following structure: For some scalar $g$, functions $F(z)$ and $h(z)$ defined on $[0,\bar{z}]$ , and a non-linear operator $\phi(F,z)$ (in reality, $F$ and $h$ are vector ...
0
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2answers
126 views

ode45 solver MATLAB improve time

I am numerically simulating the Mathieu equation using ODE45 and I have to keep changing the parameters delta and epsilon for each simulation to get the required peiodic solution. Following is the ...
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2answers
161 views

wavelet for numerical partial differential equations

Is there a good introduction into wavelet Galerkin schemes for numerical partial (and ordinary) differential equations?
3
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2answers
113 views

scipy odeint: sum of conservative ode equations does not remain zero as it is being solved — is this normal?

Let us say we have the following equations: dy1/dt = f(y1, t) [1] dy2/dt = g(y2, t) [2] The equations are such that they are "conservative", i.e. the ...
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1answer
69 views

Heuristic help with preconditioning large system ODEs

I'm looking for some general insight on preconditioning. In particular, relevant references/resources/comments would be greatly appreciated. Note, I have been through some of the literature, but am ...
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2answers
256 views

How to model a fishing rod (or a rope)?

I wish to model a fishing rod (or a rope) by joining short segments. (The segments may have equal (short) length but each segment should be assigned its own individual mass.) One segment will ...
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2answers
115 views

How to get ODE solution at specified time points?

The code below basically illustrates my problem. It is a test code for a pendulum. I solve it using a method suggested on ...
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0answers
21 views

Error analysis and the Model Problem [closed]

In numerical methods for ODE's, the model problem y' = cy where c is complex is regarded as sufficient in performing error analysis for different methods in ...
2
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1answer
142 views

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 ...
0
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1answer
81 views

Initial estimate of coupled variables while applying implicit euler to solve nonlinear system of odes

I am trying to solve a system of coupled odes having the following three equations. $$ \begin{align} \frac{dn_A}{dt} & = e\left[j(t) - f\, θ_H\sinh\left(\frac{g\,n_A}{T}\right)\right] \\ ...
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97 views

Strategy for solving a non-trivial differential equation

I would like to numerically solve an equation of a type as shown below. Does anyone of you have an idea how to approach such a problem? Any links to literature or for further reading would be greatly ...
3
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0answers
29 views

Solving ODE boundary problem with additional conditions

I need to solve two point bondary problem for ODE. The solution of the problem itself is very easy. The real issue is that when solving arising non linear system of equations it always converges to ...
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2answers
96 views

Solving system of differential equations with interconnected boundary conditions

I am trying to solve the following system of differential equations numerically over the domain $x=0$ to $x=D$. The main difficulty is that the boundary conditions are interconnected and depend on the ...
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1answer
281 views

Python: Grid with step control ODE solver

I have a problem in physics formulated via an ODE. Now I like to solve it numerically using Pythons scipy.integrate and the therein complex_ode. I figured out how and it works but now I like to ...
3
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1answer
223 views

Algorithm for solving an ODE with time-dependent parameter numerically

Would anyone please explain me what is the mathematical algorithm to solve a IVP system of ODE with a time-dependent parameter. e.g. ...
3
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1answer
99 views

Options for solving ODE systems on GPUs?

I would like to farm out solving systems of ODEs onto GPUs, in a 'trivially parallelisable' setting. For example, doing a sensitivity analysis with 512 different parameter sets, or solving the ...
2
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2answers
403 views

Number of equations and precision of SciPy's integrate.odeint()

Is there any reason why SciPy's integrate.odeint() should become less precise when the number of equations increase? I'm trying to solve these two sets of differential equations: $\frac{dy_1}{dx} = ...
3
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3answers
337 views

4th Order Runge-Kutta Method for Coupled Harmonic Oscillator

I'm attempting to write a c program to gather values from a coupled spring system where there is a wall, connected to a mass m1 by a spring, then this mass is connected to a second mass m2 by another ...
5
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2answers
137 views

Stability of the first-order exponential integrator method

The question is about the first-order exponential integration method described in this article. Consider a system of ordinary differential equations $$y'(t) = -A\,y(t) + \mathcal{N}(t, y), \qquad ...
6
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1answer
330 views

Why am I getting so much error for my Runge Kutta Fehlberg solver?

My current project is a reprogramming of a protein folding model involving the solution of thousands of ODEs in C++. I've been making some stop and start progress as I'm writing the solver to run ...
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0answers
63 views

Increase convergence of non-linear equations resulting from ODEs

I am trying to solve a set of couple ODEs: $V_l(r) - r W_l(r) - f1(r) W_l' = 0\tag 1$ $r^2 h''_l(r) + f2 r h_l'(r) + f3 h_l(r) - f4 U_l(r) = 0 \tag 2$ $\kappa (U_l + h_l) + V_{l+1} + W_{l+1} = ...
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2answers
184 views

How can i solve this first order system of differential equations

Edit: I am trying to solve $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu\frac{\partial^{2}u}{\partial x^{2}}\\ u(0,t)=0\nonumber \\ u(1,t)=0\nonumber \\ u(x,0)=\sin(\pi x)\\ ...
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1answer
138 views

Can the method of lines technique be used to solve a ODE directly for the steady-state value?

Say we have a discretised a coupled nonlinear system of two PDEs to give a system of ODEs which approximates the original system, $$ \frac{\partial u}{\partial t} = F_1(t,\boldsymbol{u,v}) \\ ...
4
votes
2answers
171 views

Convert an implicit ODE system to an explicit ODE set to use Runge-Kutta

For my series on mechanical systems with Lagrange, I would like to add the double pendulum and a rolling pendulum. I set up the Lagrangian $L$ and solved for the ODE of motions, and got the following: ...
6
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1answer
159 views

Numerically solving systems of about 100 ODEs

I am looking to solve large systems of non-linear ODEs. There appears to be a very large list of methods available varying in complexity, and I have a hard time searching through them and picking one. ...
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0answers
51 views

Stochastic Collocation for time evolving ODE

For an Stochastic Differential Equation, e.g., $$ \frac{du}{dt} = \alpha*\sin(u*t) $$ where $\alpha$ is normally distributed with nonzero mean, I am trying to use a stochastic collocation approach ...
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1answer
117 views

ode45 usage in this case?

The background is that I'm solving a problem in numerical analysis and I seem to be halfway having solved the non-linear system and now getting to the part where I should apply solving differential ...
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1answer
193 views

Non-linear ordinary differential equation in the modeling of the oscillation of a meniscus

I am trying to model the oscillation of a fluid miniscus in a straw when the miniscus is displaced from its equilibrium level. The results was the following non-linear ODE: $$y''= 1/y - 1.$$ This ...
5
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2answers
164 views

ODE: How to measure stiffness if the Jacobian has zero eigenvalues?

Say you have a system of ODE's where the Jacobian has one zero eigenvalue; what does that tell you about the stiffness of the system? This case doesn't seem to be discussed in the cases I have been ...
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0answers
45 views

Energy anomolies in many body simulation

I am trying to simulate the gravitational interaction between many bodies. I am using a direct PP force calculation and a 4th order symplectic integrator with a variable step size. The energy of ...
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2answers
669 views

How do I solve a boundary value ODE in MATLAB?

Specifically, ode15i. I have ode15i solving a system of 5 first order implicit odes in 5 variables with an initial condition (made consistent by decic). It's great for what I need, except I need to ...
4
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2answers
196 views

How to decide stability of Runge-Kutta method for non-linear ODE?

I'm working on a parameter study of Duffing's equation $\ddot x + \delta \dot x + \alpha x + \beta x^3 = \gamma \cos{\omega t},$ where $\delta, \alpha, \beta, \gamma$ and $\omega$ are real ...