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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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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27 views

Radial wave-function using Finite Differences

I want to solve an equation of the form $$\left[\frac{d^2}{d r^2} + \frac{1}{r}\frac{d}{d r}\right]\psi = 0$$ where $\psi$ is a wave function using finite difference method. The equation is more ...
3
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1answer
102 views

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 ...
3
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1answer
100 views

Solve an ODE with positivity-preserving property unconditionally

I have an ODE for a scalar function $u=u(t)$ of the form: $$ \frac{du}{dt}=L(u). $$ Here the function $L=L(u)$ satisfies: $$ L(0)=0, \quad L'(u)\le0. $$ Then it is easy to see that the solution $u=u(t)...
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1answer
129 views

Computational time not proportional to integration interval in ODE-solver?

I am running octave and i have been trying ode45, ode54, ode23 etc to integrate the equation ` $$Q''(t) = B\cos(Q)\sin(\omega t)$$ $$Q(t=0)=0.$$ When the time interval to be integrated increases, the ...
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0answers
67 views

Stiffness emerges as number of ODEs increases

I want to solve a system of ordinary differential equations with Matlab. I need this to solve a mechanical engineering related problem. If $n$ is the number of degrees of freedom of my mechanical ...
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0answers
23 views

How come the use of delay differential equations in model parameter estimation better than ordinary differential equations? [closed]

in systems biology why is the use of delay differential equations better than ordinary differential equations i.e. compartment models in delay modelling? is there a data independence angle in models ...
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3answers
384 views

ODEs vs DAE vs ADE?

I am totally confused between ODEs which I am familiar with, and differential algebraic equations (DAE) and Algebraic Differential Equations (ADE). Are they the same but just different names or what ...
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2answers
67 views

Integrating a dynamical system until an algebraic condition is satisfied

I have a model given by a system of differential equations $$ \frac{dy}{dt}=f(y)$$ with $y=(y_1,y_2,y_3)$ and $f:\mathbb{R}^3 \to \mathbb{R}^3$. The system works as follows : integrate the ode's ...
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2answers
165 views

Determine numerical infinity for Schrodinger equation $−\psi''(x) + x^ 2 \psi(x) = E\psi(x)$

Consider the following Schrodinger equation for the harmonic oscillator with real $x$: $$ −ψ''(x) + x^ 2 ψ(x) = Eψ(x). $$ I solve the last equation using shooting method and implicit Runge-Kutta ...
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0answers
51 views

How to solve bring my implicit equation to closed form?

I have a simulated system as shown in the following image: $L_0$ is attached to two other bodies $L_1$ and $L_2$. Furthermore, body $L_3$ is also in the simulation (it is attached to $L_2$ but that ...
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0answers
60 views

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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1answer
75 views

Why is this method for simulating a system of springs and masses unstable?

I have a computer simulation system of bodies connected by springs, so their movement is governed by: $x_{n+1} = x_n-\Delta tk(x_n-r)$ Where $r$ is the idea distance between every two bodies, and $\...
3
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1answer
132 views

Methods for solving $x'=Ax+b$ for small, sparse, singular $A$

I am in the process of building a robotics physics engine. I have been using the Linear ODE $x' = Ax + b$ for the core of my physics integration, but have never found a really good solution method for ...
3
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1answer
58 views

Problem with Richardson extrapolation method for weak convergence in SDE

I have implemented the Richardson extrapolation of the Euler-Maruyama method to 4th order, to estimate the moments of SDE. The Euler-Maruyama works, and I would expect the Richardson extrapolation to ...
5
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1answer
37 views

Projection on Stiefel manifold after integration step

A few days ago, I asked how constraints like $A^T A = I$ can be implemented if one wishes to integrate differential equations of the form $\dot{A}=f(A,t)$. Kirill was so kind to point out that a ...
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23 views

How to apply asymptotic matching condition in bvp4c?

I want to know how to write the code to apply asymptotic matching condition? In the paper, it is stated that they apply asymptotic matching condition for the range of ...
3
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1answer
93 views

Integration of differential equation with orthogonality constraint

Lets say I have a system of differential equations which has the form $$\dot{C}_{\alpha,\beta,m} = f_{\alpha,\beta,m}(C_{\alpha,\beta,1},\ldots,C_{\alpha,\beta,N};t).$$ The $f$s are some functions of ...
3
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0answers
86 views

How to obtain the reduced model from a subspace projection method?

I have a system of ordinary differential equations (ODEs). It is a large system that has dozens of equations and hundreds of parameters. I wish to reduce its size so it becomes computationally more ...
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1answer
66 views

Solving stiff equations in Mathematica

I have problem to solve stiff equations. Any idea on how to solve this? I have tried "StiffSwitching" but it didnt work. Im solving this using Mathematica 10. Here is my code. Im sorry if I wrote the ...
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1answer
100 views

Solving second order SDE with Gaussian white noise for first time derivative in Matlab

I'm having trouble solving a second order differential equation with Gaussian white noise. The equation I'm solving follows the form: $$Ax'' + Bx' + \sin(x) = i + i_{n}$$ where $i_{n}$ is the ...
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80 views

Solving first order system in MATLAB

I have problem to solve this in MATLAB. The equation already in first order system, yet I still confused to write the code and solve it. Or this equation cant be solved by using bvp4c perhaps? ...
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28 views

Shooting method MATLAB upper order non linear ODE [duplicate]

How can I solve a system of nonlinear differential equations using Matlab? I know I need to use the shooting method but how should I do it? I know I have to control the value of f'' so that it ...
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59 views

Numerically solving geodesic differential equations with a priori knowledge of the Riemann curvature tensor

The geodesic differential equations are given as \begin{align} \frac{d^2 x^j}{ds^2} + \Gamma^{\phantom{h}j}_{h\phantom{j}k}\frac{dx^h}{ds}\frac{dx^k}{ds} = 0, \end{align} where the $\Gamma^{\phantom{...
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1answer
155 views

Shooting method - Matlab ODE

I'm trying to solve these equations of hypersonic adiabatic flow over a flat plate. I did all the simplifications and got these equations for the stagnation point flow. $$\left(Cf''\right)' + f f'' = \...
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1answer
150 views

How can I solve stiff equations by method of line (MOL)?

I want to solve 7 coupled equations.I use method of line(MOL) and discrete the equation in Length and radius and convert them to a system of ODEs in time.and use ode15s to solve them in MATLAB. But an ...
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1answer
49 views

Fixed-step ODE solver with variable order?

I am interested in fixed-step simulation of an ODE. The methods I know of are either variable-step with prescribed error tolerance or fixed-step without error control. Are there methods known which ...
9
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1answer
162 views

Which numerical methods preserve time reversal symmetry?

If I have a physical system which contains a time reversal symmetry (for example a Hamiltonian $H(x,p)=p^2/2m + V(x)$ with $V(x)$ real) and I want to solve the differential equations which describe ...
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35 views

maltab ode solver- user defined criteria to stop calculations

is there a way to add a user defined convergence criteria to an ode solver so that the solution is stopped? I know that Matlab uses absolute and relative tolerances but would that suffice in solving ...
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52 views

Matlab help with differential equations [duplicate]

For my coursework I have to model human hair in which I have $$\frac{d^2\theta}{ds^2} = s f_g \cos(\theta)+s f_x \cos(\phi)\sin(\theta)$$ Where $\theta$ is a function of $s$. I am given the conditions:...
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1answer
117 views

Solving an ODE beyond existence. What's happening?

As an example for an ODE course I used the ODE $$ y' = \frac{y}{x} + \frac{1}{\cos(\tfrac{y}{x})} $$ to illustrate domains of existence. Standard substitution $z=y/x$ turns the equation to $$ z' = \...
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2answers
332 views

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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0answers
30 views

Boundary conditions shooting method

I am trying to solve the differential equation $\frac{d^{2}y}{dr^2}+(\frac{1}{r}+1)y=0$ with the boundary conditions $y(r) \rightarrow r \frac{dy}{dr}(0)$ as $r \rightarrow 0$ and $y(r) \rightarrow ...
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1answer
55 views

Solving an ODE while maintaining weak positivity and weak monotonicity

I have a system of $N$ ODEs of the form, $$ M(z,F(z)) \cdot F'(z) = \Phi(z,F(z)) $$ where the mass matrix is $M(z,F): R\times R^N \to R^{N\times N}$ and $\Phi(z,F):R\times R^N \to R^N$ is (potentially)...
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1answer
95 views

BDF methods for implicit-explicit method

Are there BDF formulas like the ones given here but one that can be used with implicit-explicit discretization? The right hand side in those formulas is supposed to be implicitly discretized at the ...
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0answers
50 views

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 ...
3
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0answers
123 views

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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97 views

Computing solutions with singularities using MATLAB ODE45

I am new to solving numerically ODES and thus it is difficult for me to judge the reliability/trustworthiness of the results that I have produced for the following problem: I am dealing with a 2nd ...
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67 views

Techniques to solve this complicated ODE

I have to solve the following ordinary differential equation: $$A(\rho, \Phi)\Phi'' + C(\rho, \rho', \Phi)\Phi' - D(\rho, \rho')\Phi^2 - \lambda \rho^4 \Phi^3 = 0 \enspace .$$ Here, prime $(')$ ...
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2answers
129 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 ...
2
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2answers
81 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 ...
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0answers
89 views

ODE events to switch between 5 equations (friction model)

I am modelling a 1 dof spring-mass-damper system with friction. As first attempt I modelled the friction according to the simple Coulomb model (figure A here http://article.sapub.org/image/10.5923.j....
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1answer
48 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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102 views

Solving constrained BVP, singular Jacobian

The boundary value problem is $$ \begin{cases} \dot{x}_i = \begin{cases} (0.5D^{-1}\psi)_i, \text{ if }(0.5D^{-1}\psi)_i \le 0 \\ 0 \text{, otherwise} \end{cases} \\ \dot{\psi} = 2\Sigma x \\ x(0) =...
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1answer
271 views

Numerical solution of Geodesic differential equations with Python

I have made a solver based on the SymPy.diffgeom library, where I use Scipy.Integrate to solve the following system of second order differential equations : \begin{align} u'' &+ \Gamma^0_{00}(u')...
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1answer
194 views

How can I solve coupled equations by the method of line(MOL)?

I want to solve 3 coupled PDEs equations. They depend on time, radius and length. I used the method of lines (MOL) and converted them to a system of ODEs in time. Now I want to solve them using MATLAB....
0
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1answer
97 views

How can i define algebraic equation in differential function in MATLAB?

I want to solve 7 pde's that are functions of time, radius(j) and length(i). I used the method of lines and converted them to a system of odes in time and it becomes something like this: $$dy/dt=((y(i,...
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1answer
89 views

What is the meaning of this error in MATLAB?

Warning: Failure at t=6.137539e-04. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (1.734723e-18) at time t. In ode15s (line 730) In ...
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47 views

Time dependent self-consistent equations

I am facing the following problem. I need to solve numerically a set of coupled equations $$i\frac{d}{dt}f_{n}^{(i)}(t) = \left[U\cdot n(n-1) + \mu\cdot n\right]f_{n}^{(i)}(t) - \sqrt{n+1}\Phi_i^{*}\...
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1answer
52 views

stirred tank model; DAE versus ODE model

I do have a stirred tank reactor with two inlets and one outlet. Several components enter the reactor at inlet 0 and particles at inlet 1. All component from inlet 0 adsorb on the particles from inlet ...