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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.

9
votes
0answers
165 views

Inverse problem in linear ODE

I have a linear ordinary differential equation (ODE) with a system matrix with constant coefficients: $$\dot{y}(t) = \mathcal{A}\; y(t), \quad y(0) = y_0$$ with $y(t) \in \mathbb{R}^{n \times 1}$ and $...
6
votes
0answers
160 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 ...
4
votes
0answers
66 views

Solution of constrained system of ODEs

Can someone point me in a direction to solve this kind of integral constrained system of ODEs. \begin{align} &\int_0^{1/2}\dot{y}^2(t)=p\\ &2\lambda_1\ddot{y}(t)+\pi cos(\pi y(t))=0\\ &y(...
4
votes
0answers
311 views

Convergence rate of Picard iterations

Given a first order ODE $y'(x)=f(x,y)$ with the initial condition $y(x_0)=x_0$ such that it satisfies Picard thoerem of existence and uniquness, one can compute the solution by Picard iterations : $$ ...
4
votes
0answers
74 views

Large residual when integrating 2nd order ode close to singularity with SciPy ode / ODEPACK

I am trying to integrate a 2nd order ODE with a singularity at close to the initial condition. Why do I get large residuals when I plug-in the result of my integration back into the ODE? The equation ...
3
votes
0answers
47 views

Calculate the Bloch wave

The eigenvalue problem $$\frac{d^2u}{dx^2}+2i k\frac{du}{dx}-[k^2-6\sin(x)^2]u(x)=-\mu u(x)$$ gives the first five eigenvalues with $k=0$ or $k=1$ which are $2.06$, $2.26$, $5.16$, $6.81$, and $7....
3
votes
0answers
55 views

Solve ODE with non-negative and maximization constraints

My task is to solve $$\eta_k\frac{d^2C_k}{dz}(z)=-e_k, k = 1,2,3$$ $$C_k\ge0$$ $$C_1(0)=0, C_2(0)=A, C_3(0)=0$$ $$C_1(L)=B, \frac{dC_2}{dz}(L)=0, \frac{dC_3}{dz}(L)=0$$ with $$e_1 = -\beta_1-\beta_3$...
3
votes
0answers
143 views

Avoiding divergent solutions with `odeint`? shooting method

I am trying to solve an equation in Python. Basically what I want to do is to solve the equation: $$ \frac{1}{x^2}\frac{d}{dx}\left(Gam \frac{dL}{dx}\right)+L\left(\frac{a^2x^2}{Gam}-m^2\right)=0 $$ ...
3
votes
0answers
98 views

How to account for the interface between two different phases in a discretized diffusion model?

I have tried to set up a model for the diffusion of a gas into a liquid. The two media are next to each other and the geometry is spherical because the system should simulate the diffusion out of a ...
3
votes
0answers
98 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{...
3
votes
0answers
186 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 ...
3
votes
0answers
144 views

Solving an ODE using shooting method

I have been trying to solve the following nonlinear ordinary differential equation: $$-\Phi''-\frac{3}{r}\Phi'+\Phi-\frac{3}{2}\Phi^{2}+\frac{\alpha}{2}\Phi^{3}=0$$ with boundary conditions $$\Phi'(...
3
votes
0answers
85 views

Is it normal to expect the error of simulation of a damped harmonic oscillator to decrease as the damping factor decreases?

I am simulating a damped harmonic oscillator using the RK4 method of numerical integration. I am comparing the simulated results with the analytical ones (for the free evolution case) and obtaining ...
3
votes
0answers
39 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 ...
3
votes
0answers
81 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 ...
3
votes
0answers
89 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 ...
2
votes
0answers
51 views

How to implement adaptive step size Runge-Kutta Cash-Karp?

Trying to implement an adaptive step size Runge-Kutta Cash-Karp but failing with this error: ...
2
votes
0answers
83 views

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} &...
2
votes
0answers
60 views

Finite differenced eigenvalue prob. of inhomogeneous boundary conditions?

I am basically asking about eigenvalue problems of differential equations using some finite difference method (FDM). Usually the system is subject to some boundary conditions (BC), e.g., Dirichlet or ...
2
votes
0answers
782 views

Using RK2 Method to solve the simple harmonic oscillator of a horizontal mass on a spring (1D)

Being new to numerical analysis techniques, in particular RK2, I decided the best way to jump in is by using python to solve the well known mass-spring oscillator using RK2 techniques. My problem is ...
2
votes
0answers
431 views

Scaling step size in adaptive runge-kutta method

I'm developing my own generic Runge-Kutta solver, and I'm currently implementing the adaptive step-size routine. I say generic because I want to be able to test different RK implementations by only ...
2
votes
0answers
147 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 ...
2
votes
0answers
208 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....
2
votes
0answers
303 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) =...
2
votes
0answers
124 views

Rank deficient Jacobian in discretized periodic solutions to autonomous ODE

I'm trying to numerically find periodic solutions to different systems of autonomous nonlinear ordinary differential equations. I decided to use a finite difference scheme and solve the resulting ...
2
votes
0answers
89 views

How to tell if symplectic integrator is more suitable for my problem, and what are downsides?

This question follows another one that I have already asked. My intention was to use a classical Runge-Kutta 45 method to solve ODEs of my system. However, I have seen recommendations for using ...
2
votes
0answers
134 views

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 ...
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 ...
1
vote
0answers
40 views

Computing Trajectory Equations of Kerr Geodesics

I want to numerically solve the trajectory equations of a Kerr geodesic given by wikipedia in Matlab. The trajectories look like: I implemented the equations and solved it with the standard Runge-...
1
vote
0answers
38 views

Finite dimensional optimization problem over dynamical system

I am interested in solving numerically the following mathematical problem Consider an ode of the form $$ \dot q(t) = f(q(t),t_1,\ldots, t_N),\qquad t\in [0,T], $$ where $q\in \mathbb{R}^n$ is the ...
1
vote
0answers
16 views

Translating grid with extrusion speed

I am putting into MATLAB code the equations that describe a plastic extrusion process. From a paper, I found I should use a spatial grid that translates with the extrusion speed, being the reference ...
1
vote
0answers
76 views

SDE solver in python: manual determination of integrator step size (dt)

Aim: I am trying to solve a system of SDEs, while using the SDEint package in python 3.x. It is a system of SDEs adapted from and inspired by the Zombie Apocalypse ...
1
vote
0answers
88 views

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 ...
1
vote
0answers
353 views

Which solvers for BVP in python are the best? Is there something better that scipy.integrate.solve_bvp?

I am trying to solve a boundary value problem with Python. I have been using scipy.integrate.solve_bvp but the result that it is giving me is completely wrong. Basically my code is as follows: ...
1
vote
0answers
51 views

Detecting stability in solutions to coupled nonlinear equations in aerodynamics,

I'm studying a quasi-steady force model (published in a fluid dynamics journal) that consists of coupled, nonlinear ODEs that describe unsteady aerodynamics -- recently, my advisor and I have found ...
1
vote
0answers
398 views

One Dimensional Schrodinger's Equation solution using Numerov Method

I have been trying to solve Time Independent Schrodinger's equation in one dimension using Numerov Method as discussed in this excellent lecture notes I found on net. The Numerov method can solve an ...
1
vote
0answers
43 views

Solve ODE with two unknown functions

I want to solve a diffusion-convection problem numerically. I start from a PDE of the form $$\partial_t f(x,t) = \partial_x(K(x)\;f(x,t))+\partial_{xx}\;f(x,t)^{m}\;.$$ It is possible to calculate a ...
1
vote
0answers
154 views

How to implement chaotic sender and receiver with ordinary differential equations?

I am referring this paper, and trying to implement chaotic sender and receiver, to decode message, as given in section $V^{th}$ Chaotic signal masking. The process that I want to implement is figure ...
1
vote
0answers
68 views

Convert scipy integration with one step to matlab integration

Scipy integration allows us to do ode integration one adaptive timestep at a time and do something to it. However, matlab ode needs us to specify a timespan , and determine the adaptive timestep ...
1
vote
0answers
84 views

The Schrodinger equation for time-dependent Hamiltonian after one timestep, taking exponential or use ode solver?

The Schrodinger equation for time-dependent Hamiltonian is $$i\hbar\frac{d}{dt}\psi(t) = H(t)\psi(t) \, .$$ Assuming I knew $\psi(t)$, I want to know $\psi(t+\Delta t)$. Should I take exponential ...
1
vote
0answers
110 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 ...
1
vote
0answers
77 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 ...
1
vote
0answers
306 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 ...
1
vote
0answers
82 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^{*}\...
1
vote
0answers
54 views

should boundary conditions be effecting moving mesh results?

I have a question on the use of moving mesh to solve the inviscid euler equations. I have solved the following equations: $$\frac{\partial}{\partial t}\left[\begin{array}{c} \rho\\ \rho u \end{array}\...
1
vote
0answers
48 views

Can variational formulations be solved using series solutions?

What I specifically mean is, given some functional $F\left[\mathbf{x}\right]$ which is stationary with respect to $\dot{\mathbf{x}}=f(\mathbf{x})$ and some boundary or initial conditions, can one ...
1
vote
0answers
2k views

How to solve an ode with stochastic time-dependent input

I am trying to repeat an example I found in a paper. I have to solve this ODE: $25 \ddot{x} + 15 \dot{x} + 330000 x = p(t)$ where $p(t)$ is a white noise sequence band-limited into the 10-25 Hz ...
1
vote
0answers
84 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} = 0\...
1
vote
0answers
224 views

Block Backward Differentiation Formula (BBDF), on order 4 formula

I am trying to implement a program the numerical method to solve ODE called Block BDF as explained in this article: https://waset.org/journals/waset/v38/v38-49.pdf As it is variable step-size, I need ...
1
vote
0answers
30 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 ...