Questions tagged [differential-equations]
For questions about solving, analyzing, or creating differential equations to model some system. If possible, include specific tags about the type of differential equation (e.g. [tag:pde], [tag:ode], [tag:stochastic-ode]).
187
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9
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Numerical code to solve LLG is not preserving norm
I am new to this thread. I am trying to do a simple exercise on solving LLG equation. The equation reads:
$\frac{d\vec{m}}{dt} = -\gamma(\vec{m} \times\vec{H})$.
The expected output for an input state ...
- 1
0
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0
answers
35
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Can the Crank-Nicolson Method Be used to Solve The Schrodinger Equation with a Time Varying Potential?
I have been following an excellent article about how to use the Crank-Nicolson method to solve the Schrodinger equation. In the article it starts with a $V(x, y, t)$ but the potential seems to become ...
- 11
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0
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23
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Numerov Method with Time Varying Potential
Is it possible to use the Numerov method to solve the Time Dependent Schrodinger Equation ($\frac{i\partial\Psi(x, y, z, t)}{\partial t} = \nabla^2\Psi(x, y, z, t) + \Psi(x, y, z, t)V(x, y, z, t)$) ...
- 11
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0
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20
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Translating poisson equation to sfepy code
As the title suggests, I am trying to convert the Poisson equation:
Into sfepy code that can solve the differential euqation problem numerically.
I am trying to modify the tutorial on lienar ...
- 165
3
votes
1
answer
89
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unconditionally stable schemes better than conditionally stable ones in accuracy?
Let's consider two finite difference schemes for PDEs/ODEs. One is conditionally stable, the other is unconditionally stable. People always prefer unconditionally stable ones to conditionally stable ...
- 187
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votes
1
answer
114
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Nonlinear Robin boundary condition involving square root
If you have a nonlinear second-order boundary value problem where the domain of the problem is $x \in [a,b]$, the boundary conditions imposed are the Robins condition at $x=a$ and the Dirichlet ...
- 153
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0
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89
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ValueError: array must not contain infs or NaNs; When using solve_ivp in the scipy library
I am solving an initial value problem using solve_ivp. The problem consists of computing the concentration profile of a set of reactions over time, given the initial concentrations and some of the ...
- 23
1
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0
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45
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Solving PDE on a non-uniform grid with Crank-Nicolson scheme
I am solving a 1D diffusion-type equation with the finite-difference Crank-Nicolson (CN) scheme, and I need to densify the spatial grid around the central point. One could change the spatial variable ...
- 21
1
vote
0
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59
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Finite-difference produces a derivative off by one order
I have a nonlinear 2nd order boundary value differential equation where I used finite-difference method (central finite-difference) to solve it.
$$z''(x)-\frac{\frac{1}{100} z(x)^4 \left(2 z'(x)^2+12\...
- 153
0
votes
1
answer
137
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Solving a partial integro-differential equation numerically
I am trying to find the solutions for a probability density $p(x,t)$, governed by,
$$
\frac{\partial p(x,t)}{\partial t} =\int_{-\infty}^\infty dx' \; \Lambda(x-x')\frac{\partial^2 p(x',t)}{\partial x'...
- 201
1
vote
2
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109
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Calculate the derivative of the finite-difference method result
I have a nonlinear boundary ODE,
$$y'' + 3 y y' = 0, \qquad y(0) = 0, y(2)=1 $$
I want to solve this using the finite-difference method. I obtained the result for the data set of $y(x)$ (including the ...
- 153
3
votes
1
answer
99
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Wrong Boundary Conditions Result Using Wavelet Collocation
I have a functional $S$,
$$S = \int_{x_0}^{x_b} dx \frac{1}{z(x)^d} \sqrt{1 + \frac{z'(x)^2}{f(z)}}, \qquad f(z) = 1-\left(\frac{z(x)}{z_h}\right)^{d+1}
$$
where $d=3$ is the dimension and $z_h$ is ...
- 153
4
votes
3
answers
937
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How can I numerically integrate the Kepler problem?
I tried to solve a simple Kepler problem numerically.
I have discrete time steps, a starting position $(x_0,y_0)$ and starting velocity $(u_0, v_0)$.
I used this iteration by calculating the forces ...
- 151
2
votes
1
answer
92
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Deterministic SIR metapopulation model and coupling behavior
Context:
I am trying to reproduce a figure from Keeling 2007 that illustrates time lags that can occur between the peaks (maximum) of the infected solutions for two subpopulations of a metapopulation ...
- 171
2
votes
1
answer
206
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Python bifurcation diagram of seasonally forced epidemiological models
TL:DR
How can one implement a bifurcation diagram of a seasonally forced epidemiological model such as SEIR (susceptible, exposed, infected, recovered) in Python? I already know how to implement the ...
- 171
1
vote
1
answer
122
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(Regular) Coulomb wave function
I'm looking for a way to implement the regular Coulomb wave function in python. This function is a solution to
\begin{align}
\frac{\text{d}^2\,u}{\text{d}z^2}+\left(1-\frac{2\eta}{z}-\frac{\ell(\ell+1)...
- 85
2
votes
0
answers
101
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Numerical precision on tricky coupled nonlinear boundary value problem on infinite interval
I am trying to solve with high precision the following coupled system $(f,h)$ on $[0,\infty]$:
$$-h''-\frac{1}{r}h'+\lambda_c h(f^2-1)+4\lambda_h h^3=0$$
$$-f''-\frac{1}{r}f'+\frac{1}{r^2}f+ f(-\...
- 33
1
vote
1
answer
64
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Boundary value problem solver fails on trivial case
I am trying to solve a boundary value problem on $[0, \infty]$, using scipy's scipy.integrate.solve_bvp and I am seeing that the solutions are not converging even ...
- 33
4
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0
answers
64
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SINDy Vs standard methods for system identification
I have been trying to understand the recently proposed Sparse Identification of Nonlinear Dynamics SINDy. Despite several attempts, I seem to fail to understand the difference between SINDy and the ...
- 810
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1
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80
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Passing additional arguments to `odeint` from `torchdiffeq` to solve an IVP
In Python I use the package torchdiffeq (as provided here) to solve initial value problems. Given an arbitrary function ...
- 3
2
votes
1
answer
53
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Solving constrained odes's using inbuilt solvers in Matlab/Octave
I would like to solve a set of coupled second order differential equations using inbuilt Matlab/Octave subroutines. These equations arise when trying to model sliding of mass ($m_2$) over a wedge of ...
0
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0
answers
67
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Solving a system of PDEs with an ODE
I want to solve the following system of equations which consists two PDEs and one ODE:
\begin{align}
\rho_t+v\rho_x &= 0; \newline
Y_t+vY_x &= 0 ;\newline
v_t &= -\frac{1}{(\...
- 11
1
vote
1
answer
55
views
How to solve advective equation with source term depending on variable
I have the following equation
$$
\dfrac{\partial s}{\partial t} + \nabla \cdot \left( \vec{v} s\right) = f(s)
$$
Where $f(s)$ is an explicit source term that depends on $s$, e.g., ($\sin(s)\;cos(s)$).
...
- 21
1
vote
1
answer
122
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Differential equation for radioactive cooling in fortran
Today I'm trying to evaluate this differential equation for internal energy in a gas in Fortran:
$$
\frac{du}{dt} = - \frac{n_H^2}{\rho}\frac{\Lambda}{n_H^2}
$$
Where nH is the density of hydrogen in ...
- 121
7
votes
1
answer
186
views
How does non-dimensionalization improve the behavior of ODE solvers?
I have a set of coupled ODEs that I'm solving numerically. The independent variable is time and runs from values of $10^{15}$ to $10^{17}$ in units of seconds. The state variables in their usual ...
- 193
2
votes
0
answers
97
views
Numerical solution to integro-differential equation
The time dynamics of an atom interacting with a reservoir of spectral density $J(\omega)$ are obtained by solving the following integro-differential equation:
$$ \frac{\mathrm{d}c(t)}{\mathrm{d}t} = - ...
- 21
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0
answers
115
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Solving differential equations with fast oscillations using odeint
I have wrote this code to solve an equation , I know the behavior of this function has very rapid oscillations, when I RUN it gives bogus values for some "m[x]" and some "t"'s, ...
- 11
2
votes
1
answer
71
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Numerical computation of Lyapunov exponents: how to find convergence or non-convergence efficiently?
I am wondering what are the standards for convergence of Lyapunov exponents (and Kaplan-Yorke dimension)? For example, I have a MATLAB code to calculate Lyapunov exponents for the classic Lorenz ...
- 21
1
vote
0
answers
29
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How to classify ODE equilibria that are stable but slowly changing in value with time?
I'm numerically solving a system of coupled ODEs where time is the independent variable. At each time, I can solve for the equilibrium values of my state variables where their respective derivatives ...
- 193
1
vote
1
answer
92
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Accurately solving system of differential equations
So I am trying to solve two equations simultaneously. The goal is to find values for $\frac{de}{dt}$ and $\frac{d}{dt}$ which are the rates of change of the variables $a$ and $e$. I am then ...
- 13
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vote
0
answers
68
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Implementation of integration schemes for ordinary differential equations in Python and peformance comparison
I look for a book/manual where I can find implementations of different integration schemes for ordinary differential equations (like 4-th order Runge-Kutta) in Python with Numba.
To be more specific, ...
- 171
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0
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78
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How to estimate stability and stiffness of a system of coupled ODEs?
I'm running into issues with Python/Julia ODE solvers requiring prohibitively small timesteps to evolve a system of 4 coupled ODEs (the order of magnitude of the state variables and time unit span ~40-...
- 193
2
votes
1
answer
234
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2nd order differential equation coupled to integro-differential equation in python
I'm trying to solve the following equations numerically in python
$$\begin{align}
12\pi\int_0^\infty drf(r)\phi(r)r^4&=E\\
f(r)-\frac{1}{2\mu}\bigg(\frac{d^2\phi(r)}{dr^2}+\frac{2}{r}\frac{d\phi(r)...
- 85
3
votes
1
answer
436
views
Boundary value problem with singularity and boundary condition at infinity
I'm trying to solve the following boundary value problem on $[0,\infty]$:
$$f^{\prime \prime}=-\frac{1}{r} f^{\prime}+\frac{1}{r^{2}} f+m^{2} f+2 \lambda f^{3}$$
$$f(0)=0 \ ; f(\infty)=\sqrt{-m^2/(2\...
0
votes
1
answer
211
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How to deal with solving coupled ODE systems where variables are updated multiple times within each timestep?
I'm solving a system of coupled ODEs using Euler integration for simplicity. To make this concrete, please see the (extremely simplified) minimal working example below in Python. Imagine we have a box ...
- 193
1
vote
0
answers
79
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Well-conditioned pseudospectral for computing eigenvalues to (partial) differential equations
I am working on writing a Chebyshev pseudospectral method (see for example "Chebsyhev and Fourier spectral methods" by John Boyd) to solve for the eigenvalues of differential equations of ...
0
votes
0
answers
20
views
Transparent Boundary Conditions relationship with intermediate BCs in ADI-PR method
I have read some materials about ADI - PR method with the aim to understand how to put boundary conditions in my 2D scheme which solves the Time-Dependent Schrodinger Equation.
All the theory I read ...
- 131
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0
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138
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Finite difference solver for the 2D Poisson's equation with an integral boundary condition
I wanted to attempt an implementation of a finite-difference-based solver for the 2D elctrostatic Poisson equation when metallic objects are present. Also, I hope to take as input, the location of ...
2
votes
1
answer
195
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solving a Algebraic Differential Equation in Julia using modelingToolKit.JL
I'm trying to solve a differential algebraic equation in Julia's modelingTookKit.JL, where the vector field has the form f(X) = 0.
I found an example of a DAE in the below link
modelingToolkit.JL DAE](...
- 53
2
votes
1
answer
134
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DG method for solving Hyperbolic Partial Differential Equation with Dirichlet Boundary Conditions
Consider the following partial differential equation
\begin{align}
\frac{\partial u}{\partial t}+\frac{\partial f}{\partial x} &= g(x,t), \ \ x\in \Omega = [x_{L},x_{R}] \\
u(x,0) &= u_{0}(x) ...
- 83
3
votes
1
answer
397
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How to solve a BVP with known parameters?
I need to solve a boundary value problem (BVP) of second order, where the equation depends on several know parameters, which are geometric parameters and material constants.
I would like to solve this ...
- 570
3
votes
1
answer
215
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Solving DAE in Julia using GPUs
I'm trying to solve a Differential Algebraic Equation (DAE) in Julia which is very computationally expensive using GPUs. I'm brand new to Julia and don't have much experience coding with GPUs. The ...
- 53
0
votes
0
answers
92
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Solve simultaneous differential equations with embedded functions and a parameter estimation
The aim is to solve the below equations and plot $m$ with time, i.e. $\frac{dm}{dt}$
$k$ is unknown and needs to be estimated. For the parameter estimation, the below values in the table for m versus ...
3
votes
2
answers
340
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Is there a Python version of the ODE tool pplane?
This is the same question as this one, except for Python instead of Mathematica. Basically, the MATLAB software PPLANE is a staple in ODE courses. Is there a Python equivalent?
I don't know much about ...
2
votes
1
answer
97
views
Implicit integrator for ODE with quadratic right-hand side
I have an ODE for an unknown $x(t):[0,\infty)\to\mathbb R^n$ of the following form:
$$
x_i'(t)=a_i^\top x(t) + x(t)^\top Q_i x(t),
$$
for $i\in\{1,\ldots,n\}$. Here, the vectors $a_i\in\mathbb R^n$ ...
- 1,341
1
vote
2
answers
675
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Recommendations for ODE solvers for stiff equations
I'm continuing the research of a former Ph.D. student in my group requiring the solution of a system of ODEs. On a technical note, they wrote:
The system of Boltzmann equations behaves numerically ...
- 105
2
votes
0
answers
121
views
Floquet theory for periodic delay differential equations: current numerical routines
I would like to determine the stability of a system of periodic delay differential equations (a seasonal host-parasite model). I've tried to implement the method described in Lemma 2.5 in this paper:
...
- 21
2
votes
1
answer
2k
views
solve_ivp from scipy does not integrate the whole range of tspan
I'm trying to use solve_ivp from scipy in Python to solve an IVP. I specified the tspan ...
1
vote
1
answer
164
views
Logistic growth curve using scipy is not quite right
I'm trying to fit a simple logistic growth model to dummy data using Python's Scipy package. The code is shown below, along with the output that I get. The correct ...
9
votes
1
answer
754
views
How to solve a second order differential equation (diffusion) with boundary conditions using Python
I am having trouble implementing a model from a publication.
Huang, K-L.; Holsen, T.M.; Selman, J.R. Ind. Eng. Chem. Res. 2003, 42, 15, 3620–3625
scihub link: https://sci-hub.se/10.1021/ie030109q
I ...
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