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]).
169
questions
0
votes
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32
views
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
1
vote
1
answer
42
views
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
votes
0
answers
57
views
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
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votes
1
answer
50
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., ($\sin(s)\;cos(s)$).
...
- 11
-1
votes
0
answers
52
views
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 ...
7
votes
1
answer
138
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
89
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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votes
0
answers
89
views
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
54
views
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
26
views
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
88
views
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
1
vote
0
answers
64
views
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
0
votes
0
answers
71
views
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
165
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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)...
- 23
3
votes
1
answer
238
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\...
- 205
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votes
1
answer
134
views
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
73
views
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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votes
0
answers
103
views
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
124
views
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
121
views
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
247
views
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
159
views
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
82
views
Solving an apparently tricky geodesic BVP in Matlab
I want to be able to solve the BVP
$$\ddot \mu_k = -\frac{\mu_k}{2} \left ( \sum_{i=1}^n \frac{\dot \mu_i^2}{\mu_i} - \frac{\dot \mu_k^2}{\mu_k^2} + \frac{ \left [ \sum_{i=1}^n \dot \mu_i \right ]^2}{\...
- 123
0
votes
0
answers
64
views
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
221
views
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
91
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 $a_i\in\mathbb R^n$ ...
- 1,331
1
vote
2
answers
522
views
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
118
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
1
vote
1
answer
998
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
98
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
675
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 ...
- 91
3
votes
4
answers
195
views
Solving the eigenvalue from a set of coupled second order differential equation numerically
I met a problem in solving a set of coupled differential equation, as shown below:
$$A_1\psi_1(z)+A_2\frac{d^2\psi_1(z)}{dz^2}+A_3\frac{d\psi_2(z)}{dz}=\lambda\psi_1(z)$$
$$A_4\psi_2(z)+A_5\frac{d^2\...
- 131
3
votes
0
answers
89
views
Numerical Soultion to Background equations of cosmology
I am trying to solve the background equations of cosmology numerically using Runge-Kutta Dormand Prince method with simplified assumption $8\pi G=1$ and $c=1$. The equations are
$$\ddot a = - \frac{1}{...
- 39
0
votes
0
answers
30
views
How to solve nonlinear second order ODE in Matlab? [duplicate]
I am working on simulating a car suspension system using Matlab.
Specifically, I have to derive equation of motion using the Lagrange method and then use ode 45 to solve it. However, while using ...
Luận Trần Công
7
votes
0
answers
161
views
Is there a graphical interpretation or explanation of automatic differentiation compared to numerical differentiation
I have been looking at automatic differentiation for solving differential equations lately. I understand the basic ideas of using Dual numbers and such for finding derivatives, etc. However, I feel ...
- 255
0
votes
1
answer
48
views
How to decrease error in (FTCS) forward time centered space method?
I am using the FTCS method for solving differential equations. I know that the condition for stable output is
$$
\frac{\alpha \Delta t}{\Delta x ^2} < \frac{1}{2}
$$
But when I use the distance ...
- 13
4
votes
1
answer
111
views
Finite Volume on Cubed Sphere
The US weather model uses an uncommon (?) discretization called 'Finite Volume on Cubed Sphere'.
To avoid the singularities that occur at the poles when using lat/lon discretization, they instead ...
- 43
5
votes
2
answers
613
views
Specifying ode solver options to speed up compute time
I'm specifying the 'JPattern', sparsity_pattern in the ode options to speed up the compute time of my actual system. I am sharing a sample code below to show how I ...
- 447
3
votes
2
answers
773
views
Solving numerically the 1D Kuramoto-Sivashinsky Equation using spectral methods
Lately, I've been trying to solve numerically the 1D Kuramoto-Sivashinsky Equation using spectral methods.
Let $\nu$ be the viscosity and $[0,L]$ the domain. The 1D equation is,
$$
u_t + uu_x + u_{xx} ...
- 153
4
votes
1
answer
92
views
Geometric integrators besides midpoint/Crank-Nicolson?
I have a first-order ODE
$$
\dot{x} = a(t) \times x, \quad x(0) \in\mathbb{R}^3.
$$
with $\|a(t)\| = 1 \;\forall t$.
Consequently, $\|x(t)\|=\|x(0)\|$ for all $t>0$. I would like this to be ...
- 3,058
3
votes
2
answers
436
views
How to solve second order coupled non linear differential equations
For a project I am doing, I have to solve the following system of differential equations numerically using my own code:
$$
x^2K'' = KH^2 + K(K^2-1)
$$
and,
$$
x^2H'' = 2K^2H + \alpha H(H^2-x^2)
$$
...
- 41
1
vote
1
answer
99
views
Numerical integrator for $a'(t)=e^{-a(t)}f(t)$
Suppose I know a function $f(t)$ and all its derivatives in $t$ in closed form. Given $a(0)$ and some $t_0>0$, I'm looking for an explicit integrator that can estimate $a(t_0)$, where $a(\cdot)$ ...
- 1,331
1
vote
1
answer
359
views
Finite Element Method for 1D Poisson Equation with Inhomogeneous Boundary Conditions
Im trying to solve the Poisson equation in 1D: $$-u_{xx} = f(x), \hspace{6mm} u(a) = d1, \hspace{2mm} u(b) = d2$$Assuming a uniform partition such that $x_n = a + nh$, where $h = (b-a)/N$ and $n \in [...
- 123
2
votes
0
answers
166
views
Find time step for Euler method in numerical solving of a system of non linear differential equations
I have a system of non linear differential equations in the form :
$$\frac{dy_i}{dt}=\sum_j a_{ij} y_i y_j $$.
I first tried to solve it with Python suing ...
- 171
0
votes
1
answer
75
views
2 point BVP solver: how to compute errors
Background
I am working with chapter 2 in LeVeque's book: https://faculty.washington.edu/rjl/fdmbook/
I have build my own solver in Python to solve the 2 point BVP:
$$
\epsilon u''+u(u'-1) =0 , \\
u(0)...
- 255
1
vote
0
answers
103
views
Stochastic differential equation system (SDE) : overflow encountered in double-scalars
I'm trying to integrate the following SDE system from Dekker et al. [1]
$$\begin{cases}
\frac{dx}{dt}=a_1x^3+a_2x+\phi+\zeta_x\\
\frac{dy}{dt}=b_1z+b_2(\kappa(x)-(y^2+z^2))y+\zeta_y\\
\frac{dz}{dt}=...
- 153
5
votes
2
answers
1k
views
CUDA & Python for numerical integration and solving differential equations
Can anyone please suggest some libraries which allow use CUDA in Python for numerical integration and/or solving of differential equations?
My goal is to solve large (~1000 equations) of coupled non-...
- 171
3
votes
0
answers
90
views
Difference between wave vector and density matrix in numerical calculation of Schrödinger equation
I solved Schrödinger equation for a following tow-level time-dependent Hamiltonian numerically in two ways:
...
- 31
0
votes
0
answers
440
views
Numerov method for solving Schrödinger equation
I have just begun learning computer science to apply it to Physics and I am trying to write a code for solving Schrödinger's equation of the harmonic oscillator (setting $V=\frac{x^2}{2}$) in one ...
- 27