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]).
240
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50
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Step size constraint in Euler backward
I am dealing with an assignment in MATLAB. It has to do with 'self-driving' cars which are driving in-front/behind eachother. Assuming M cars on a single-lane road, each car adjusts its speed based on ...
1
vote
0
answers
41
views
Solution to the Liouville-Gibbs equation
What would be the approach to numerically solve for $\rho(x,t)$ the following equation with some initial conditions
$$\frac{\partial\rho}{\partial t}
+\sum_{i=1}^n\left(\frac{\partial(\rho g_i)}{\...
- 395
1
vote
0
answers
30
views
Implementation of operator splitting method for Wigner equation
I am dealing with the integro-differential equation for Wigner function,
$$\frac{\partial f}{\partial t}+p\frac{\partial f}{\partial x}+\\+\frac{1}{\chi}\left\{\int_{-\pi}^{+\pi}dy\,\int_{-\infty}^{+\...
- 181
1
vote
0
answers
136
views
FEM for nonlinear first-order ODE
Currently I am trying to solve nonlinear Ricatti equation using FEM (Matlab language):
$$\frac{d r(z)}{dz} = i k(z) r(z) + \frac{i k(z)}{2}(\epsilon(z) - 1)(1 + r(z))^2$$
$z = [-h/2, h/2]$ and $r(-h/2)...
- 31
1
vote
2
answers
111
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Is there any way to reduce an RK4 method's dependence on step size?
I am working in the sphere of orbital simulations, where orbital trajectories are computed using the differential equations describing gravity. Due to the great timescales of orbits, a step size of ...
- 133
1
vote
1
answer
69
views
Why does a two-body simulation result in no change of the y-component?
I've been attempting to create a model of a heliocentric orbit based on Newton's law of gravitation:
$$
\frac{d^2 \vec r}{dt^2} = -\frac{GM}{|r|^2} \hat r
$$
This is what I have so far:
...
- 133
2
votes
2
answers
926
views
Numerical implementation of ODE differs largely from analytical solution
I am trying to solve the ODE of a free fall including air resistance.
I therefore defined my ODE as:
def f(v, g, k, m):
return g - k/m * v**2
which in my ...
- 123
1
vote
0
answers
131
views
Using Sundials CVODE in MATLAB
I'm currently using ode15s to solve a set of stiff differential equations.
I am trying to use the MATLAB profiler to understand the section of the ode solver code which calls BLAS routines.
Since the ...
- 411
4
votes
1
answer
648
views
Is my differential equation solving code wrong?
I am trying to simulate LLG equation without damping. The equation is
$$\frac{d\vec{m}}{dt} = \vec{m}\times\vec{H}$$
I am solving in spherical coordinates as LLG equation is known to have problems in ...
- 141
1
vote
0
answers
37
views
How does the time needed for force propagation increase with the discretization (using symplectic Euler schemes)?
I am trying to model a physical system by (lets say the system is a long deformable object, on which I can apply forces). It can be described by Cosserat Rod theory and discretized, by modelling it ...
- 11
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votes
1
answer
85
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Computing discrete laplacian matrix for mesh fairing
I asked this question on the math stack exchange and got an answer, but I am just as utterly confused as before. My fundamental goal is to actually construct the matrix, that is, a series of steps I ...
- 263
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votes
0
answers
34
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Constructing generalized Laplacian matrix?
I am staring intently at this paper by Botsch and Kobbelt.
In particular, I want to make the matrix specified in equation 5. I am trying to understand the specific computations I must instruct a ...
- 263
1
vote
0
answers
75
views
Solving perturbed Einstein Boltzmann equations using RK4
I'm trying to learn to numerically solve the perturbed Boltzmann-Einstein equations in cosmology using the RK4 method.
These are the equations:
$$\dot{\Theta}_{r,0}+k\Theta_{r,1}=-\dot{\Phi}$$
$$\dot{\...
- 11
3
votes
1
answer
191
views
First derivative central differences with reflecting boundary conditions
I have the following problem in 1-D:
\begin{equation}
\partial_t u(t,x) = v(x)\partial_x u(t,x) + \kappa(x) \partial_{xx} u(t,x),\,x\in\Omega;\quad \partial_{\nu}u(t,x) = 0, \,x\in\partial\Omega.
\end{...
- 1,271
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0
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58
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Algorithm for 1-dimensional minimal surfaces
Consider a set of points. For simplicity, let's say that those are 2D points (although the problem works in higher dimensions as well). The goal is to find the minimum possible length of a connected 1-...
0
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0
answers
101
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What is the difference between approximations of mixed derivative and how to implement it
currently I am solving 2D nonlinear second order differential equation containing mixed derivative. I started searching how to descretisize it and found two formulas for 4th order approximation. First ...
- 31
1
vote
1
answer
258
views
Euler's Method for fast moving particle trajectory
I'm trying to figure out how a magnet affects the trajectory of a particle travelling near the speed of light downwards toward the ground. The equation for the force of the magnetic field is pretty ...
- 11
0
votes
1
answer
115
views
computing Lyapunov exponents numerically
I am trying to compute numerically the Lyapunov exponents of an ODE. I follow the method described in Parker, Chua "Practical Numerical Algorithms for Chaotic systems"
There is also relevant ...
- 101
0
votes
0
answers
37
views
Does a warning in solve_bvp mean that the solution has to be discarded?
I am trying to solve a nonlinear and discontinuous fourth order BVP using the solve_bvp function of SciPy. My equation is $y^{(4)}=cf(y)$, where $f(y)$ is a ...
- 33
2
votes
2
answers
228
views
Solving IVP backward in time via python
I'm having difficulty solving an initial value problem (IVP) in Python backwards in time.
The code is at the end of this post.
First, please let me state my simplified problem.
The forward IVP is ...
- 21
1
vote
1
answer
102
views
Isolating decaying solutions to nonlinear second-order ode
I need to solve a nonlinear ODE of the form
$$
\frac{d^2 y}{dx^2} + \frac{1}{x}\frac{dy}{dx}-\frac{1}{x^2}\sin(y)\cos(y)+\frac{2}{\alpha}\frac{\sin^2(y)}{x}-\sin(y)=0
$$
numerically, subject to the ...
- 11
1
vote
1
answer
174
views
How to solve coupled differential equations numerically?
I've just started a project, trying to do simulation of electrodynamics using the well-known Maxwell equations:
$$
\nabla \cdot \mathbf E = \rho \\ \; \\
\nabla \cdot \mathbf B = 0 \\ \; \\
\nabla \...
- 111
5
votes
2
answers
639
views
How do people know the classic Lorenz attractor is actually deterministically chaotic at infinite time?
Correct me if I am wrong, but I take it that people actually think the classic Lorenz system is fully characterized, i.e. we know what the attractor looks like and the various fractal dimensions of it,...
- 95
1
vote
1
answer
112
views
how to solve a coupled nonlinear partial differential equations(Boundary Value Problem)
So far, I have been looking into linear and nonlinear differential equations(Boundary value problem) that look like the following:
$\frac{d^2 \theta}{dz^2} + \sin(\theta) = 0$
I am now familiar with ...
- 35
5
votes
1
answer
93
views
Prediction of sphere (i.e. roast) core temperature heated in an oven
The real-life problem
Assume I put a spherical roast with initially constant temperature of start_temp=25 (°C) into an oven with ...
- 161
1
vote
1
answer
61
views
Once Lyapunov exponents have converged, can they diverge again?
I know the strength of attraction for a single attractor might vary from place to place. Say, if I calculated the Lyapunov exponents for a small portion of the attractor and they have already ...
- 95
2
votes
2
answers
550
views
Problems solving 2D heat equation using physics-informed neural networks
I am trying to solve 2D heat equation using the physics-informed neural networks approach. The training loss is decreasing, but my final network outputs make no sense. I am using Python/Pytorch.
2D ...
0
votes
1
answer
99
views
finding weak form of nonlinear differential equation for FEM simulation
The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential ...
- 35
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votes
0
answers
39
views
Energy potential involvign pressure and viscoscity?
Making a really long explanation short. I am looking for an energy density function for an incompressible isotropic fluid that involves both viscoscity and pressure.
The reason is, there is a way to ...
- 263
0
votes
0
answers
54
views
Solving a minimization problem without flattening inputs?
I am reading this paper on improving time steps for solving simulation problems:
https://www.math.ucla.edu/~jteran/papers/GSSJT15.pdf
The authors developed this energy function:
$$E(x) = \frac{1}{2\...
- 263
1
vote
3
answers
226
views
Partial derivatives for triangular meshes (in 3D)
A grid offers an obvious definition for the partial derivatives at a grid point, given
$x$ the value of a point $p$ in an $n$ dimensional grid, the forward partial derivative that point for coordinate ...
- 263
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votes
0
answers
40
views
General dimensional solution to fast sweeping quadratic equation
I am reading Hongkai Zhao's paper The Fast Sweeping method. I have implemented the method in 2D and now I want to move onto 3D.
However section 2.6 confuses me.
The article says:
But I have no idea ...
- 263
1
vote
0
answers
38
views
How can i study stability for a new method that solves second degree non lineara differential equations?
I developed a new method to solve this equations $\frac{d{y}^{2}}{dx^{2}}=g(x,y,\frac{dy}{dx})$ for a general g(x,y,z), wich their solution is a function f(x)=y. Obviously you need $y(x_0)=y_0$ and $y'...
- 111
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votes
0
answers
74
views
Encountering blow-up when solving the one-way heat equation using Lax-Wendroff
This is my first time attempting to implement a finite difference method for a PDE in Python, and I am having a bit of trouble. The PDE I am trying to solve is as follows:
$$
\begin{cases}
...
- 153
0
votes
1
answer
49
views
Why is the maximum potential energy greater than the maximum kinetic energy?
I was plotting the energy variation in a mass-spring system. If I define the initial conditions to be at maximum displacement from the origin, the potential energy is plotted correctly but kinetic ...
- 123
0
votes
0
answers
55
views
Numerical method for space fractional derivative in 1 dimension
I am very new to the subject of fractional derivatives which arise while characterizing the anomalous transport of passive scalar in turbulence.
I have found an equation of the following form, to ...
- 87
0
votes
1
answer
76
views
How to get a normalized gradient with FreeFem++?
I am trying to use FreeFem++ to solve the heat geodesics algorithm.
The algorithm is:
solve $\dot u = \Delta u$ at a specific time $t$.
compute $X = \frac{\nabla u_t}{|\nabla u_t|}$
solve $\Delta\phi ...
- 263
-1
votes
1
answer
122
views
Solving Transcendental equation involving special functions becoming nightmare any one can help?
Simply i want to solve for schrodinger equation for finite potential well problems in spherical coordinates. For case in which l=0 . It is simple but when l changes. The solution are spherical ...
- 13
3
votes
1
answer
162
views
Numerical scheme for the level set equation that solves inverse mean curvature flow problems
I am considering the problem of simulating the evolution of an interface given as a curve in 2D (or surface in 3D) that evolves according to a velocity specified at the interface of the form:
$$\vec{v}...
- 33
0
votes
1
answer
73
views
On solving a first order nonlinear differential equation
It all starts with this Cauchy problem:
$$
\begin{cases}
\sin(2x(t)) -\cos(3x'(t)) = x(t) + x'(t) \\
x(0) = 1 \\
\end{cases}
\quad \quad \text{with} \; t \in [0,10]\,.
$$
Not knowing which way to turn,...
- 103
2
votes
0
answers
61
views
Symplectic (or alike) integrator for system with Coulomb singularity and time-dependent potentials
I am trying to calculate classical trajectories for a single positive ion and a single electron inside an RF trap. Therefore, I am dealing with a two-body system that possesses:
Coulomb potential ...
- 21
2
votes
0
answers
100
views
Scipy.root not converging even when provided with initial guesses very close to solution
I've made a previous question here and also in SO wondering why only the fsolve solver converges for the simple one dimensional unsteady conduction problem
$$ \frac{\partial T}{\partial t} = \alpha \...
- 133
0
votes
2
answers
305
views
Why is this scipy.root code not converging?
I'm running a test problem to set up larger problems. Solving the simple unsteady heat equation via finite differences:
$$ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2T}{\partial x^2}$$
$\...
- 133
1
vote
0
answers
84
views
How to include zero flux boundary conditions?
I am trying to solve the following differential equation in the domain of $\theta \in [0, 2 \pi]$ using finite differences scheme:
For $0< \theta \leq \pi$
\begin{align}
\rho_i^{n+1}=\rho_i^{n}+D\...
- 111
1
vote
2
answers
194
views
C^1 continuous element for a triangle?
I am looking for an element for FEM that is piecewise $C^1$ continuous across triangles (i.e. $C^1$ continuous on the edge separating 2 triangles of the mesh).
I have heard about the Bell element:
...
- 263
0
votes
0
answers
81
views
It is possible to solve integro-differential equations using in Fenics?
I am interested in solve the following integro-differential equation:
\begin{align}
\frac{\partial{\rho(\theta, t)}}{\partial{t}} = D \frac{\partial{\rho(\theta, t)}}{\partial{\theta^2}} - \beta \...
- 111
0
votes
1
answer
139
views
FreeFEM++ converting equation into code
I am trying to solve the following problem nuemrically:
$$u_t = \Delta u + \sin t$$
To that effect I scanned the documentation of FreeFEM, the closest example I can find to my problem is the Thermal ...
- 263
1
vote
1
answer
65
views
Solving a boundary value problem with variable number of coupled equations
Let's assume the equation
$$
\nabla^2u_n(\vec{r})+a_n(\vec{r})u_n(\vec{r})=\sum_{m=1}^{N}b_{nm}u_m(\vec{r}),\quad n=1,2,\dots,N,\quad \vec{r}\in\varOmega\tag{1}\label{eq1},
$$
is to be solved for $u_n$...
- 194
0
votes
0
answers
100
views
Pyton get fem solving simple 2D differential equation?
I need to solve the differential equation:
$$u_t = \Delta u + \sin t$$
On a 2D domain with homogeneous Dirichlet boundary conditions. Tod o this I am trying to use the python package getfem.
I am ...
- 263
0
votes
1
answer
141
views
Solve 1st order ODE in using `scipy`
I've been trying to solve the following equation
$$
y(t)=-A\cdot\frac{\mathrm{d} y}{\mathrm{d} t}+B\cdot\left(\frac{\mathrm{d} y}{\mathrm{d} t}\right)^{2}+C
\\
y(t=0)=y_{0}\\
$$
where $A$, $B$, and $C$...
