Questions tagged [differential-equations]
The differential-equations tag has no usage guidance.
67
questions
1
vote
1answer
62 views
How to Break Coupled ODEs down to first order for Runge-Kutta
My question might seem a bit simple. I am trying to solve a system of ODEs using Runge-Kutta method. I am having difficulty breaking down the equations into a system of first order ones required ...
1
vote
1answer
62 views
Numerical methods for non-linear diffusion
I have the following non-linear diffusion equation, for $\ z(x,t)$:
$\ z_t = -C(\sin(\omega t))^m x^{hm}(hm x^{-1}(z_x)^n + n z_{xx} (z_x)^{n-1}) $
Any advice for numerical (or analytical) solutions?...
1
vote
0answers
23 views
Book Recommendation: Analysis and design of mechanistic models - such as pharmacokinetics or hydrology models
I have been looking at an interesting book "Pharmacokinetic-Pharmacodynamic Modeling and Simulation" by Peter Bonate on pharmacokinetic models: the models of how medical drugs work their way through ...
2
votes
0answers
48 views
Numerical integration of SDE: choice of $dt$ and algorithm
I am working on the following Stochastic Differential Equation (SDE) in the Quantum Mechanics context:
$$dX_{t} = a X_{t} dt + b X_{t} dW$$
where $X_{t}$ is my stochastic varible, $dt$ is my ...
0
votes
0answers
63 views
How one can simulate a system given by differential equation?
I want to simulate a diffusion environment given by the differential equation
$$\frac{\partial u(x,y,t)}{\partial t}=D\left(\frac{\partial^2 u(x,y,t)}{\partial x^2}+\frac{\partial^2 u(x,y,t)}{\...
1
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0answers
50 views
Governing equations vs Transport equations
This is a basic question. But I did not find any explanations for this. How are governing equations, like mass, momentum, energy conservations equations, different from 'Transport equation'?. Is a ...
0
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0answers
57 views
What are the differences between these different forms of equation?
What are the differences between Conservative differential form, Non-conservative differential form, Conservative Integral form and Non-conservative integral form of differential equations? I know ...
1
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0answers
71 views
Solved : Damped spring-mass system, wrong position, correct speed and acceleration
I am modulating a spring-mass system with gravitation and aero drag, with python programming. The spring is hanging vertically and attached a weight. The user then selects a length to drag it down ...
4
votes
1answer
130 views
Fast and free server for computing
I have to calculate a huge differential equation. With my laptop, it's going to be computed for several days.
Is there a free (I need just for 3 days) fast server for scientific calculations?
My ...
2
votes
1answer
45 views
Symplectic linear multistep method?
I'm doing a gravitational n-body simulator and I'm thinking of implementing linear multistep methods like Adam-Bashforth. But is there any symplectic multistep methods?
1
vote
1answer
54 views
Online Parameter Estimation using steepest descent
I have a first order system which is described by the following differential equation:
dx/dt = -a*x + b*u where u is the input <...
4
votes
2answers
106 views
Finite difference for a highly nonlinear equation - The wind within the forest
Based on the Navier-Stokes equations and a few parameterizations, the horizontal steady-state wind $u(z)$ within a forest of height H satisfies:
$$
a\left(\frac{du}{dz}\right)^2 + b\frac{du}{dz} \...
3
votes
2answers
98 views
Damped Harmonic Oscillation. Efficient algorithm to find the parameters resulting in threshold oscillation amplitude
Let's assume, that we have damped harmonic oscillation of a body in the form of a cone, immersed in a liquid.
Equilibrium condition of the body is:
$$m\overrightarrow{a} = \overrightarrow{F_\text{...
1
vote
0answers
30 views
Why does the correlation function of this stochastic differential equation starts at different points?
I am working with the following differential equation:
The equation is $$x=\beta +\sqrt{2D} \xi(t)$$ where $\xi(t)$ is a white noise term, with a reflecting wall boundary conditions.
After solving ...
2
votes
0answers
61 views
How to implement adaptive step size Runge-Kutta Cash-Karp?
Trying to implement an adaptive step size Runge-Kutta Cash-Karp but failing with this error:
...
1
vote
1answer
142 views
Solving differential equation in Python with variable coefficients (I just know the coefficients numerically)
I am trying to implement a routine to solve a differential equation in Python. Basically the kind of equation that I am interested in solving is of the form:
$\displaystyle \frac{d}{dx^2} \left(x y(x)...
0
votes
1answer
69 views
Stability of PDEs
I am currently trying to solve some PDEs with FiPy. At page 56, the manual mentions (https://www.ctcms.nist.gov/fipy/download/fipy-3.0.pdf).
The largest stable timestep that can be taken for this ...
0
votes
1answer
81 views
Actual global error vs theoretical global error: How to combine theory with practice
I have implemented an Adams Bashforth 4 method to solve an Initial Value Problem for an ODE and I am testing it against the test equation:
$y'=\lambda y$ with $y(0)=1$ with the exact solution: $y(t)=...
0
votes
1answer
96 views
Unexpected solutions solving an ODE using odeint
I am trying to solve a system of 8 coupled differential equations using scipy's odeint. I have already written my code and it runs fine, but the solutions I get are completely different from what I ...
0
votes
0answers
47 views
Determining the pseudo-time period of a system of $n$-pendulums via Kane's method in Python
We can use Kane's method to integrate the equations of motion for a system of $n$ pendulums with arbitrary masses and lengths (see derivation). In particular, if $(x_i,y_i)$ denotes the Cartesian ...
1
vote
1answer
41 views
Wrong results for $2$ stage multistep method $y_{n+2} - y_n = h\left[(1/3)f_{n+2} + (4/3)f_{n+1} + (1/3)f_n\right]$
I need to fix a code to utilise the $2$ stage multistep method :
$$y_{n+2} - y_n = h\left[(1/3)f_{n+2} + (4/3)f_{n+1} + (1/3)f_n\right]$$
Since this is an implicit method, I used a Newton-Raphson ...
1
vote
0answers
39 views
Numerically solving a system of parabolic PDEs and 1st order ODEs
I'm trying to solve the following system of differential equations numerically. What are the available finite difference approaches and matlab solvers to solve such a system? Other approaches to solve ...
5
votes
1answer
126 views
Numerical solution of two coupled nonlinear eigenvalue problems
I would like to numerically solve the following system of coupled nonlinear differential equations:
$$
-\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a +
\left( g_a |...
2
votes
1answer
89 views
Numerically solving a partial differential equation
I am trying to numerically solve the following PDE,
$$\frac{\partial u^A}{\partial t} = c_1\frac{\partial^2 u^A}{\partial^2x} \,,$$
where $c_1$ is a constant. The above can be discretized using the ...
0
votes
0answers
78 views
why I cannot find explicit finite difference for elliptic equation
Let us think on the Poisson equation $\nabla^2 u(\bf{x})=\rho(x)$ with Neumann boundary conditions, with $\bf{x}=\it (x,y)$ in 2D.
Here is a stencil with central differences in both $x$ and $y$ (...
1
vote
0answers
90 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 ...
2
votes
1answer
72 views
PML boundary conditions
I set up two one-way wave equations for constant velocity $c$ in one-dimension. When I implement them I get a highly unstable (divergent) solution. I wonder if someone could give me a suggestion about ...
4
votes
3answers
162 views
Finite difference for 1D wave equation: why the spike initial data results in a noisy output?
I am using a second-order finite difference in space and time approximation for the 1D wave equation.
No source but initial data: $I(x)=\mathrm{e}^{-400 (x-0.5)^2}$.
Velocity $c=1$, $nx=501$, $nt=...
1
vote
1answer
193 views
How write a integration loop in fortran, leapfrog scheme to solvind PDE (advection)?
I want to resolve numerically this equation using of difference finite method with Leapfrog Scheme
$$\frac{\partial{u}}{\partial t}+ v \frac{\partial{u}}{\partial x}= 0 $$
I'm trying to write a code ...
1
vote
1answer
145 views
Can a second-order ODE be “inconsistent” with its boundary conditions?
I am trying to solve a set of coupled, nonlinear ODEs. The only dependent variable is a 1-dimensional spatial coordinate, let's call it $x$. For now, I've managed to approximate away some of the ...
0
votes
1answer
50 views
Calculate forces on atoms from potential energy of system and position of atoms
Background
I am using a neural network to calculate the potential energy of atoms in a configuration and then adding energy of all atoms to compare it with the true energy of the configuration(label) ...
4
votes
1answer
71 views
Symplectic Algorithms for Hamilton’s Equations as opposed to just Volume-Preserving
this might be a silly question, but if we’re trying to numerically solve Hamilton’s equations with some discrete scheme, sometimes when the scheme preserves phase space volume (Hamilton’s eqns are ...
2
votes
2answers
57 views
Discrete-time input matrix when one of the eigenvalues of the system matrix is zero
If we have a continuous time state-equation,
$$ \dot{x}(t) = A x(t) + B u(t)$$
where $A \in \mathbb{R}^{n \times n}, x \in \mathbb{R}^{n\times1}, B \in \mathbb{R}^{n \times m}, u \in \mathbb{R}^{m \...
1
vote
0answers
57 views
fourth order Poisson iterative solver --in Matlab
I want to calculate the stream function $\psi$ starting from a velocity field $(u,v)$ (such that $u=-\frac{\partial\psi}{\partial y}$ and $v=\frac{\partial\psi}{\partial x}$). I thus calculate the ...
3
votes
1answer
37 views
Apart from initial discontinuities, what is tricky about neutral DDEs?
Background
A neutral delay differential equation is one where the derivative does not only depend on its past state, but also the derivative at a past point:
$$ \dot{y}(t) = f\big(t, y(t), y(t-τ_1), ...
24
votes
2answers
2k views
What does “symplectic” mean in reference to numerical integrators, and does SciPy's odeint use them?
In this comment I wrote:
...default SciPy integrator, which I'm assuming only uses symplectic methods.
in which I am refering to SciPy's odeint, which uses ...
1
vote
0answers
16 views
Procedure to identify characteristic properties of unknown functions in a DAE model
I have a system of 1st order odes given by
$$
\dot{x_1}(t) = \alpha_1 f_1(x_1,t) + \beta_1 u(t) \\
\dot{x_2}(t) = \alpha_2 f_2(x_2,t) + \beta_2 u(t)
$$
They are constrained by an algebraic equation
...
2
votes
1answer
79 views
Jump-Diffusion process: practical solver beyond Euler method?
A jump-diffusion process is a stochastic process where both continuous noise (in my case complex Wiener noise $dZ,dZ^*$ such that $dZ^2=dZ^{*2}=0,|dZ|^2=dt$) and discrete Jumps (in my case Poissonian $...
1
vote
1answer
53 views
Integrating a nonlinear ordinary differential equation
I am solving an equation of the form
$(*)$ $0 = a(f) (\partial_rf)^2 + b(f) (\partial_rf) + c(f),$
where $f$ is a real function of $r\in \mathbb{R}$, and $a,b,c$ are real functions of $f$. The ...
0
votes
1answer
481 views
Crank–Nicolson method for nonlinear differential equation
I want to solve the following differential equation from a paper with the boundary condition:
The paper used the Crank–Nicolson method for solving it. I think I understand the method after googling ...
0
votes
1answer
93 views
Runge-Kutta timestep in atomic units
I'm using 4th order RK to solve the schroedinger equation in atomic units. Say I want to simulate 400fs in intervals of h=10fs, then in atomic units this is h=413a.u and 400fs=16500a.u. 4RK involves ...
5
votes
2answers
147 views
Algorithm for finding initial conditions of differential equations given trajectory
Let's say I'm given a system of three first-order differential equations in three variables, where all of the equations are known, and we additionally know the trajectory of two of the variables at a ...
1
vote
0answers
46 views
Second-order PDE with seven variables
I need to solve the following partial differential equation in seven variables with four boundary conditions. I don't think Mathematica has the capacity to solve this differential equation.
Do you ...
1
vote
0answers
48 views
Eigenvalue ODE in Spherical Coordinates--Numerical
I wish to solve an eigenvalue problem:
$$\nabla^{2}f=Ef $$
If I assume spherical symmetry $f(r,\theta,\phi)=f(r)$, I can reduce the problem to 1D:
$$(\frac{2}{r}\frac{d}{dr}+\frac{d^{2}}{dr^{2}})f=...
1
vote
1answer
72 views
finite difference for a second order ode
I saw in a code for discretization of something like $\frac{d^2T(x)}{d^2x}$ , ( $x = sin(\theta)$ ) tries
...
1
vote
0answers
121 views
Solving complicated coupled ODE using RK4/ODE45 in Matlab
I have the following coupled differential equations also known as Guiding Center Approximation. It is used to explain the position- and velocity change of particles (electrons and protons, N = 1000) ...
3
votes
1answer
71 views
SIRS Model doesn't depend on initial conditions?
So i have been working (as an undergrad, by working i mean "Redoing a few things my professor does") in a SIRS model for epidemies. SIRS stands here for:
Susceptible -> Infected -> Recovered -> ...
1
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0answers
90 views
Code for solving the heat equation on the semi-infinite rod
Cross posted in mathematica.SE.
Question : I want to test the solution which is given below is right by Matlab/Maple/Mathematica.
Please look the post in mathstackexhange
or
Please look below.
...
2
votes
1answer
120 views
Some questions on Trace (operators) on the boundary in the context of PDEs
Background: The solution space of original problem (which requires a fine enough mesh to resolve the microstructure) can be split into a macroscale solution space and microscale solution space. This ...
3
votes
3answers
110 views
Numerically finding constants of motion
Given a set of ODE's $ \dot{z} = f(z) $ (or discrete time $ z_{t+1} = f(z_t) $), is there a way to numerically find constants of motion?
For $ f(z_t) \approx M z_t $, diagonalizing the matrix $ M $ ...
