Questions tagged [differential-equations]
The differential-equations tag has no usage guidance.
86
questions
1
vote
0answers
17 views
Optimality conditions for optimal control: BVP - DAE
I am solving an optimal control problem of the form
$$
\min_u \qquad\int_0^T \langle u(t), u(t) \rangle \, \mathrm{d}t \\
s.t. \quad \dot{x} = \tilde{f}(x) + u, \quad x(0)=x_0 \\
\qquad \tilde{\Phi}(...
4
votes
2answers
51 views
MInimizing cost function using iterative search for a minimum method
I want to estimated the parameters $\ \hat{\theta} $ of a model using an iterative search for the minimum of a cost function. The cost function is defined as follows:
$$ V_N(\hat{\theta}) = \frac{1}{...
1
vote
2answers
95 views
Solving a parameter estimation problem using trajectory optimization
This is a follow-up to my previous question here
I've the following system of equations for studying information flow in the below graph,
$$ \frac{d \phi}{dt} = -M^TDM\phi + \text{noise ...
2
votes
1answer
70 views
Using Implicit Euler with second order differential equations
We can numerically integrate first order differential equations using Euler method like this:
$$y_{n+1} = y_n + hf(t_n, y_n)$$
And with Implicit Euler like this:
$$y_{n+1} = y_n + hf(t_{n+1},y _{n+...
-1
votes
0answers
35 views
Solving a 2-time Volterra integro-differential equation
In non-equilibrium field theory one usually encounters the so called Kadanoff-Baym equations, which are Volterra integro-differential of 2 times
$$
i \partial_t f(t,t^\prime) = h(t) f(t, t^\prime) + \...
1
vote
1answer
37 views
How to properly compute weights for Weighted Least Squares (WLS)?
I want to apply the weighted least squares method in order to identify parameters of a dynamic process. The process is described by a second order differential equation of the form:
$$ \ddot{y}+a_1\...
2
votes
0answers
50 views
Numerical errors due to terms of the form $\frac{1}{r}$ (r goes to 0 at the boundary) while using finite difference method
I am trying to solve a system of differential equations using finite difference method.
There are few terms of the form $\frac{A(r)}{r}$, both $A(r)$ and r go to zero at the boundary. Analytically ...
1
vote
0answers
59 views
Are linear, CTCS codes always stable?
I would like to solve some equations which basically look like this
$$\frac{\partial u}{\partial x}=F\left(v,\frac{\partial v}{\partial y},\frac{\partial^2 v}{\partial y^2}\right),$$
$$\frac{\partial ...
0
votes
0answers
19 views
Online Parameter Estimation
I have a system $\ y = θ^*u+d(t) $ where $\ y $ is the output , $\ u $ is the input which satisfies $\ u\epsilon L_{\infty} $, $\ θ^* $ is an unknown but constant parameter and $\ d(t) $ is an unknown ...
4
votes
1answer
147 views
Solving coupled differential equations in Python, 2nd order
I have a system of coupled differential equations, one of which is second-order. I am looking for a way to solve them in Python. I would be extremely grateful for any advice on how can I do that!
$k$...
1
vote
1answer
49 views
Dealing with boundary conditions using Fourier spectral methods
I am currently working on a project where I need to use Fourier spectral methods to solve the KS equation. I found this code which is using the Fourier spectral methods to solve the classic 1D heat ...
0
votes
0answers
135 views
Numerically solving a partial differential equation in python with Runge Kutta 4
I'm supposed to solve the following partial differential equation in python using Runge-Kutta 4 method in time.
$$
\frac{\partial}{\partial t}v(y,t)=Lv(t,y)
$$
where $L$ is the following linear ...
2
votes
1answer
120 views
Numerical Solution to Rayleigh Plesset Equation in Python
I have been trying to numerically solve the Rayleigh-Plesset equation for a sonoluminescence bubble in Python. You can read about this phenomenon here: https://iopscience.iop.org/article/10.1088/0143-...
1
vote
0answers
54 views
Stably solve transport equation with source term
I am trying to solve a transport equation of the form for the variable $\psi(t,r)$
\begin{equation}
\partial_t\psi-\alpha(r)\partial_r\psi-\beta(r)^2\psi-f(t,r)=0
,
\end{equation}
where I am solving ...
1
vote
0answers
20 views
Using nondimensionalization to solve an ode in MATLAB [duplicate]
I am trying to solve an ode that uses some extremely large numbers and some extremely small numbers, namely
$$
e = 1.6\times 10^{-19}\\
E = 10^6\\
\tau = 6\times 10^{-24}\\
m = 9.1\times 10^{-31}\\
c ...
2
votes
1answer
75 views
MATLAB's ode45 not dealing with initial conditions well [RESOLVED]
*Concern highlighted in yellow
*Solution at bottom
I have a differential equation to solve for the motion of an electron:
$$
\frac{d^2v}{dt^2} = \frac{1}{\gamma^6}\left( \frac{eE}{\tau m} - \left( \...
1
vote
1answer
79 views
Crank-Nicholson for diffusion-advection vs diffusion equation
Let's consider the following 1D diffusion equation:
$\frac{\partial u}{\partial t} = xk \frac{\partial}{\partial x}(\frac{1}{x}\frac{\partial u}{\partial x})$
where we assume that the diffusion ...
1
vote
1answer
112 views
Numerical integration in 2D
I would like to solve the following problem
$$
\vec{v}(x,y)= k\, \nabla \theta(x,y)
$$
with respect to the unknown function $\theta$. Parameter $k$ is just a real constant quantity.
I have two ...
0
votes
0answers
21 views
Offline Parameter Estimation for second order system - Ordinary Least Squares
I have a second order system which is described by the following differential equation:
$\ \ddot{y}+α_{1}*\dot{y}=b_{0}*u $
where $\ y $ is the output of the system and $\ u $ is the input of the ...
3
votes
1answer
128 views
Solving an SDE with time-dependent parameter in R
I am trying to solve a system of SDEs in R using the Diffeqr package.
Let's reduce the system to a simple ODE:
...
1
vote
1answer
75 views
How to Break Coupled ODEs down to first order for Runge-Kutta
My question might seem a bit simple. I am trying to solve a system of ODEs using Runge-Kutta method. I am having difficulty breaking down the equations into a system of first order ones required ...
1
vote
1answer
73 views
Numerical methods for non-linear diffusion
I have the following non-linear diffusion equation, for $\ z(x,t)$:
$\ z_t = -C(\sin(\omega t))^m x^{hm}(hm x^{-1}(z_x)^n + n z_{xx} (z_x)^{n-1}) $
Any advice for numerical (or analytical) solutions?...
1
vote
0answers
28 views
Book Recommendation: Analysis and design of mechanistic models - such as pharmacokinetics or hydrology models
I have been looking at an interesting book "Pharmacokinetic-Pharmacodynamic Modeling and Simulation" by Peter Bonate on pharmacokinetic models: the models of how medical drugs work their way through ...
2
votes
0answers
60 views
Numerical integration of SDE: choice of $dt$ and algorithm
I am working on the following Stochastic Differential Equation (SDE) in the Quantum Mechanics context:
$$dX_{t} = a X_{t} dt + b X_{t} dW$$
where $X_{t}$ is my stochastic varible, $dt$ is my ...
0
votes
0answers
65 views
How one can simulate a system given by differential equation?
I want to simulate a diffusion environment given by the differential equation
$$\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
vote
0answers
52 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
votes
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
vote
0answers
80 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
137 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
62 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
67 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 <...
3
votes
2answers
112 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:
$$
a\left(\frac{du}{dz}\right)^2 + b\frac{du}{dz} \...
3
votes
2answers
112 views
Damped Harmonic Oscillation. Efficient algorithm to find the parameters resulting in threshold oscillation amplitude
Let's assume, that we have damped harmonic oscillation of a body in the form of a cone, immersed in a liquid.
Equilibrium condition of the body is:
$$m\overrightarrow{a} = \overrightarrow{F_\text{...
1
vote
0answers
38 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
79 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
296 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
75 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
84 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
122 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 ...
1
vote
1answer
44 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
135 views
Numerical solution of two coupled nonlinear eigenvalue problems
I would like to numerically solve the following system of coupled nonlinear differential equations:
$$
-\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a +
\left( g_a |...
2
votes
1answer
96 views
Numerically solving a partial differential equation
I am trying to numerically solve the following PDE,
$$\frac{\partial u^A}{\partial t} = c_1\frac{\partial^2 u^A}{\partial^2x} \,,$$
where $c_1$ is a constant. The above can be discretized using the ...
0
votes
0answers
81 views
why I cannot find explicit finite difference for elliptic equation
Let us think on the Poisson equation $\nabla^2 u(\bf{x})=\rho(x)$ with Neumann boundary conditions, with $\bf{x}=\it (x,y)$ in 2D.
Here is a stencil with central differences in both $x$ and $y$ (...
1
vote
0answers
93 views
How to solve an implicit ODE with forward Euler?
Consider the implicit ODE
$$
M(y)\dot{y} = F(t,y)
$$
If $M$ is non-singular for all $y$
How to use the forward-Euler method to numerically solve for $y$ without inverting $M(y)$?
I only came out ...
2
votes
1answer
87 views
PML boundary conditions
I set up two one-way wave equations for constant velocity $c$ in one-dimension. When I implement them I get a highly unstable (divergent) solution. I wonder if someone could give me a suggestion about ...
4
votes
3answers
205 views
Finite difference for 1D wave equation: why the spike initial data results in a noisy output?
I am using a second-order finite difference in space and time approximation for the 1D wave equation.
No source but initial data: $I(x)=\mathrm{e}^{-400 (x-0.5)^2}$.
Velocity $c=1$, $nx=501$, $nt=...
1
vote
1answer
273 views
How write a integration loop in fortran, leapfrog scheme to solvind PDE (advection)?
I want to resolve numerically this equation using of difference finite method with Leapfrog Scheme
$$\frac{\partial{u}}{\partial t}+ v \frac{\partial{u}}{\partial x}= 0 $$
I'm trying to write a code ...
1
vote
1answer
150 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
51 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) ...
