Questions tagged [differential-equations]
The differential-equations tag has no usage guidance.
126
questions
0
votes
1answer
33 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 [...
2
votes
0answers
83 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 ...
0
votes
1answer
51 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:
$$
\epsilon u''+u(u'-1) =0 , \\
u(0)...
1
vote
0answers
53 views
Stochastic differential equation system (SDE) : overflow encountered in double-scalars
I'm trying to integrate the following SDE system from Dekker et al.
To do that I'm using the Euler-Maruyama method which is basically a forward-Euler scheme plus a Gaussian noise term where the mean ...
3
votes
2answers
268 views
CUDA & Python for numerical integration and solving differential equations
Can anyone please suggest some libraries which allow use CUDA in Python for numerical integration and/or solving of differential equations?
My goal is to solve large (~1000 equations) of coupled non-...
3
votes
0answers
43 views
Difference between wave vector and density matrix in numerical calculation of Schrödinger equation
I solved Schrödinger equation for a following tow-level time-dependent Hamiltonian numerically in two ways:
...
0
votes
0answers
75 views
Numerov method for solving Schrödinger equation
I have just begun learning computer science to apply it to Physics and I am trying to write a code for solving Schrödinger's equation of the harmonic oscillator (setting $V=\frac{x^2}{2}$) in one ...
-2
votes
1answer
38 views
Solving differential equation by setting vectorization `on` in MATLAB
This is a follow up to my previous question posted here.
I've set up an ode system in MATLAB and I'm trying to vectorize the code to increase the speed of computation.
The follow is the code for my ...
1
vote
1answer
46 views
solving differential equations with jacobian pattern
I'm trying to compare the simulation time for solving a system of differential equations with and without jacobian pattern for a toy model using ode15s in MATLAB.
...
2
votes
0answers
66 views
Linearising Nonlinear Coupled Partial Differential Equations - Alfvénic Diffusion
I am trying to solve the following coupled partial differential equations with a finite difference scheme:
$$\partial_tf+v\partial_zf+\partial_z\frac{1}{W}\partial_zf=0$$
$$\partial_tW+v\partial_zW-\...
0
votes
2answers
113 views
Solve a system of coupled differential equations in Python
I have a system of two coupled differential equations, one is a third-order and the second is second-order. I am looking for a way to solve it in Python.
I would be extremely grateful for any advice ...
7
votes
1answer
147 views
Which finite difference better approximates $uu'$?
I want to approximate $uu'$ with a finite difference. On the one hand, it seems to be
$$(uu')_i=u_i\frac{u_{i+1}-u_{i-1}}{2\Delta t}=\frac{u_iu_{i+1}-u_iu_{i-1}}{2\Delta t}$$
On the other hand,
$$(uu')...
-1
votes
1answer
133 views
ODEintWarning: Excess work done on this call (perhaps wrong Dfun type)
I was messing around with some numerical integration functions. I wrote an arbitrary differential equation to test my understanding, the code is as follows:
...
1
vote
0answers
53 views
Book recommendation on numerical methods for solving Integro-Differential equations
I was wondering if anyone could recommend a good book or resource on numerical methods for solving integro-differential equations? Of course I am familiar with the methods for solving ODEs and PDEs ...
1
vote
1answer
138 views
Trouble with backwards time integration in Python
I am struggling with a rather basic numerical integration task: Using Python's scipy.integrate.solve_ivp module to integrate an ODE sytem backwards in time. As a test, I am using the following ODE ...
2
votes
0answers
83 views
How to derive the adjoint sensitivity equations for a least squares objective function gradient
The Problem
I would like to determine the gradient of a least squares objective function which depends on a vector of 40 parameters $p$, and the solution of a system of 32 differential equations. In ...
1
vote
1answer
62 views
Solution of Coupled Differential equation for a 2d linear flow using RK4 method in python 3
I want to study the dynamics of a 2d linear flow, whose dynamical equation is- $\begin{pmatrix} \dot{x_1}\\ \dot{x_2}\\ \end{pmatrix}=\begin{pmatrix} 1 & 1\\ 4 & -2\\ \end{pmatrix}\begin{...
1
vote
1answer
64 views
2-DOF Robotic Manipulator Trajectory Tracking Simulation
I am trying to simulate a 2-DOF planar robotic manipulator (have its joints follow a predefined trajectory) that's described by its dynamic model:
$$ M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q) = \tau $$
...
1
vote
2answers
325 views
How to set up the differential equation system to speed up computation?
I've set up a system of differential equations, obtained after discretizing pde, in the following way
...
1
vote
1answer
297 views
scipy odeint: excess work done on this call and very sensitive to initial value
I am trying out odeint and received the error
'Excess work done on this call (perhaps wrong Dfun type).'.
The values returned are also super sensitive to small ...
0
votes
1answer
68 views
Odeint error for nonlineal differential equations
I receive the following error when I run the code.
ODEintWarning: Excess work done on this call (perhaps wrong Dfun
type). Run with full_output = 1 to get quantitative information. warnings.warn(...
7
votes
1answer
249 views
Numerical computation of Lyapunov exponent
I'm trying to compute the Lyapunov exponent for a smooth continuous time dynamical system(say, $\dot{\bar{x}} = f(\bar x)$). I using the QR decomposition method. Here are the steps that I follow.
...
3
votes
1answer
140 views
How to get a more accurate cancelation
I shall try to get to the point, so let me know if there is something left and you need more details.
I am solving a couple of equations that are not coupled explicitly, but their corresponding ...
2
votes
1answer
121 views
A problem with Poisson equation
I'm computing the Hartree potentials of atoms by solving the Poisson equation and I use hydrogen atom as a test case. The Poisson equation for hydrogen atom in atomic units is given by
$$\nabla^2 V_H =...
0
votes
0answers
28 views
Set of integrators do not consistently solve an equation in Python
I must solve the following second order differential equation:
$\delta \phi^{''}_{\mathbf{k}}+(3-\epsilon)\delta \phi^{'}_{\mathbf{k}}+\left(\frac{k^2}{a^2 H^2}+\frac{V_{,\phi\phi}}{H^2}-6\epsilon +4\...
3
votes
1answer
411 views
Solving the heat diffusion equation with source term
I am trying to solve the 1-D heat equation numerically with a variable source term. The system is basically a tank containing styrene in which it polymerizes to liberate heat. I have assumed that the ...
2
votes
0answers
39 views
How to increase the stability of a DAE solver?
I am trying to solve a set of linear PDEs of the form
$$F\left(\vec{y},\frac{\partial \vec{y}}{\partial x},\frac{\partial^2 \vec{y}}{\partial x^2},\frac{\partial \vec{y}}{\partial t}\right)=0.$$
To ...
1
vote
0answers
37 views
Discretization formula for a system of two differential equations. “Solution to one of these is the initial condition of the other”. In which sense?
Consider the following stochastic differential equation
\begin{equation}
dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1}
\end{equation}
where $A$, $B$ and $C$ are parameters ...
2
votes
2answers
76 views
Methods for solving discrete PDEs using algorithmic differentiation results
I'm looking for a method to solve a 20000 variable, 20000 residual non-linear PDE with a Galerkin method.
I have Fortran subroutines for:
The residuals: $\vec{r}(\vec{x})$;
Their Jacobian multiplied ...
0
votes
0answers
56 views
Solver for large dense BVP system in python
I have a large system of boundary value problems of the form
$$ \frac{d^2 y }{dt^2} = C(t) y + b(t), $$
where the variable $y$ is a vector that has anywhere from 50 to around 500 components, $C$ is a ...
3
votes
2answers
245 views
Finite difference method having a discontinuity
I am trying to understand the FDM which is a widely used method solving differential equations by using approximation below.
$$\dfrac{\partial u}{\partial x}=\dfrac{u(i+1)-u(i-1)}{2\Delta x}$$
How can ...
0
votes
1answer
109 views
Two RK4 method in one program
I want to solve this integral using RK4 by coding in Fortran:
$$R=∫1/a(t) dt → dR/dt=1/a(t) =f(t)$$
Initial point: t=0 (or a=0.001) and R=0
And I have to get a(t) by solving another ...
0
votes
0answers
34 views
Simulating the response of nonlinear system with stiff differential equations
I want to simulate the response of a nonlinear system given in the following form:
$$ \dot{x_1} = f_1(\bar{x_1})+g_1(\bar{x_1})x_2, \ x_1(0) = 0.2 $$
$$ \dot{x_2} = f_2(\bar{x_2})+g_2(\bar{x_2})x_3, \...
1
vote
1answer
101 views
Partial differential equation FEM application
I have a PDE which looks like Helmholtz wave equation on one dimensional domain.
$$\dfrac{d^2u(x)}{dx^2}+\pi^2u(x)=f(x)$$
where $-\infty <x<\infty $
Also, $f(x)= 1$ for $-0.25<x<0.25$, I ...
0
votes
1answer
47 views
MATLAB ode45 doesn't start at initial conditions
I wrote a code in MATLAB to solve a system of differential equations, but my solution doesn't seem to take into consideration the initial conditions I specified. I am not sure how to interpret this ...
1
vote
3answers
143 views
Runge-Kutta method for an ODE with initial value which is root of denominator
I wrote a code in Fortran to solve this differential equation using RK4 method:
$$
\frac{dy}{dx}=A\sqrt{\frac{B}{y}+\frac{C}{y^2}}
$$
$A$, $B$, and $C$ are some known constants. The problem is that ...
0
votes
0answers
73 views
Applying boundary Conditions on FEM
I have a partial differential equatons as shown below.
$$\dfrac{d}{dx}((1+x)\dfrac{du(x)}{dx})=0$$
With the following boundary conditions.
$$u(0)=0, u(3)=10$$
To solve it using FEM, I multiplied the ...
0
votes
0answers
12 views
Differential parameterized inequalities
Let $H$ be an Hamiltonian and denote $\vec{H}$ the associated Hamiltonian vector field.
I am interested in solving numerically the following problem
$$
\dot z(t) = \vec{H}(z(s),t_1,\ldots, t_p) \...
2
votes
0answers
71 views
Numerical method for harmonic oscillator with jumping constant
Let $k_1 \neq k_2$ be positive reals, $t_0 > 0$ and consider the following Cauchy problem in $[0,+\infty)$:
\begin{cases}
y(t) + k(t)y''(t) = 0 \newline
y(0) = 1/\sqrt{k_1} \newline
y'(0) = 0,
\end{...
2
votes
1answer
149 views
Solving numerically a linear ODE
I start by saying that I do not have a strong background in numerical analysis, so I may miss some basic things or make trivial mistakes.
Motivated by some problems in digital signal processing, I ...
2
votes
1answer
280 views
Lambdifying a symbolic matrix in Julia
If I have a symbolic matrix defined as T below, is there any way to lambdify this as function of variables, say σ..., and return ...
2
votes
0answers
55 views
Modelling of Stefan Maxwell equation
I am trying to solve Maxwell Stefan's equation over a membrane to get the transient mole fraction distribution over the membrane thickness 'z'. But somehow I am not able to code it using ODE45, more ...
0
votes
0answers
58 views
How to solve odd-order differential equations in FEM? Petrov-Galerkin?
I've recently learned about using weighted residuals with the Galerkin method to numerically approximate even-order differential equations (for linear elements, I'm still a beginner). It seems for odd-...
1
vote
0answers
29 views
Optimality conditions for optimal control: BVP - DAE
I am solving an optimal control problem of the form
$$
\min_u \qquad\int_0^T \langle u(t), u(t) \rangle \, \mathrm{d}t \\
s.t. \quad \dot{x} = \tilde{f}(x) + u, \quad x(0)=x_0 \\
\qquad \tilde{\Phi}(...
4
votes
2answers
75 views
MInimizing cost function using iterative search for a minimum method
I want to estimated the parameters $\ \hat{\theta} $ of a model using an iterative search for the minimum of a cost function. The cost function is defined as follows:
$$ V_N(\hat{\theta}) = \frac{1}{...
1
vote
2answers
119 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,
$$ \frac{d \phi}{dt} = -M^TDM\phi + \text{noise ...
2
votes
1answer
101 views
Using Implicit Euler with second order differential equations
We can numerically integrate first order differential equations using Euler method like this:
$$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
vote
1answer
49 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:
$$ \ddot{y}+a_1\...
2
votes
0answers
55 views
Numerical errors due to terms of the form $\frac{1}{r}$ (r goes to 0 at the boundary) while using finite difference method
I am trying to solve a system of differential equations using finite difference method.
There are few terms of the form $\frac{A(r)}{r}$, both $A(r)$ and r go to zero at the boundary. Analytically ...
1
vote
0answers
65 views
Are linear, CTCS codes always stable?
I would like to solve some equations which basically look like this
$$\frac{\partial u}{\partial x}=F\left(v,\frac{\partial v}{\partial y},\frac{\partial^2 v}{\partial y^2}\right),$$
$$\frac{\partial ...
