Questions tagged [differential-equations]

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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
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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 ...
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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)...
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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
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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
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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
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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
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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
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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
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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-\...
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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
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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
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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
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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
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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
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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
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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
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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: $$ M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q) = \tau $$ ...
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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
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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
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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
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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
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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
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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
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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
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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
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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 ...
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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
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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
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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
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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
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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 ...
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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
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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
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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
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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 ...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ...