Questions tagged [nonlinear-equations]
Solution of nonlinear systems of equations. The equations might be algebraic or differential equations.
293
questions
5
votes
0answers
51 views
Continuous limit and nonlinear functional analysis
I have a kind of general question about approximation schemes in nonlinear functional analysis. Given a nonlinear map $\Phi$ from an open set (in an infinite dimensional Banach space) of functions to ...
0
votes
0answers
46 views
Solving two coupled non-linear second order differential equations numerically using python [closed]
Please help with the following Python code for a nonlinear coupled oscillator, I am getting an error, I don't understand the problem.
...
-1
votes
0answers
13 views
Finding Fixed Points of Hodgkin-Huxley Equations
I'm trying to find the fixed points of the Hodgkin-Huxley model, given by the system of equations:
$$
dV/dt=f_V(V,m,h,n)=\frac{1}C_M[I_{ext}-\bar{g}_{Na}m^3h(V-V_{Na})-\bar{g}_Kn^4(V-V_K)-\bar{g}_l(V-...
-1
votes
0answers
29 views
Python/Matlab - fsolve in Fisher context : Sensitivity to guess
I have 2 Fisher matrices F1 and F2. I can diagonalise these 2 matrices which give D1 and <...
0
votes
0answers
47 views
Solve non-linear equation in R
I need to solve the following equation for $x$ in [0, 1]. Assume $0<\alpha<1$ and $0<\lambda$.
$$(1 - x)^{\alpha+1} - \lambda (x+1)^{\alpha+1} = -2\lambda (\alpha + 1) x^\alpha$$
Would very ...
2
votes
1answer
70 views
How to derive the simplified Newton iteration in the TR-BDF2 ODE integration scheme
The Problem
The TR-BDF2 explained in this paper [1], is quite a popular numerical scheme used to integrate $\dot{y} = f(t,y)$, consistent of the following two stages:
\begin{align}
y_{n+\gamma} &...
2
votes
2answers
136 views
Numerical Methods of solving a non-linear ODE?
I want to solve the nonlinear equation $\frac{d^2x}{dt^2} + k\sin x = 0$, numerically. I found that solving this elliptic integral would be cumbersome, so is there a numerical method i could use to ...
1
vote
0answers
71 views
Optimize speed for calculating the approximate inverse of a large matrix
I am searching for a faster method to calculate an approximate inverse of a large matrix (up to 32000x32000) resulting from a discrete non-linear system of partial differential equations. I'm using C++...
15
votes
1answer
3k views
Comparing Jacobi and Gauss-Seidel methods for nonlinear iterations
It is well known that for certain linear systems Jacobi and Gauss-Seidel iterative methods have the same convergence behavior, e.g. Stein-Rosenberg Theorem. I am wondering if similar results exist for ...
5
votes
0answers
59 views
2nd-order TVD criteria for flux-limiter
Consider a nonlinear hyperbolic conservation equation:
$$
\partial_{t}u = -\partial_{x}f(u)
$$
The spatial derivative of $f(u)$ may approximated after a spatial discretization by $x_{j}=j\Delta x$
$$
\...
0
votes
1answer
71 views
Interpreting multivariable root-finding results from Matlab's fsolve algorithm
Edit: So I was able to get the same value of r that's given, when coding up the sum of squares of function values directly in the script file, rather than on the Command Window. So, maybe there's a ...
1
vote
0answers
26 views
Numerically estimate the Jacobian from a scalar time-series
I'm trying to numerically estimate the Jacobian from a time-series. Following the paper, https://journals.aps.org/pra/abstract/10.1103/PhysRevA.43.2787.
Consider that I have a scalar time series $x = (...
2
votes
1answer
108 views
Solving system of nonlinear vector functions
I am trying to figure out how to implement a solver for a system of nonlinear equations of the form
\begin{align*}
u_1 &= y_n + h\left(a_{1,1}f(t_n + c_1 h, u_1) + a_{1,2}f(t_n + c_2 h, u_2)\...
1
vote
1answer
48 views
How to effectively find a starting point solving a non-linear equation?
I have the following equation (the Kurz-Giovanola-Trivedi model [1])
$$
v^2 \frac{\pi^2 \Gamma}{P^2 D^2} + v \frac{mC_0(1-k)\xi}{D[1-(1-k)Iv(P)]} + G = 0,
$$
where $Iv(P)=P \cdot \exp(P) \cdot E(P)$, $...
0
votes
0answers
69 views
Derivative-free ill-conditioned non-linear least squares
I am looking for a package which can solve (non-linear) least squares problems without the use of derivatives (because of an expensive model), but which also deals with ill-conditioning well (such as ...
2
votes
2answers
335 views
Jacobians with automatic differentiation
I have an objective function F: Nx1 -> Nx1, where N>30000. There are many sparse matrix/tensor multiplications in this function, so taking an analytic Jacobian by paper and pen is cumbersome.
...
3
votes
1answer
94 views
How avoid square shape with Laplacian operator in reaction diffusion calculations?
I have used different variants of the Laplacian operator (div grad) using 4, 8, 12, 20 and 24 of the closest points. I get problems due to the chosen coordinate system and the discretization of the ...
1
vote
1answer
58 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 ...
1
vote
0answers
73 views
RK4-method starts oscillating above certain input parameters
I am trying to solve an equation of the following type
$$\partial_zE(z)=-c_0J$$
with $$J=c_1\beta E^3(z)$$
using the boost::odeint-framework and a fixed time stepper, with $c_0$, $c_1$ and $\beta$ ...
1
vote
1answer
73 views
Issue solving nonlinear equation containing a quotient
I have a coupled set of PDEs that need to be solved as part of a larger problem. I am currently approaching this by computing spatial derivatives with finite differences and using PETSc's nonlinear ...
0
votes
1answer
100 views
Method of Lines Runge-Kutta nonlinear stability and behavior
I have a system of 4 nonlinear 1st-order PDEs. I want to solve them numerically by method of lines, first discretizing space. This leads to the system of $N\times 4$ coupled ODEs.
$$
\mathbf{u}_{i} =...
2
votes
2answers
74 views
Methods for solving discrete PDEs using algorithmic differentiation results
I'm looking for a method to solve a 20000 variable, 20000 residual non-linear PDE with a Galerkin method.
I have Fortran subroutines for:
The residuals: $\vec{r}(\vec{x})$;
Their Jacobian multiplied ...
0
votes
0answers
52 views
Non-linear differential equation
I have this equation
$$y\left(\dot y^2+1\right)=m + \Lambda y^3,$$
where $\Lambda=1.1\cdot 10^{-52} $ (Cosmological constant). I want to get the graph of the solution of this equation (2-parametric ...
3
votes
0answers
88 views
Is the matrix exponential and the Jordan canonical form actually useful for solving differential equations?
All of my yearlong graduate-level Linear Algebra course notes from my professorāan algebraist/representation theoristāshows his love for the exponential map $e^A$ and the Jordan canonical formāand one ...
0
votes
0answers
22 views
How to multiply 2 decision variables and a matrix using python
So, basically our agenda is to assign tour guides to tour groups based on this equation and that will be done by these 2 decision variables z(u,g) and y(g,p) where z(u,g) will be 1 if tour guide 'u' ...
0
votes
0answers
34 views
Estimating the dimension of a solution space in nonlinear least squares
Suppose I have a nonlinear least squares problem,
$$
\min_{\mathbf{x}} || \mathbf{f}(\mathbf{x}) ||^2
$$
with $n$ residuals and $m$ parameters, so that $\mathbf{x} \in \mathbb{R}^m$, and $\mathbf{f} \...
2
votes
1answer
116 views
Numerical methods that can be written in flux conservative form
I have a system of non-linear PDEs that I expect to have shocks as well as the appearance of Gibbs phenomena (spurious oscillations that form near the shock) for 2nd-order methods or higher. I have ...
3
votes
0answers
63 views
Conjugate Gradient for nonlinear equation system
Is it possible to apply adaptions of the conjugate gradient algorithm i.e. Fletcher-Reeves, Polak-Ribere or others to systems of nonlinear equations? How should the equation system be adjusted so one ...
1
vote
0answers
334 views
How to solve a nonlinear diffusion equation?
Consider a thin film with a perpendicular applied magnetic field $H_a$ in $z$-axis. The nonlocal relation between $H_a$, the self-field $H_\text{self}$ (generated by the eddy current $J$) and the ...
0
votes
1answer
78 views
FDM on nonlinear PDEs
I'm working with a 2D Navier Stokes PDE in the unstabilized version - the equation is a linear equation of the type $\frac{āu}{āt} = F(u,t)$.
In order to perform time discretization with FDM (finite ...
0
votes
1answer
22 views
Python: Getting second output variable from minimizing a computationally intensive function on first outputs
I have a function in python that is quite computationally expensive to evaluate, of the form:
...
0
votes
1answer
36 views
Coupled pdes of the first order
May question is about possible approaches to solve the following system
$$
\begin{array}{rcl}
\nabla{n}&=&n\,\mathbf{E},\\
\nabla\cdot\mathbf{E}&=&1-n,
\end{array}
$$
in general with ...
1
vote
0answers
64 views
Bifurcation points on homotopy path by numerical continuation?
I am trying to implement an algorithm that finds (possibly) all solutions of a system of nonlinear polynomial equations $$F(X) = 0$$ I thought about using the (convex) homotopy $$H(X,T) = TF(X) + (1-T)...
3
votes
3answers
2k views
Solving a linear equation system with pure Neumann condition
I am trying to solve a linear equation system $\textbf{A}\textbf{x}=\textbf{b}$, e.g. a Poisson equation discretized in strong form, using biCGstab method.
Since there are only natural Neumann ...
7
votes
4answers
764 views
Large-scale nonlinear optimization problem
I want to solve a nonlinear optimization problem of the following form
\begin{equation}
\min\left(\sum_i d^{x_i}c_{i}\right)\\
0 \leq x_{i} \leq a\\
\sum_{i} x_{i} \leq b
\end{equation}
$a$, $b$, $...
4
votes
2answers
71 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:
$$ V_N(\hat{\theta}) = \frac{1}{...
1
vote
0answers
80 views
Linearize non-linear PDE with BCs to hyperbolic problem: How does linearization affect BCs?
I am working with the Shallow Water equations that is a system of non-linear PDEs that simulate water waves propagation on some domain, in my cases the $x$-axis. I have here a GIF showing the results (...
1
vote
1answer
88 views
Full approximation scheme - smoothers - literature recomendation
I would like to use full approximation scheme(FAS) for solving nonlinear PDE. I am looking into practical implementation of FAS in parallel(MPI) settings. I noticed the most common/effective smoother ...
0
votes
0answers
75 views
Solving a nonlinear problem with a very small components with finite element method
In solving nonlinear hyperelastic solid mechanics problems, to converge to the correct solution we need to do step-by-step loading which makes the deformation at each step very small (for my ...
0
votes
1answer
121 views
Numerically solving nonlinear parabolic stochastic PDEs
For a project I'm doing, I have to numerically solve a nonlinear parabolic stochastic partial differential equation, of the form
$$
u_t = u_{xx} + f(u)(u_x)^2 + a(u) + b(u)W(t, x),
$$
where primes ...
0
votes
1answer
65 views
Solve convection-diffusion equation with a non-linear source term
I would like to solve this equation (which is adapted in my case, a plug flow reactor when a reaction occurs):
$ \frac{df}{dt} = D \frac{dĀ²f}{dzĀ²} - u \frac{df}{dz} + r(z,t) $
with $r(z,t)= - k f^{n}...
2
votes
2answers
665 views
Which SciPy nonlinear solver when Jacobian is analytically known and sparse?
I have a nonlinear function fun with n inputs and n outputs. I also have a function jac which calculates the Jacobian, which is ...
1
vote
0answers
74 views
Incorporating radiation boundary condition at the edge in finite difference
I am trying to solve the 2-d heat equation on a rectangle using finite difference method. I am confused as to how to incorporate non linear radiation boundary condition at the edge.
$-k\frac{\partial ...
3
votes
1answer
44 views
Nonlinear least squares resolution matrix
For a linear least squares problem, it is possible to define a resolution matrix, relating the estimated model parameters to the true model parameters. If we are solving a regularized problem,
$$
\...
1
vote
1answer
81 views
How to choose between compact finite differences and spectral methods
For a project in my advanced numerical method class I have to solve the 1D Kuramoto-Sivashinsky equation.
$$
u_t + u u_x + \lambda u_{xx} + \eta u_{xxxx} = 0.
$$
As explained here I will solve it ...
1
vote
1answer
58 views
Initial condition for Kuramoto-Sivashinsky
For a project in my advanced numerical method class I have to solve the 1D Kuramoto-Sivashinsky equation of which I know little. I just know that it was derived the equation to model the diffusive ...
2
votes
1answer
63 views
Solution method of nonlinear heat transfer analysis
The governing equation of transient heat transfer analysis is described as follows:
$$C \frac{dT}{dt}+K T = Q$$
When using backward difference scheme for the discretization of the time we get the ...
3
votes
0answers
69 views
Computation of Troullier-Martins pseudowavefunctions
The computation of Troullier-Martins pseudowavefunctions has been
described in [1].
The pseudowavefunction $R^{\textrm{PP}}_l$ is defined by
$$
R^{\textrm{PP}}_l(r) =
\left\{
\begin{array}{ll}
R^{\...
2
votes
1answer
580 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-...
2
votes
0answers
55 views
References to solve system of differential equations which describe the evolution of sandpile surface using the finite element method
I want to solve the following nonlinear system in 1D
\begin{cases}
\dot{R} + v \frac{\partial R }{\partial x} - \frac{\partial }{\partial x}\left( D \frac{\partial R }{\partial x} \right) -\Gamma =...
