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Questions tagged [nonlinear-equations]

Solution of nonlinear systems of equations. The equations might be algebraic or differential equations.

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Mathematica Package for validating effective string theory solution

I am asking for Mathematica package that given an input of: symmetric matrix $G_{\mu\nu}$, antisymmetric matrix $B_{\mu\nu}$ and a scalar function $\Phi$ will check whether it is a solution to the one-...
Daniel Vainshtein's user avatar
1 vote
0 answers
84 views

An alternative to Levenberg–Marquardt algorithm

When trying to solve for a (over)determined non-linear least square method: $$\underset{x}{\min}||f(x)||^2_2, f: \mathbb{R}^n \rightarrow \mathbb{R}^m, x\in \mathbb{R}^n, m\geq n$$ we use the Gauss-...
William Lin's user avatar
4 votes
3 answers
618 views

Analysis of convergence of Newton method

I often used the Newton-Raphson method in material calculation, where I had to solve a small set of nonlinear equations (size=1..5). In most cases, it worked. However, convergence failure is often ...
kstn's user avatar
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How to ensure solution evolves forward in Modified Riks Method by Crisfield?

I am writing a Matlab based script for solving nonlinear FEA problems using Modified Riks method by Crisfield. As a starting point, I am solving $y = x^3 - 2 x^2 - x + 2$. Around the function minima ...
frustrated_engineer's user avatar
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68 views

Solving coupled 2nd-order differential equation

I would appreciate it if you could help me solve the following coupled equation numerically $$ [-\frac{1}{2} \partial_r^2 + v_0(r) -E]\psi_{\ell} + v_1(r) \psi_{1-\ell}(r) = 0, $$ where $\ell = 0 , 1$ ...
Ghoti's user avatar
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5 votes
2 answers
134 views

Cheap way to keep parameter matrices orthogonal during optimization?

TLDR; I can keep matrix variables approximately orthogonal by taking a single gradient step in the direction of "effective rank" of matrix at each step of iterative solver, is there a more ...
Yaroslav Bulatov's user avatar
2 votes
0 answers
69 views

Solving systems of the form $y_i=UW x_i$ for $U,W$

I'm looking for pointers/examples of solving system of equations $y_i=f_W(f_U(x_i))$ for $W,U$ where $f_M(x) \approx M x$ $U,W$ are updated simultaneously $i\in (0, 10^{12})$ Simplest example is ...
Yaroslav Bulatov's user avatar
1 vote
1 answer
113 views

How to impose boundary conditions when solving a nonlinear dynamical system given by the FEM solver

I am solving a nonlinear dynamical system given by a nonlinear elastic problem which takes the following form: $$ \boldsymbol{M} \ddot{u} + \boldsymbol{K}_{\textrm{NL}}u = 0 ,$$ here $u \in \mathbb{R}...
Saddam N Y Hijazi's user avatar
0 votes
1 answer
82 views

Non-dimensionalizing the Ideal MHD System

Non-dimensionalization is a really frustrating topic for me, and I imagine many others, because in school it was glossed over while being really important to implementing a simulation. I'm writing a ...
Ashamandarei's user avatar
2 votes
0 answers
53 views

Solving a system of non-linear equations to find relationship between arguments

I have a program that implements a multivariate function, call it $f = \mathcal{Q}(Z,v)$ that I can compute given $Z,v$. The $v$ variable is related to the $f$ variable by another relation, call it $v ...
haricash's user avatar
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How to embed linear elasticity/constrain solver in non-linear soft-body dynamics

I often do simulation of dynamics of mass points connected by strings (e.g. molecular dynamics, soft body dynamics etc.). Typically I do it simply by integration of equations of motion by e.g. verlet ...
Prokop Hapala's user avatar
1 vote
0 answers
33 views

Visualizing a low-dimensional torus in a high-dimensional system

In the 4D Henon-Heiles system, it is well-known for certain parameters the attractor is a 2D torus. I am wondering how can we plot this actual torus (embedded in 3D) by somehow projecting all 4 ...
Axel Wang's user avatar
  • 197
3 votes
1 answer
67 views

Period-doubling bifurcation, quasi-periodicity and dimension of torus

This is more of a conceptual question but closely related with how non-linear dynamics simulations results should be interpreted. I am confused about the relationship of "period" in the ...
Axel Wang's user avatar
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Convergence of Modified Crank-Nicolson Scheme

I'm dealing with a particular reaction-diffusion equation having the form $$ c_t = \alpha \nabla^2 c + F(c,x,t). \tag{1}$$ where $F$ is nonlinear. I would like to solve (1) with a finite-difference ...
zaccandels's user avatar
2 votes
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Numerical solution for inviscid Burgers' equation seems to have no breaking time?

So I'm trying to use the Lax-Friedrichs method to solve the inviscid burgers' equation with initial condition $$u(x,0) = \sin(x)$$, using $$u_m^{n+1} = \frac{1}{2}(u_{m+1}^n + u_{m-1}^n) - \frac{\...
Applesauce44's user avatar
2 votes
1 answer
94 views

Numerical calculation of Lyapunov exponents using SciPy's built-in solve_ivp

I have previously successfully implemented the QR decomposition method in MATLAB to calculate Lyapunov exponents for Lorenz equations. See here. This method integrates the stacked system, i.e. the ...
Axel Wang's user avatar
  • 197
2 votes
1 answer
330 views

Large set of nonlinear equations in Sympy

I have a set of 6 nonlinear equations, and using Sympy I find the values of the 6 unknowns. This works perfectly and it directly gives the exact solution, using sympy.solve to be specific. Now I ...
je2703's user avatar
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1 vote
0 answers
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FEM for nonlinear first-order ODE

Currently I am trying to solve nonlinear Ricatti equation using FEM (Matlab language): $$\frac{d r(z)}{dz} = i k(z) r(z) + \frac{i k(z)}{2}(\epsilon(z) - 1)(1 + r(z))^2$$ $z = [-h/2, h/2]$ and $r(-h/2)...
Andrew's user avatar
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1 answer
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How can I find the current for a nonlinear electrical circuit using the Shockley equation, in Octave?

For this electrical circuit: The voltage $ V_D $ can be found by solving a nonlinear equation: $$ \frac{V_{DD}}{R} - \frac{V_D}{R} - I_se^{V_D/V_T} = 0 $$ In this example, let $R=1000$, $V_T = 0.025$,...
Astor Florida's user avatar
4 votes
1 answer
120 views

Numerical continuation of all eigenvalues of small, dense matrices

Consider a one-parameter family of matrices $A(q)\in \mathbb{R}^{n\times n}$, $q\in\mathbb{R}$. For my applications, $n$ is typically between $5$ and $50$ and $A(q)$ is generally dense, so direct ...
whpowell96's user avatar
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Algorithm to solve system of nonlinear equations

I would like some tips in figuring out a good algorithm to find the solution of the following system. Let $\theta$ be a constant in $(0,1)$, let $i,l=1,...,N$, let $a_{l}$ and $b_{i,l}$ be some ...
Andres's user avatar
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2 votes
1 answer
131 views

Spot redundant equations within nonlinear system of equations

Is there a general procedure to detect if in a system of m-nonlinear equations (also non polynomial) of n-unknowns some of the equations are redundant? Can the rank of the Jacobian matrix tell me ...
aSpagno's user avatar
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1 vote
2 answers
225 views

Optimization Problem with Array Index as decision variable

I am trying to formulate an optimization problem where the decision variable is an index of the input array as part of the formulation. For example, I have the following term (this is simplified): $A[...
Kasparov92's user avatar
0 votes
1 answer
62 views

Oscillation in non-linear porous flow solved by finite difference

I am trying to solve numerically the flow of a gas through a porous, spherically symmetric body. The non-dimensional equations read: $$ \frac{\partial\rho}{\partial t}+\tau\frac{1}{r^2}\frac{\partial}{...
MaximeMaurice's user avatar
1 vote
1 answer
142 views

how to solve a coupled nonlinear partial differential equations(Boundary Value Problem)

So far, I have been looking into linear and nonlinear differential equations(Boundary value problem) that look like the following: $\frac{d^2 \theta}{dz^2} + \sin(\theta) = 0$ I am now familiar with ...
Hari Sam's user avatar
3 votes
0 answers
205 views

Python code of explicit method of a nonlinear a BVP

I am trying to have a Python code for the following nonlinear BVP: $$\frac{\partial N}{\partial t}=\frac{\partial^2 N}{\partial x^2}+N(1-N)-\sigma N$$ $$N(0,x)=\sin(2\pi x)$$ $$N(t,0)=0 \hspace{3mm}N(...
Peachy April's user avatar
1 vote
1 answer
75 views

Once Lyapunov exponents have converged, can they diverge again?

I know the strength of attraction for a single attractor might vary from place to place. Say, if I calculated the Lyapunov exponents for a small portion of the attractor and they have already ...
Axel Wang's user avatar
  • 197
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1 answer
127 views

finding weak form of nonlinear differential equation for FEM simulation

The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential ...
Hari Sam's user avatar
1 vote
0 answers
38 views

How can i study stability for a new method that solves second degree non lineara differential equations?

I developed a new method to solve this equations $\frac{d{y}^{2}}{dx^{2}}=g(x,y,\frac{dy}{dx})$ for a general g(x,y,z), wich their solution is a function f(x)=y. Obviously you need $y(x_0)=y_0$ and $y'...
martín canullán's user avatar
1 vote
1 answer
140 views

Which analogs of Newton's multivariate method are faster?

Currently, I am studying a 2D nonlinear Schroedinger equation and searching for the fastest method. $$ \begin{equation} i \frac{\partial \psi}{\partial t} = [-\frac{1}{2} \nabla^2 + V_0(r) - i\gamma ...
Andrew's user avatar
  • 31
2 votes
0 answers
173 views

Poisson equation solution in a semiconductor structure

I am trying to solve the $\textbf{1-D}$ Poisson equation for a semiconductor structure at equilibrium (There is no external bias applied). $\textbf{Background}$ \begin{equation} \frac{d^2V}{dx^2} = -\...
0-0's user avatar
  • 33
0 votes
1 answer
79 views

On solving a first order nonlinear differential equation

It all starts with this Cauchy problem: $$ \begin{cases} \sin(2x(t)) -\cos(3x'(t)) = x(t) + x'(t) \\ x(0) = 1 \\ \end{cases} \quad \quad \text{with} \; t \in [0,10]\,. $$ Not knowing which way to turn,...
Monster's user avatar
  • 113
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1 answer
85 views

Solving $\sum_i^d a_i \exp(-q_i k)=b_0$ for $k$

Suppose $a_i,q_i,b_0$ are positive real numbers. I need to solve the following equation for $k$ $$\sum_i^d a_i \exp(-q_i k)=b_0$$ Is this a well-known problem? One my special cases has $a_i=q_i$ In my ...
Yaroslav Bulatov's user avatar
1 vote
0 answers
72 views

Strategies to solve an equation with a polynomial and a numeric function

I have to solve numerically an equation of the following form: $$ \sum_{n=0}^m c_n x^n = f(x) x^k $$ Where the $c_n$ are real values, $k$ is an integer and $f$ can only be evaluated numerically. The ...
WIP's user avatar
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Accuracy of the Crank-Nicolson method for non-linear, inhomogeneous heat equation

I am currently coding a solution to the following PDE: $\frac{\partial T }{\partial t} =\frac{\partial}{\partial \theta}(A(\theta ,\phi )\frac{\partial T }{\partial \theta}) +\frac{\partial }{\partial ...
mathbruh67's user avatar
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0 answers
79 views

Is it possible to globally solve a general coordinate-wise monotone nonlinear system 3x3?

Consider a system $$ F_i(x, y, z) = 0, \quad i = 1, 2, 3 $$ with $F_i$ monotonic w.r.t. $x, y$ and $z$. The system 2x2 can be easily solved with alternating direction method that will find all its ...
jokersobak's user avatar
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0 answers
155 views

Discontinuous Galerkin failing to converge Euler equations under p-refinement

I am solving the steady state compressible Euler equations in conservation form in 2D in a rectangular domain with a discontinuous Galerkin (DG) code I have developed. As a test, the boundary ...
Wil's user avatar
  • 63
3 votes
0 answers
139 views

Splitting system of equations into linear and nonlinear part and solving separately

I was working on a problem recently (calculating all flows in a network given input and output flows, basically what Hardy-Cross tries to achieve) which can be formulated as a well-determined system ...
nardi's user avatar
  • 131
2 votes
1 answer
339 views

Numerical computation of Lyapunov exponents: how to find convergence or non-convergence efficiently?

I am wondering what are the standards for convergence of Lyapunov exponents (and Kaplan-Yorke dimension)? For example, I have a MATLAB code to calculate Lyapunov exponents for the classic Lorenz ...
Axel Wang's user avatar
  • 197
2 votes
1 answer
102 views

Time Integrators for Water Wave Simulation

I am interested in using a numerical wave tank (NWT) to study the performance of various water wave dampers using MATLAB. I am looking at nonlinear water waves. My current NWT (2d, periodic) is based ...
Gary Moon's user avatar
  • 123
3 votes
0 answers
172 views

Numerically solving a 6th order non-linear differential equation in Matlab

I've posted yesterday a question about solving a non linear equation : it was not clear so I am reformulating my question. I am trying to solve a high-order non linear differential equation presented ...
Wiss's user avatar
  • 33
1 vote
1 answer
1k views

Easy way to perform solver over pandas dataframe

I'm moving from Excel to Python and I'm trying to solve these equations: $$\begin{align} X_1&=\bigg[\big(3.47-\log(X_2)\big)^2+\big(\log(c)+1.22)^2\bigg]^{0.5}\\ X_2&=\frac{a}{101.32}\bigg(\...
JCV's user avatar
  • 113
1 vote
0 answers
68 views

Preserving conservation properties across time-integration schemes

I am interested in deriving an accurate, implicit discretization of a nonlinear advection-diffusion equation $$ \partial_t f = \frac{1}{v^2} \partial_v J(f,v) \equiv F(f,v) \tag{$\star$} $$ with flux ...
Endulum's user avatar
  • 735
2 votes
1 answer
215 views

Global convergence behavior of several Krylov solvers in scipy.opt

In the context of mechanical simulation, where I solve the stationary action principle directly (i.e. $\nabla S = 0$ for some scalar function $S$), I use the wrapper ...
G. Fougeron's user avatar
6 votes
1 answer
340 views

Why do we solve non-linearity in hyperbolic PDEs that way?

I am used to solve parabolic or elliptic non-linear PDE and the common methods to tackle non-linearity are Picard's iteration and Newton's method. I am a bit confused by the way things are done with ...
Iddingsite's user avatar
1 vote
0 answers
204 views

Fitting data with a Voigt function

I have some data, (xrd data), that I would like peak fit with a pseudo-Voigt function, a combination of a Gaussian and a Lorentzian function. These are the functions $G(x) = I \exp\left( -\frac{4\ln(2)...
Peter's user avatar
  • 33
4 votes
1 answer
204 views

Method to linearize highly nonlinear partial differential equation

I have a set of coupled pdes which I want to solve using finite-difference, of which one is nonlinear. The three linear pdes for quantities $T_f$, $T_s$ and $c$ are convection-diffusion-reaction-like ...
DozerD's user avatar
  • 81
0 votes
0 answers
56 views

Best way to solve system of quadratic forms

I have a system of equations that have the following structure. Let $x\in\mathbb{R}^m$ and let $x_k$ be the $k$-th element of $x$. Let $H_k\in\mathbb{R}^{m\times m}$ for $k=1,\ldots, m$. I need to ...
5d41402abc4's user avatar
1 vote
0 answers
101 views

Is it possible to use a fixed point iteration for solving this nonlinear system?

Consider the following differential equation \begin{align} \frac{\partial f(u)}{\partial x} &= g(x), \ \ x\in [x_{L},x_{R}] \label{Eq2.2} \\ u(x_{L}) &= g_{1} \end{align} where $f(u)$ is a ...
TheComander's user avatar
5 votes
1 answer
154 views

Analysis of nonlinear finite element methods

I have been doing a lot of reading on the development of finite element methods and their analysis using, e.g., functional analysis. I am clear on the formulation of the weak form of a PDE and ...
Wil's user avatar
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