Questions tagged [nonlinear-equations]
Solution of nonlinear systems of equations. The equations might be algebraic or differential equations.
338
questions
2
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1
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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 ...
- 21
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vote
0
answers
136
views
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)...
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votes
1
answer
65
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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 $R=1000$, $V_T = 0.025$,...
- 103
4
votes
1
answer
99
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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 ...
- 2,054
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votes
0
answers
111
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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 ...
- 1
2
votes
1
answer
125
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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 ...
- 23
1
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2
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164
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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[...
- 131
0
votes
1
answer
57
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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:
$$
\frac{\partial\rho}{\partial t}+\tau\frac{1}{r^2}\frac{\partial}{...
1
vote
1
answer
114
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how to solve a coupled nonlinear partial differential equations(Boundary Value Problem)
So far, I have been looking into linear and nonlinear differential equations(Boundary value problem) that look like the following:
$\frac{d^2 \theta}{dz^2} + \sin(\theta) = 0$
I am now familiar with ...
- 35
3
votes
0
answers
152
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Python code of explicit method of a nonlinear a BVP
I am trying to have a Python code for the following nonlinear BVP:
$$\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(...
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1
vote
1
answer
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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 ...
- 95
0
votes
1
answer
100
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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 ...
- 35
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0
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40
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General dimensional solution to fast sweeping quadratic equation
I am reading Hongkai Zhao's paper The Fast Sweeping method. I have implemented the method in 2D and now I want to move onto 3D.
However section 2.6 confuses me.
The article says:
But I have no idea ...
- 263
1
vote
0
answers
38
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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'...
- 111
1
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1
answer
137
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Which analogs of Newton's multivariate method are faster?
Currently, I am studying a 2D nonlinear Schroedinger equation and searching for the fastest method.
$$
\begin{equation}
i \frac{\partial \psi}{\partial t} = [-\frac{1}{2} \nabla^2 + V_0(r) - i\gamma ...
- 31
2
votes
0
answers
148
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Poisson equation solution in a semiconductor structure
I am trying to solve the $\textbf{1-D}$ Poisson equation for a semiconductor structure at equilibrium (There is no external bias applied).
$\textbf{Background}$
\begin{equation}
\frac{d^2V}{dx^2} = -\...
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0
votes
1
answer
73
views
On solving a first order nonlinear differential equation
It all starts with this Cauchy problem:
$$
\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]\,.
$$
Not knowing which way to turn,...
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0
votes
1
answer
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Solving $\sum_i^d a_i \exp(-q_i k)=b_0$ for $k$
Suppose $a_i,q_i,b_0$ are positive real numbers. I need to solve the following equation for $k$
$$\sum_i^d a_i \exp(-q_i k)=b_0$$
Is this a well-known problem? One my special cases has $a_i=q_i$
In my ...
- 2,273
1
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0
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Strategies to solve an equation with a polynomial and a numeric function
I have to solve numerically an equation of the following form:
$$
\sum_{n=0}^m c_n x^n = f(x) x^k
$$
Where the $c_n$ are real values, $k$ is an integer and $f$ can only be evaluated numerically.
The ...
- 153
1
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0
answers
88
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Accuracy of the Crank-Nicolson method for non-linear, inhomogeneous heat equation
I am currently coding a solution to the following PDE:
$\frac{\partial T }{\partial t} =\frac{\partial}{\partial \theta}(A(\theta ,\phi )\frac{\partial T }{\partial \theta}) +\frac{\partial }{\partial ...
- 11
0
votes
0
answers
77
views
Is it possible to globally solve a general coordinate-wise monotone nonlinear system 3x3?
Consider a system
$$
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 ...
- 281
0
votes
0
answers
152
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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 ...
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3
votes
0
answers
124
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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 ...
- 131
2
votes
1
answer
204
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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 ...
- 95
2
votes
1
answer
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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 ...
- 123
3
votes
0
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159
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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 ...
- 33
1
vote
1
answer
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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(\...
- 113
1
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0
answers
67
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
...
- 725
2
votes
1
answer
203
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 ...
- 253
5
votes
1
answer
324
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 ...
- 199
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0
answers
181
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)...
- 33
4
votes
1
answer
187
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 ...
- 81
0
votes
0
answers
55
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 ...
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0
answers
100
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 ...
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5
votes
1
answer
146
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 ...
- 63
1
vote
1
answer
53
views
Verification of coupled system of equations for light propagation
I am trying to simulate the propagation of light in material using the non-linear schrödinger equation (NLSE):
$$\partial_zE=\frac{i}{2k_0}\nabla^2_\perp E+\frac{ik_0n_2}{n_0}\vert E\vert^2E-0.5\beta^{...
- 543
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votes
0
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189
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How to treat nonlinear radiation term in heat equation using Finite-element method?
I am trying to solve the time-dependant non-linear heat equation with radiation. This equation is coupled to the radiative transfer equation but for the purpose of my question, this does not matter ...
- 31
0
votes
0
answers
57
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Discretizing Multi-species Ion Exchange Equations by Finite Volume Method
I'm solving a system of multispecies ion exchange equations (diffusion+drift fluxes) in 1-d spherical domain using finite volume method to obtain the ion concentrations at the next time step. After ...
- 1
3
votes
0
answers
115
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Strange Picard iteration
I am interested in solving the equation
$$
\begin{aligned} \nabla \cdot\left(\nabla \phi-\frac{\nabla \phi}{|\nabla \phi|}\right) &=0 & & \text { in } \Omega \\ \phi &=0 & & \...
- 601
2
votes
1
answer
207
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How to discretize a non-linear PDE with boundary conditions and intial value
Consider this non linear PDE:
$$u_t + c(u^2)_x =\alpha u_{xx} \text{ , } -1<x<1 , t>0 $$
with
$$u(-1,t) = g_L(t) \text{ , } u(1,t) = g_R(t) \text{ and } u(x,0)=f(x)
$$
where the 3 functions(...
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votes
0
answers
131
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How does the error work for the Strang Splitting?
We know in Strang splitting that the splitting error in the steady state solution is proportional to $h^2$. I want ask 2 things:
If this error in the steady state solution is the global error?
If we ...
0
votes
1
answer
192
views
Discretization of a non-linear ODE using FDM isn't grid indepenent
I am trying to solve the ODE :
$\frac{d^2T}{dx^2} = \omega_1 T+\omega_2 T^2$
+
using different numerical methods. I have tried the following discretizations so far and none of them seem to be grid ...
1
vote
0
answers
36
views
Approximation in the derivation of the Arc Length method
I am studying the proof of the Arc Length method in section 2.2 of this thesis. In equation (2.2) the author introduces the supplementary conditions
$$
(\Delta {\bf u} + \delta {\bf u})^T \cdot (\...
- 255
1
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0
answers
215
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Solving coupled PDEs with self-consistency condition
I am figuring out how to attack a problem (the Usadel equations of superconductivity) in which I need to solve a set of nonlinear PDEs for the fields $\{G_i (r)\}$
$$ U(G_i(r), \nabla G_i(r), \Delta(r)...
- 161
1
vote
0
answers
111
views
Stochastic differential equation system (SDE) : overflow encountered in double-scalars
I'm trying to integrate the following SDE system from Dekker et al. [1]
$$\begin{cases}
\frac{dx}{dt}=a_1x^3+a_2x+\phi+\zeta_x\\
\frac{dy}{dt}=b_1z+b_2(\kappa(x)-(y^2+z^2))y+\zeta_y\\
\frac{dz}{dt}=...
- 173
1
vote
0
answers
126
views
Solving Laplace equation with constraint on boundary
I have found the following PDE problem in a paper:
Essentially, we have a rectangular domain where there is a unknown interface $z=\xi(x)$ (liquid-air interface) separating the domain into two medium ...
- 227
0
votes
2
answers
150
views
diffusivity matrix assembly in nonlinear finite element analysis
I want to solve a diffusion analysis using finite elements. According to fick's law, governing equation is
$$\frac{\partial h}{\partial t} = D \frac{\partial^{2} h}{\partial x^{2}}$$
. h is relative ...
- 13
2
votes
0
answers
85
views
Linearising Nonlinear Coupled Partial Differential Equations - Alfvénic Diffusion
I am trying to solve the following coupled partial differential equations with a finite difference scheme:
$$\partial_tf+v\partial_zf+\partial_z\frac{1}{W}\partial_zf=0$$
$$\partial_tW+v\partial_zW-\...
- 21
1
vote
0
answers
141
views
Improve code of logarithmic quantizer
I am implementing a logarithmic quantizer which is defined as follows:
$$ q(u) = \begin{cases} u_i , \frac{u_i}{1+\delta} < u < \frac{u_i}{1-\delta} \\
0 , 0 \leq u \leq \frac{u_o}{1+\delta} \\
-...
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3
votes
1
answer
322
views
Nonlinear root solving libraries which accept a Jacobian in band-storage
I'm in search for a library for solving large systems of non-linear equations, similar to MINPACK, but unlike MINPACK, can accept a Jacobian in band-storage.
My Jacobian is sometimes not invertible, ...
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