Questions tagged [nonlinear-equations]
Solution of nonlinear systems of equations. The equations might be algebraic or differential equations.
104
questions with no upvoted or accepted answers
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views
Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system
Problem
Solving a non-linear system of equations.
The number of variables is the same as the number of equations.
When I fix a set of variables (say $\vec{y}$) and keep another set free (say $\vec{...
- 163
9
votes
0
answers
136
views
Review of modern homotopy methods and practical techniques
I'm hoping someone can recommend recent literature concerning homotopy methods for solving systems of nonlinear equations. Already by the time of Layne Watson's 1986 paper there were a lot of methods,...
- 297
6
votes
0
answers
673
views
Implementing a Hill-Type Muscle Model
I'm interested in implementing the muscle model used in Geijtenbeek and Wang et al's work.
Both papers link to the paper by Geyer and Herr, which describes this model:
However, the paper on this ...
- 323
5
votes
0
answers
154
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 $x_{j}=j\Delta x$
$$
\...
- 95
5
votes
0
answers
159
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numerical analysis of a partial integro-differential equation
I have to numerically solve a nonlinear partial integro-differential equation. This is my equation,
$$\frac{\partial y(x,t)}{\partial t}=\int_{-1/2}^{1/2} \frac{\pi\cos u}{\sin\pi u-\sin\pi x} \frac{\...
5
votes
0
answers
754
views
Finding roots of systems of equations with a Jacobian that is singular everywhere
Given $a\in\mathbb{R}^{mn\times n}$, find a $C\in\mathbb{R}^{n}$, $x\in\mathbb{R}^{m\times n}$ such that
$$
0 = f_{k}(\boldsymbol{C}, \boldsymbol{x}):=\sum_{i=1}^{m} C_{i} \left(\prod_{j=1}^{n} a_{kj}^...
- 529
5
votes
0
answers
195
views
Time-stepping for coupled nonlinear PDEs
What are good references for time-stepping of the coupled incompressible Navier-Stokes-heat equation (Boussinesq flow),
$$
\begin{cases}
\rho\left(\dot{\mathbf{u}} + \mathbf{u}\cdot\nabla \mathbf{u}\...
- 3,108
4
votes
0
answers
61
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 ...
- 141
4
votes
0
answers
100
views
Computation of Troullier-Martins pseudowavefunctions
The computation of Troullier-Martins pseudo-wavefunctions has been
described in [1].
The pseudo-wavefunction $R^{\textrm{PP}}_l$ is defined by
$$
R^{\textrm{PP}}_l(r) =
\left\{
\begin{array}{ll}
R^{\...
- 331
4
votes
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answers
129
views
Integrators for Nonlinear/Stiff PDE
It was suggested I ask this question in this section. Anyway:
I have a particular nonlinear PDE of the form
$$
u_t(x,t)=iu_{xx}(x,t)+f(x,u(x,t))
\tag{1}
$$
Where f is some nonlinear function. With ...
- 41
4
votes
0
answers
333
views
Nonlinear least squares and regularization
Consider the nonlinear least-squares minimization of a vector of $n$ residuals $\mathbf{f}$ in $p$ parameters $\mathbf{x}$:
$$
\min_{\mathbf{x}} || \mathbf{f}(\mathbf{x}) ||^2
$$
This can be done with ...
- 1,016
4
votes
0
answers
219
views
Poisson equation in frequency domain
I need some help in numerically solving the nonlinear Poisson's equation for electrons in frequency domain. The steady-state equation is:
\begin{equation}
\nabla.(\epsilon\nabla\varphi) = q\left(n_i\...
- 430
3
votes
0
answers
82
views
Python code of explicit method of a nonlinear a BVP
I am trying to have a Python code for the following nonlinear BVP:
$$\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(...
- 29
3
votes
0
answers
95
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 ...
- 131
3
votes
0
answers
150
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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
3
votes
0
answers
158
views
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
3
votes
0
answers
106
views
Strange Picard iteration
I am interested in solving the equation
$$
\begin{aligned} \nabla \cdot\left(\nabla \phi-\frac{\nabla \phi}{|\nabla \phi|}\right) &=0 & & \text { in } \Omega \\ \phi &=0 & & \...
- 601
3
votes
0
answers
95
views
Is the matrix exponential and the Jordan canonical form actually useful for solving differential equations?
All of my yearlong graduate-level Linear Algebra course notes from my professor—an algebraist/representation theorist—shows his love for the exponential map $e^A$ and the Jordan canonical form—and one ...
- 31
3
votes
0
answers
83
views
Conjugate Gradient for nonlinear equation system
Is it possible to apply adaptions of the conjugate gradient algorithm i.e. Fletcher-Reeves, Polak-Ribere or others to systems of nonlinear equations? How should the equation system be adjusted so one ...
3
votes
0
answers
96
views
Large-scale optimization of nonlinear equations
I'm looking to find a computationally efficient solution to a large system of nonlinear equations. I'm trying to maximize the following function:
$$ f(\vec{x}) = \sum_i^N C_i (x_i-A_i)x_i^{\epsilon_{...
- 193
3
votes
0
answers
36
views
Detecting blocks in non-linear system of equations
When solving systems of non-linear equations using Newton's method, it is often observed that the system has an independent sub-system, e.g. :
$$ f(x,y) = 0 $$
$$ g(x,y) = 0 $$
$$ h(x,y,z) = 0 $$
If ...
- 53
3
votes
0
answers
92
views
PETSc SNES for user defined state
How to use PETSc SNES (scalable nonlinear equation solver), when the solution is not a vector but a user defined state?
I am implementing a non-linear mechanics problem (geometrically exact shell 5-...
- 55
3
votes
0
answers
340
views
How to avoid the Broyden's jacobian approximation becoming poorer with the number of iterations?
I have to solve many times a nonlinear system of the form
$$f(x) = b^{(n)}$$
inside a loop.
The function $f$ is expensive to compute and I do not have its jacobian, so I have tried the good Broyden's ...
- 439
3
votes
0
answers
205
views
Solving a nonlinear poisson equation via variational minimization
I am kind of new in finite elements and I am solving simple "Poisson nonlinear" problem.
$- \nabla ((1 + u^2) \nabla u) = f$
$u = 0 \ \text{on} \ \Omega $
I am using Newton solver, where I have ...
- 31
3
votes
0
answers
104
views
Applications of algorithm for solving systems of equations with uncertainty
We have been developing algorithms for detecting "robust" zeros of multidimensional functions $f: X\to\Bbb R^n$ where $X$ is an $m$-dimensional domain in $\Bbb R^m$. More precisely, for a given $f$, ...
- 101
3
votes
0
answers
194
views
Integration of nonlinear PIDE via spectral methods
At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction $\...
- 203
3
votes
0
answers
81
views
Estimating eigenvalues from time-dependent non-linear operator
I have a very sparse non-linear system $N(u) = 0$ that can be solved as a time-dependent ODE, $\frac{du}{dt} = N(u)$, and explicitly integrated until $\frac{du}{dt} = N(u) = 0$, e.g. by forward euler, ...
- 2,293
2
votes
0
answers
87
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} = -\...
- 33
2
votes
0
answers
117
views
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 ...
2
votes
0
answers
78
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:
$$\partial_tf+v\partial_zf+\partial_z\frac{1}{W}\partial_zf=0$$
$$\partial_tW+v\partial_zW-\...
- 21
2
votes
0
answers
72
views
Multilevel minimization - boundary conditions
I am interested in minimizing
$$min_{x \in R^{n^l}} f^l(x),$$
where $f^l(x)$ is nonlinear objective function arising from discretization of PDE. I would like to use nonlinear multilevel minimization ...
2
votes
0
answers
559
views
Understanding MP-PIC implementation in OpenFOAM
The multiphase particle-in-cell (MP-PIC) method is characterized by mapping particle properties from the Lagrangian coordinates to the Eulerian grid. However, the implementation of this method in ...
- 442
2
votes
0
answers
86
views
How to solve $y(x) y'''(x)=f(x)$
I have a PDE of the form $\partial_t y(x,t)+\partial_x(y(x) y'''(x)-f(x))=0$, where $f(x)=\cos(x)$.
Suppose a stable equilibrium exists, and I want to find the steady-state solution $y(x) y'''(x)=f(x)...
- 217
2
votes
0
answers
111
views
Nonlinear system with diagonal nonlinearity
Consider a nonlinear system of the form $\boldsymbol{f}(\boldsymbol{x}) = \boldsymbol{0}_{\mathbb{R}^n}$ for $\boldsymbol{x} \in \mathbb{R}^n$, where the function $\boldsymbol{f}$ is given by
\begin{...
- 176
2
votes
0
answers
2k
views
Solve system of polynomial equations with Python
I have 5 at most 4th order polynomials in 5 variables,
$$p_i(x_1,x_2,x_3,x_4,x_5) \qquad i = 1, \ldots, 5$$
where all coefficients are either rational or floating point. I'd would like to get the ...
- 121
2
votes
0
answers
183
views
Neumann boundary conditions in the Maccormack scheme
I am trying to solve the viscous Burger equation
$$
\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \mu \frac{\partial^2 u}{\partial x^2}
$$
with Neumann boundary conditions. I am
...
- 21
2
votes
0
answers
291
views
Convergence of a very large non-linear least squares optimization
(note: I also posted this question on stackoverflow before finding this community here, which seems a better place for it)
I'm trying to solve the following problem: I have a lot (~80000) surface ...
- 31
2
votes
0
answers
59
views
Nonlinear Sylvester-Like Equation
Maybe you can point me to some results already developed for this.
I have to solve for $X$ the following "Sylvester-like" equation:
$$ AX - XB = F(X)$$
where $A\in\mathbb{R}^{a\times n}$, $B\in\...
- 121
2
votes
0
answers
151
views
Precision not improving by decreasing step-size in nonlinear Schrödinger
I tried to simulate soliton propagation by solving the nonlinear Schrödinger equation using the split-step Fourier method. The following is an example of the Matlab code copied from a textbook.
...
- 217
2
votes
0
answers
111
views
Backing out a function of parameters from system of nonlinear equations
I have a system of equations that cannot be solved for in closed form:
$F_1(x_1,x_2,\beta)=0 ~\&~ F_2(x_1,x_2,\beta)=0 $
I want to solve for functions $x_1=x_1(\beta) ~\&~ x_2=x_2(\beta)$
...
- 121
2
votes
0
answers
53
views
A test suite of large systems of nonlinear equations
I am looking for a kind of modern test set of large nonlinear problems. The only option I managed to find so far is rather dated: http://folk.uib.no/ssu029/Pdf_file/Testproblems/testprobRheinboldt03....
- 1,842
2
votes
0
answers
362
views
Finite difference scheme for solving nonlinear least-squares problem
I am dealing with following problem:
$$
\min_{u,\gamma}\Bigg\{ \frac{1}{1000} \iint_{S_2} {\gamma (x,y)^2 dxdy} + \iint_{S_2} {[u(x,y) - u_0 (x,y)]^2 dxdy} + \iint_{S_2} {[\Delta u(x,y) - \gamma (...
2
votes
0
answers
154
views
Rank deficient Jacobian in discretized periodic solutions to autonomous ODE
I'm trying to numerically find periodic solutions to different systems of autonomous nonlinear ordinary differential equations. I decided to use a finite difference scheme and solve the resulting ...
- 21
2
votes
0
answers
69
views
Solving a nonlinear equation with a Markov process and RVs
Assume that we have the following equation and the following assumption. The scope is to solve for some particular variables expressed later.
Update
$$E_{t}\left[ b(A_{t+1})^{1-\gamma} *R_{t+1}^{-\...
- 235
2
votes
0
answers
56
views
Solving nonlinear wave equation in a dispersive infinite waveguide
I would like to solve a three-dimensional nonlinear wave equation in an infinite cylindrical waveguide numerically. Since the waveguide is dispersive, shocks are less likely to form. Both the ...
- 103
1
vote
0
answers
36
views
How can i study stability for a new method that solves second degree non lineara differential equations?
I developed a new method to solve this equations $\frac{d{y}^{2}}{dx^{2}}=g(x,y,\frac{dy}{dx})$ for a general g(x,y,z), wich their solution is a function f(x)=y. Obviously you need $y(x_0)=y_0$ and $y'...
- 111
1
vote
0
answers
68
views
Strategies to solve an equation with a polynomial and a numeric function
I have to solve numerically an equation of the following form:
$$
\sum_{n=0}^m c_n x^n = f(x) x^k
$$
Where the $c_n$ are real values, $k$ is an integer and $f$ can only be evaluated numerically.
The ...
- 153
1
vote
0
answers
76
views
Accuracy of the Crank-Nicolson method for non-linear, inhomogeneous heat equation
I am currently coding a solution to the following PDE:
$\frac{\partial T }{\partial t} =\frac{\partial}{\partial \theta}(A(\theta ,\phi )\frac{\partial T }{\partial \theta}) +\frac{\partial }{\partial ...
- 11
1
vote
0
answers
64
views
Preserving conservation properties across time-integration schemes
I am interested in deriving an accurate, implicit discretization of a nonlinear advection-diffusion equation
$$
\partial_t f = \frac{1}{v^2} \partial_v J(f,v) \equiv F(f,v) \tag{$\star$}
$$
with flux
...
- 705
1
vote
0
answers
151
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