# Questions tagged [nonlinear-equations]

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

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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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### Backwards Difference Implicit Method for Nonlinear Parabolic PDE in Python

Original Stack Overflow Question: https://stackoverflow.com/questions/65683788/indexerror-index-31-is-out-of-bounds-for-axis-1-with-size-31?noredirect=1#comment116218335_65683788 PDE: u_t = u_xx + u(...
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### 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 ...
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### 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 ...
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### 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} &...
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### 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++...
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### 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 ...
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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 $x_{j}=j\Delta x$ \... 1 vote 0 answers 101 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 = (... • 273 0 votes 1 answer 141 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 ... 2 votes 1 answer 173 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)\... • 23 1 vote 1 answer 85 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, $$where Iv(P)=P \cdot \exp(P) \cdot E(P), ... • 111 0 votes 0 answers 79 views ### Derivative-free ill-conditioned non-linear least squares I am looking for a package which can solve (non-linear) least squares problems without the use of derivatives (because of an expensive model), but which also deals with ill-conditioning well (such as ... • 131 1 vote 2 answers 867 views ### Jacobians with automatic differentiation I have an objective function F: Nx1 -> Nx1, where N>30000. There are many sparse matrix/tensor multiplications in this function, so taking an analytic Jacobian by paper and pen is cumbersome. ... • 21 3 votes 1 answer 145 views ### How avoid square shape with Laplacian operator in reaction diffusion calculations? I have used different variants of the Laplacian operator (div grad) using 4, 8, 12, 20 and 24 of the closest points. I get problems due to the chosen coordinate system and the discretization of the ... 1 vote 0 answers 82 views ### RK4-method starts oscillating above certain input parameters I am trying to solve an equation of the following type$$\partial_zE(z)=-c_0J$$with$$J=c_1\beta E^3(z)$$using the boost::odeint-framework and a fixed time stepper, with c_0, c_1 and \beta ... • 543 1 vote 1 answer 78 views ### Issue solving nonlinear equation containing a quotient I have a coupled set of PDEs that need to be solved as part of a larger problem. I am currently approaching this by computing spatial derivatives with finite differences and using PETSc's nonlinear ... • 245 2 votes 2 answers 104 views ### Methods for solving discrete PDEs using algorithmic differentiation results I'm looking for a method to solve a 20000 variable, 20000 residual non-linear PDE with a Galerkin method. I have Fortran subroutines for: The residuals: \vec{r}(\vec{x}); Their Jacobian multiplied ... 0 votes 1 answer 139 views ### Method of Lines Runge-Kutta nonlinear stability and behavior I have a system of 4 nonlinear 1st-order PDEs. I want to solve them numerically by method of lines, first discretizing space. This leads to the system of N\times 4 coupled ODEs.$$ \mathbf{u}_{i} =...
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 ...