# Questions tagged [nonlinear-equations]

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

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### 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 ...
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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 ...
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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 ...
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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 ...
1 vote
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### 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 ...
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### 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 ...
1 vote
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### 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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### 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 ...
1 vote
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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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### 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 ...
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### 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 ...
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I'm trying to integrate the following SDE system from Dekker et al.  $$\begin{cases} \frac{dx}{dt}=a_1x^3+a_2x+\phi+\zeta_x\\ \frac{dy}{dt}=b_1z+b_2(\kappa(x)-(y^2+z^2))y+\zeta_y\\ \frac{dz}{dt}=... 1 vote 0 answers 126 views ### Solving Laplace equation with constraint on boundary I have found the following PDE problem in a paper: Essentially, we have a rectangular domain where there is a unknown interface z=\xi(x) (liquid-air interface) separating the domain into two medium ... 0 votes 2 answers 150 views ### diffusivity matrix assembly in nonlinear finite element analysis I want to solve a diffusion analysis using finite elements. According to fick's law, governing equation is$$\frac{\partial h}{\partial t} = D \frac{\partial^{2} h}{\partial x^{2}}$$. h is relative ... 2 votes 0 answers 85 views ### Linearising Nonlinear Coupled Partial Differential Equations - Alfvénic Diffusion I am trying to solve the following coupled partial differential equations with a finite difference scheme:$$\partial_tf+v\partial_zf+\partial_z\frac{1}{W}\partial_zf=0\partial_tW+v\partial_zW-\...
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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} \\ -...