# 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-...
1 vote
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### 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-...
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### 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 ...
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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 ...
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 $\ell = 0 , 1$ ...
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### 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 ...
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### 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 ...
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1 vote
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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 ...
1 vote
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### 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 ...
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### 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 ...
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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 ...
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1 vote
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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 ...
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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). $\textbf{Background}$ \frac{d^2V}{dx^2} = -\...
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### 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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### 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 ...
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1 vote
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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 ...
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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 ...
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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 ...