# Questions tagged [nonlinear-equations]

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

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### Should the Jacobian of a system of PDEs be calculated from the main equations of the discretised equation?

I am solving a coupled system of non-linear PDEs in 1D. Something like, $$u_t = F_1(u,v,w) \\ v_t = F_2(u,v,w) \\ w_t = F_3(u,v,w)$$ where each variable is a function of $x$ (the spatial dimension)...
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### Stability of numerical schemes for non-linear equations with a Jacobian with negative eigenvalues

Let us assume I have an A-stable numerical scheme. I believe that given any linear equation $y' = Ay$, it means that the numerical scheme applied to this equation is stable (and therefore convergent ...
145 views

### What is the preferred method for evolving the Nonlinear Schrödinger Equation?

I am interested in evolving the (cubic) self-focusing nonlinear Schrödinger equation, $$i\frac{\partial \psi}{\partial t}+\frac{1}{2}\frac{\partial^2 \psi}{\partial x^2}+\left|\psi\right|^2\psi=0$$ ...
164 views

### solving for unknown inside an expectation

I need to find roots for the following function: $$f(\theta) \equiv E[R(\theta;\eta)]=0$$ for some unknown $\theta$ which is deterministic, while the expectation is taken over a normally ...
229 views

### Solving a nonlinear equation with random variable

I would like to solve an equation that looks like this UPDATE $E[(R^{1-\gamma})(r_k+\theta-r_z)]=0$ , where $R=\phi r_z+(1-\phi)(r_k+\theta)$ and $\phi\in[0,1]$, $\theta$, is a random variable ...
183 views

### Non-linear root finding with positive definite Jacobian

I am dealing with a system of non-linear equations: $$f(\boldsymbol{x}) = \boldsymbol{y}, \;\;\; \boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^d.$$ And I know that the Jacobian $J(\boldsymbol{x})$ ...
3k views

### C++ alternatives for simulating dynamic systems

I'm looking for alternatives to Matlab/Simulink and Dymola for simulating a non-linear dynamic system. I know it's possible to implement the time-domain behavior without a lot of code and a good ...
134 views

### References for the nonlinear reaction-diffusion equation using Finite Element Methods

I want to study how to solve the following PDE \begin{cases} -\nabla \cdot(\ k(x,y) \ \nabla u \ ) + \beta(x,y)\ u^2 = f(x,y), \ (x,y) \in \Omega \subset \mathbb{R^2} \\ \hspace{0.5cm} u = ...
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### Calculate Transformation Matrix between two sensors

My question is if I can calculate the transformation matrix between two sensors. Each sensor provides a $4\times 4$ matrix for every timestep recorded. The sensors are moving and have some noise in ...
186 views

### Reference request: Riks method (Nonlinear FEM)

I'm struggling to find a good detailed reference explaining the Arc-length method or, more generally, Riks method and its derivations. I looked for the classical books in nonlinear mechanics (the ones ...
1k views

### Which SciPy nonlinear solver when Jacobian is analytically known and sparse?

I have a nonlinear function fun with n inputs and n outputs. I also have a function jac which calculates the Jacobian, which is ...
299 views

### Initialize arc length control in Riks' method

I'm trying to use Riks' method and I'm not sure how to set the initial values for the loading coefficient, nor the tangent vector (i.e. the derivative of the displacements and loading coefficient with ...
1k views

### Solving large, non-linear systems of ODEs numerically: what do I need to consider in order to figure out which solver to use?

I would prefer recommendations that don't require the use of proprietary tools (such as Matlab). I know of two ODE solving options for the Python ecosystem: PyDSTool (Dopri, Radau, other Runge-Kutta ...
1k views

I'm trying to solve a system non linear-equations: $$\frac{\partial K(\mathbf{\lambda})}{\partial \lambda_i} - c_i = 0$$ for $i = 1, \dots, 15$, using Newton's method: \lambda^{k + 1} = \lambda^k ... 1answer 109 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, ... 1answer 172 views ### How to derive the simplified Newton iteration in the TR-BDF2 ODE integration scheme The Problem The TR-BDF2 explained in this paper , 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} &... 1answer 131 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 ... 1answer 76 views ### Improve optimization speed for a set of similar problems: Quadratic programming with a warm start I am repeatedly solving quadratic program, x^T Q x with time dependent linear constraints Ax=b_t. Dimension of x is around 10000 and there are around 50 constraints. I want to solve the ... 1answer 116 views ### Nonlinear least squares when some parameters are linear Consider the least squares problem, \min_{\mathbf{a},\mathbf{b}} || \mathbf{f}(\mathbf{a},\mathbf{b})||^2 $$where \mathbf{a},\mathbf{b} represent the unknown parameters to be found. In my ... 3answers 2k views ### Solving nonlinear differential equations with Newton's method I have difficulties with this equation$$\frac{d^2 u}{d x^2} + u^2 - x^2 = 0 with boundary conditions: $u(0)=u(1)=0$ I do not know how to solve nonlinear differential equations with Newton's ...
I have this problem $H_i(x_1,x_2,\dots, x_N) = a_{ijk} x_j x_k + b_{ij} x_j + c_i = 0 \quad 1\leq i \leq N$ And I need to show that applying Newton-Raphson can fail to find even one real solution ...