Questions tagged [nonlinear-equations]

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

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1answer
204 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 ...
4
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2answers
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 ...
4
votes
1answer
130 views

Nonlinear least-squares solvers vs. generic minimization

A nonlinear least-squares problem with $F:\mathbb{R}^m\to\mathbb{R}^n$, $$ F(x) \to \min_x \quad (\text{in the least-squares sense}) $$ really means minimizing $$ \frac{1}{2} \|F(x)\|^2 \to \min_x. $$ ...
4
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2answers
164 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 $J(\boldsymbol{x})$ ...
4
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1answer
401 views

Coupled nonlinear PDEs with time dependence on the RHS

I would like to numerically solve the following system of 2 coupled partial differential equations for the unknown functions $\psi_X(x,y,t)$ and $\psi_C(x,y,t)$: $\partial_t \psi_X = -i\psi_C - i\...
4
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1answer
2k views

Newton's method goes to zero determinant Jacobian

I am using the Newton's method to solve $3\times3$ systems. For some particular cases, it turns out that at a given iteration, the Jacobian matrix cannot be inverted and that its determinant is very ...
4
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1answer
2k views

Numerical solution of non-linear diffusion equation via finite-difference with the Crank-Nicolson method

I want to numerically solve the non-linear diffusion equation: $$ \frac{\partial}{\partial t} T(x,t)= \frac{\partial}{\partial x}\left(T^{5/2} \frac{\partial T}{\partial x} \right) $$ I want to use ...
4
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2answers
65 views

MInimizing cost function using iterative search for a minimum method

I want to estimated the parameters $\ \hat{\theta} $ of a model using an iterative search for the minimum of a cost function. The cost function is defined as follows: $$ V_N(\hat{\theta}) = \frac{1}{...
4
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1answer
1k views

How could we solve coupled PDE with finite difference method and Newton-Raphson method?

I'm trying to solve coupled PDE by Crank-Nicolson (CN) and Newton-Raphson method with MATLAB. I have used CN method but not for coupled problem. Please if someone could help let me know to add more ...
4
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1answer
136 views

Finding self-kissing points on a plane curve?

I have a curve in the complex plane given by $$ f(t) = \sum_k r_k\exp(2\pi\mathrm{i}(t+\varphi_k)p_k). $$ Some of the parameters are specially chosen: $r_k>0$, $\sum_k r_k=1$, $p_k\in\mathbb{Z}$, $\...
4
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1answer
274 views

Fast way to repeatedly solve a small nonlinear equation system

A small nonlinear equation system (sizes around 12 ✕ 12) needs to be solved repeatedly (millions of times); each time with some variation in parameters/coefficients (although the equation set is ...
4
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0answers
108 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 ...
4
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0answers
174 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\...
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0answers
148 views

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{\...
4
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0answers
662 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}^...
4
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185 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
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2answers
5k views

Is “tangent stiffness matrix” the same as “stiffness matrix”?

I'm trying to implement nonzero Displacement Boundary Conditions in VegaFEM on a non-linear model, using the method outlined in §3.6.2 of University of Colorado's intro to FEM (modify $f = Ku$: set ...
3
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1answer
368 views

Limitations with dynamical systems vs. PDEs?

As a computational scientist, are there limitations to studying dynamical systems — systems of odes in which each state variable evolves with time — compared to studying PDEs? For instance, it seems ...
3
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1answer
590 views

Solving a system of nonlinear equations with an ODE solver is faster than with the Newton method?

This is somehow unexpected, but my recent experience with solving a system of nonlinear equations is that treating them as the right hand side of a system of ordinary equations and then evolve the ...
3
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3answers
2k views

Solving a linear equation system with pure Neumann condition

I am trying to solve a linear equation system $\textbf{A}\textbf{x}=\textbf{b}$, e.g. a Poisson equation discretized in strong form, using biCGstab method. Since there are only natural Neumann ...
3
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2answers
937 views

Slow convergence of Newton's method for finite elements

The application is a simple non-linear advection diffusion problem (steady state) using DGFEM. My error at each iteration is given by $$ e_{n+1} = ||\mathbf{J}^{-1}(\mathbf{u}_{n})\mathbf{F}(\mathbf{u}...
3
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2answers
111 views

Approximating solutions to quadratic recurrence boundary value problem

Cross-posted from Math Stackexchange: https://math.stackexchange.com/questions/2421964/approximating-solutions-to-quadratic-recurrence I have a branching process problem that has been reduced to ...
3
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2answers
474 views

Bessel EVP and fzero

I am trying to solve the Eigenvalue problem $$ x^2 y''+ x y' + x^2 y = \lambda^2 y\,,\quad x\in(0,1)\,,\quad y(0)=0\,,\quad y'(1)=y(1) $$ The differential equation is the Bessel equation. The solution ...
3
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2answers
319 views

Nanoseconds vs. picoseconds in numerical quantum problems with Matlab ODEs

Hello there and thanks for taking a look at this problem. This problem is related to my previous question and I will therefore use a similar introduction from, Choice of step size using ODEs in ...
3
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1answer
476 views

Fenics: time-independent Sine-Gordon equation

Is there a code for the equation $$ \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2} = \sin(u) $$ or for the sine gordon equation in two dimensions because I want to change some ...
3
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2answers
112 views

Finite difference for a highly nonlinear equation - The wind within the forest

Based on the Navier-Stokes equations and a few parameterizations, the horizontal steady-state wind $u(z)$ within a forest of height H satisfies: $$ a\left(\frac{du}{dz}\right)^2 + b\frac{du}{dz} \...
3
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1answer
113 views

Solving a non-linear heat equation with the galerkin method gives negative values

I am trying to solve a non-linear time-dependent heat equation $$\partial_tT=\nabla \left(k_T(T)\nabla T\right) + f$$ using the galerkin method, with neumann boundary conditions. For linearization of ...
3
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1answer
542 views

Radiation boundary condition (heat transfer)

I am looking for reference on how to implement nonlinear boundary conditions. Specifically, I am interested in implementing a radiation boundary condition for heat transfer with the FEM: $-k \frac {\...
3
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1answer
159 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 ...
3
votes
1answer
88 views

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 ...
3
votes
1answer
139 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 ...
3
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1answer
118 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 = ...
3
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1answer
266 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 ...
3
votes
1answer
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 ...
3
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1answer
905 views

Non-linear root finding when the Jacobian is almost singular

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 ...
3
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1answer
86 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 ...
3
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1answer
48 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 ...
3
votes
1answer
94 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 ...
3
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1answer
989 views

Solving system of 7 nonlinear algebraic equations symbolically

I have a system of seven nonlinear equations that I want to find their symbolic solutions. The solution will depend on the parameter K, and I should have different solutions by varying the parameter. ...
3
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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 ...
3
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1answer
538 views

PETSc or Trilinos for GPU?

I have to choose between the PETSc and Trilinos libraries for a project that involves the solution of non-linear systems of equations. From their web sites they both mention support for GPUs, ...
3
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2answers
283 views

System of quadratic algebraic equations

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 ...
3
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1answer
661 views

Split operator method

I am new to the field of computational physics and have a couple of questions regarding solving the non-linear Schrödinger equation using Operator splitting. 1) If the hamiltonian is of the form $H=\...
3
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1answer
204 views

Numerical methods for coupled stiff PDEs

I'm dealing with a set of nonlinear coupled PDEs that have the form: \begin{align} \frac {\partial y_1}{\partial t} &= y_2y_3 - y_1 \tag{1}\\ \frac {\partial y_2}{\partial t} &= y_1y_3 - y_2 \...
3
votes
1answer
176 views

Relaxation Parameters for Steady Navier-Stokes

I am working on a project involving steady solutions for the Navier-Stokes Equations. In the past I've only worked with the unsteady Navier-Stokes, so some of this is new to me. In particular, at ...
3
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1answer
1k views

Solving a Nonlinear BVP using Finite Difference Method

I am trying to write a code to solve a nonlinear BVP using the Finite Difference Method. The BVP is: $(T^2)\frac{\partial^2 T}{\partial x^2} + T \left( \frac{\partial T}{\partial x}\right)^2 + Q = 0$ ...
3
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1answer
208 views

Time Integration of a nonlinear reaction-diffusion system

I want to solve the following system of nonlinear reaction-diffusion equations (Schnakenberg Turing) using FEM methods (such as deal.ii): $$ \partial_{t} u = \Delta u + \gamma\left(a-u+u²v\right)$$ $$...
3
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1answer
201 views

Switch branch in bifurcation

I have a system of nonlinear equations $F(x,a) = 0$ and I know that at a specific point $a_c$ a bifurcation occurs, thus the Jacobian becomes singular. How can I switch branches and start following a ...
3
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1answer
87 views

How to pick a basis for the result of a non-linear function given a basis for its argument

I am trying to represent the result of a non-linear function in a small basis, given another small basis that does a good job a representing the argument of the function. More specifically, there ...
3
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0answers
86 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 ...