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5
votes
2answers
349 views

Is the shooting method the only general numerical method for solving nonlinear boundary value ODEs?

During my wandering in Mathematica.se, I gradually noticed that a certain kind of differential equation solving problem is "troubling" us all the time, that is, the boundary value problem (BVP) of ...
3
votes
2answers
110 views

Implementation of nonlinear term in FEM

Although there are similar questions, I am also struggling with the implementation of the following term in "my own code" by Finite Element Method, namely, $\nabla \phi \cdot \nabla \phi$. $\phi$ is ...
1
vote
1answer
81 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 ...
-1
votes
1answer
82 views

solving a non-linear equation with integrals involved

I would like to solve the following equation, wrt $n(e)$ $$f(n(e))=g(n(e)) + \int_{\alpha}^{e} w(n(x))dx $$ The integral there it confuses me. Any suggestion on how I can implement this on a the ...
5
votes
2answers
99 views

What are the numerical methods for huge polynomial systems?

Let a system of $n$ polynomial equations of degree $d$ with $m$ variables. I'm interested in a sparse system with $d = 3$, $n \sim 2000000$, $m \sim 50000$ and integer coefficients. What techniques ...
0
votes
1answer
21 views

Non linear system of equations with discretization on k-space

I want to numerically solve the following system of differential equations at the steady state: \begin{equation} \begin{aligned} \frac{\partial \rho_{11\mathbf{k}}}{\partial t} =& ...
0
votes
1answer
63 views

Finding roots without knowing much about the function

Consider solving numerically for roots: $( x_0, y_0): f(x_0, y_0) = 0, g(x_0, y_0) = 0$ where you only know that f, g continuously differentiable but the theoretical differentiation is not a ...
2
votes
1answer
95 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): $$ \partial_{t} u = \Delta u + \gamma\left(a-u+u²v\right)$$ ...
0
votes
0answers
18 views

Model actuators in non-linear beam element structure

I have written a non-linear finite element code using beam elements, and I'd like to represent hydraulic actuators in the model. In real terms, this would mean that the distance between two nodes at a ...
3
votes
2answers
86 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 $J(\boldsymbol{x})$ ...
2
votes
0answers
105 views

Integration of nonlinear PIDE via spectral methods

At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction ...
4
votes
1answer
255 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 - ...
0
votes
1answer
69 views

Solver for large non-linear system of equations

I am curently using R package nleqslv for solving a non-linear system of equations with 300 variables. I need to scale this to the system with ~50k variables and naturally this does not scale very ...
4
votes
1answer
96 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 ...
0
votes
1answer
53 views

MATLAB Newton non-linear equation

I have the following non-linear equation: where $w0=0.25,w0=0.5,w0=0.75$. I have to prove that if $k$ is a root, then also $−k$ is a root and that there exists only one $k∈(0,1)$ root, but my ...
0
votes
2answers
134 views

Systems of nonlinear equations

Consider the nonlinear system of equations $$ (1) \quad qk^2a_1^2E^2+wna_0a_1AE+pnka_0^2a_1E+rn^2a_0^2A^2-rn^2a_0^3A^2+qk^2a_0a_1ABE-qk^2na_0^2E^2=0, $$ $$ (2) \quad ...
1
vote
1answer
85 views

System of nonlinear equations in MATLAB

I've got some problems solving (numerically) this system of equations. \begin{array}{l} 40 \cdot \cos (2t) + 105 \cdot \cos ({\theta _3}) - 75 \cdot \cos ({\theta _4}) - 91.924 \cdot \cos ({337.62}) ...
0
votes
0answers
59 views

Solving a non linear equation and iterating for various values in python

problem:Unable to solve a unknown for multiple known values in a non-linear equation ...
5
votes
2answers
127 views

Convergence issues for a non-linear system

I have a nasty system of coupled integral equations, which I managed to discretize and recast a non-linear system, i.e. something like: $$ \vec{w} = F \left( \vec{w} \right) \hspace{32pt} w \in ...
6
votes
1answer
122 views

Non-linear optimization using approximate gradient

I'm working with non-linear optimization for imaging, such as MRI and CT. Our problem is of the form $\|Ax-b \|_2^2+\lambda \|Wx\|_1$. $A$ is never formed explicitly, so we're limited to approaches ...
7
votes
2answers
118 views

For a non-linear PDEs should the source term be discretised at $u_j$ or averaged over $(u_{j+1} + u_{j-1})/2$?

The non-linear Poisson equation in one-dimension, $$ 0 = \frac{\partial^2u}{\partial x^2} - f(u) $$ can be discretised as to give, $$ u_{j-1} -2u_{j} + u_{j+1} = h^2 f(u_j) $$ where $h$ is the ...
3
votes
1answer
125 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 ...
2
votes
1answer
64 views

Methods to solve this equation on finite fields?

Is there any analytical (exact, closed-form solution) or numerical method to solve an equation such as $p(x) = r^x$ where $p(x)$ is a polynomial whose coefficients are drawn from a finite field, ...
1
vote
2answers
125 views

finding wave function for anharmonic oscillator

I'd like to find the normalized ground state wavefunction for the anharmonic oscillator (Duffing) whose potential for which there is no analytic solution; an oscillator with a quartic potential, in ...
3
votes
2answers
193 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} = ...
10
votes
1answer
422 views

When is Newton-Krylov not an appropriate solver?

Recently I have been comparing different non-linear solvers from scipy and was particularly impressed with the Newton-Krylov example in the Scipy Cookbook in which they solve a second order ...
3
votes
0answers
221 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} ...
2
votes
2answers
183 views

How can i solve this first order system of differential equations

Edit: I am trying to solve $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu\frac{\partial^{2}u}{\partial x^{2}}\\ u(0,t)=0\nonumber \\ u(1,t)=0\nonumber \\ u(x,0)=\sin(\pi x)\\ ...
7
votes
1answer
113 views

F(x) = 0 vs. ||F(x)||^2->min

In many areas of application, one needs to solve a nonlinear system of equations $$ F(x) = 0. $$ Sometimes, the formulation $$ \|F(x)\|^2 \to\min $$ is used. Clearly, every solution $\hat{x}$ of ...
3
votes
1answer
94 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
votes
0answers
106 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 ...
2
votes
0answers
39 views

Solving nonlinear wave equation in a dispersive infinite waveguide

I would like to solve a three-dimensional nonlinear wave equation in an infinite cylindrical waveguide numerically. Since the waveguide is dispersive, shocks are less likely to form. Both the ...
4
votes
1answer
315 views

Confusion regarding the Adam-Moulton and Backwards Differentiation Formula (BDF) of the VODE solver

I am exploring the Method of Lines as a way of time stepping semi-discretised PDEs with ODE time-integration solvers. For an excellent introduction to this technique see the scholarpedia.org article. ...
9
votes
2answers
575 views

Is it possible to solve nonlinear PDEs without using Newton-Raphson iteration?

I am trying to understand some results and would appreciate some general comments on tackling nonlinear problems. Fisher's equation (a nonlinear reaction-diffusion PDE), $$ u_t = du_{xx} + \beta u ...
10
votes
2answers
147 views

Numerical method for equation solving that works on stochastically computed functions

There are many well known numerical methods for solving equations of the type $$ f(x) = 0, \quad x \in \mathbb{R}^n,$$ e.g. bisection method, Newton's method, etc. In my application $f(x)$ is ...
3
votes
1answer
136 views

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 ...
3
votes
1answer
664 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 ...
2
votes
2answers
166 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
votes
1answer
232 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 ...
0
votes
1answer
2k views

How do I extrapolate data from a NON-LINEAR (logarithmic) standard curve in Excel?

I have made a standard curve. The X-axis is logarithmic. The y-axis is linear. I have added a logarithmic trendline (y = -1.546ln(x) + 39.254; R² = 0.9906). How can I re-arrange the equation to ...
2
votes
2answers
82 views

Problems where SPD linear system arises

I know some of the places where SPD linar systems arises such as elliptic PDEs and normal equations. Can I have a more comprehensive list of scientific applications which require solving SPD linear ...
5
votes
4answers
2k views

How to find more than one root of a polynomial?

This program finds the first root of the function f, defined in the code. There are 5 roots of this function. (x=1,2,3,4,5) I wish to find all of the roots in this program and print them to the ...
4
votes
2answers
553 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 ...
2
votes
1answer
309 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
votes
1answer
213 views

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 ...
6
votes
4answers
677 views

parameters estimation

I have to estimate a parameter (K), but I don't know how I can do it. I think by a regression model (minimum least square?), but I'm not sure. The system is: ...
3
votes
2answers
126 views

Solving a nonlinear algebraic system that includes a linear term

I am trying to solve a particular system of non linear equations written as $F(x) = 0$ in an efficient way. More specifically, $$F(x) = (I - \gamma A)x - g(x) + C$$ where $\gamma$ is a ...
6
votes
1answer
295 views

Nonlinear dynamics: algorithm suggest

I've just started a thesis on nonlinear dynamics which entails numerical analysis of the Duffing oscillator (DO). It's basically just a second order ODE, or equivalently a set of ODEs. Say, after ...
8
votes
1answer
296 views

Nonlinear wave equation - Finite element or finite difference

I would like to know the which is more advantageous when it comes to solving nonlinear hyperbolic equations, Finite Element or Finite difference methods? Which method will be better in capturing ...
5
votes
2answers
159 views

How to do upwinding in finite volume schemes for nonlinear equations?

In finite difference theory, you learn, that you have to use upwinding for equations with high convection, like Burgers' equation. What does the finite volume equivalent look like? What if the ...