Questions tagged [nonlinear-equations]

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

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60 views

Rotational kinematics problem @ $\theta = 0$

I'm simulating a magnetic dipole that is subjected to an evolving magnetic field. In ISO convention ($\theta$ is the polar angle and $\phi$ is the azimuthal angle), my equation of motion that is ...
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39 views

Computing Direct Scattering Transform

I'm working on the Nonlinear Schrodinger equation (NLSE) in 1d: $$i\psi _t (t,x)+ \psi _{xx} + |\psi|^2\psi = 0 \, ,$$ for $t\geq 0$ and $x\in \mathbb{R}$. This equation is integrable, and so ...
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139 views

shallow water equation maccormack method

I am trying to make a code for 1D shallow water equation (nonlinear without source terms) using the MacCormack method for sinusoidal wave propagation. My issue is that the wave fluctuates and does not ...
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56 views

Object-oriented non-linear solving in python

I'd like to build a system in Python, consisting of (broadly speaking) objects which are internally described with (not necessarily just linear) equations, that I can connect with each other - similar ...
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85 views

Finite difference scheme for unconfined aquifer equation

For an unconfined aquifer we have this PDE for the water table position( of course after somehow making the original Boussinesq equation linearized ): $$ \frac{\partial^2(h^2)}{\partial x^2} + \frac{\...
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66 views

How to get a theoretical background in nonlinear, coupled FEM systems

I'm currently developing simulations for coupled, nonlinear, multi-region systems. Basically, I use the Finite Element Method (FEM) to model each physical quantity in each region. The obtained ...
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202 views

Breather solutions of Sine-Gordon Using Finite Differences

I'm attempting to simulate a standing breather of the form $$ u(x,t)=4\tan^{-1}\left(\sqrt{3}\cos\left(\frac{t}{2}\right)sech\left(\frac{\sqrt{3}x}{2}\right)\right)$$ for the Sine-Gordon equation $$u_{...
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92 views

Which numerical scheme should be used?

Trying to find a way to solve exponentially nonlinear elliptic equation with complex source term i manage to know about such schemes like Godunov, Lax-Friedrich, MUSCL. I still seraching for ...
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57 views

elliptic equation with exponential coefficient

I'm trying to solve the following equation $$\dfrac{\partial}{\partial x}\left(e^{au}\dfrac{\partial u}{\partial x}\right) = 0$$ Of course, this equation can be solved analytically. I am trying to ...
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79 views

Three steps of pde numerical solution and nonlinear equation

I'm very new here. I'm trying to solve nonlinear elliptic equation $$ (n(u)u')' = f(u) $$ and face with crucial misunderstanding. As I suppose, the procedure of solving some nonlinear equation ...
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308 views

Simulating a Hill-Type muscle model

I'm trying to do biologically based locomotion animation. I have a Hill-Type Muscle Model (see reference 1) that I've implemented in code. I can find the initial lengths of the tendons, then do a step ...
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178 views

Numerically solving a system of stiff nonlinear PDEs

I am attempting to numerically solve the following: \begin{align} \frac {\partial y_1}{\partial t} &= i(y_2y_3 - y_2^*y_3^*) - y_1 \tag{1}\\ \frac {\partial y_2}{\partial t} &= y_1^*y_3 - y_2 ...
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70 views

Gauss Seidel moving mesh AMR hamilton Jacobi

I was trying to implement a moving mesh algorithm using Gauss Seidel, so: I have a pde like this (Hamilton - Jacobi eq) $\begin{cases}\phi_t+H(\phi_x)=0 & [-1,1]\times [0,T]\\\phi(x,0)=\phi_0 &...
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242 views

Numerical solution of non-linear advection equation other than inviscid burgers

I am solving a non-linear advection equation of the form $u_t + f(u)_x = 0$ where $f(u)$ is a complicated function of $u$. I am solving this equation using a first order fully implicit scheme (...
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343 views

Solving system of constrained linear and non-linear equations in MATLAB

Solving system of constrained linear and non-linear equations in MATLAB I'm solving a FEM problem in MATLAB with use of the direct stiffness method. The problem is now formulated as a system of nn ...
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113 views

Solving a large system of nonlinear equations, where timeseries are the unknown

I am trying to solve a problem, which I find quite hard, like, headache-hard. I have to solve the following set of $M$ nonlinear equations: $$F(X)=\begin{bmatrix}f_1 (X)\\f_2 (X)\\...\\f_M (X)\\ \end{...
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100 views

Application of vector extrapolation methods to convergence to a steady state solution

I'm working on a fluid solver using dual-time stepping and everything works really well, except the convergence in pseudo-time is slow. I'd like to accelerate the convergence. I know multigrid methods ...
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92 views

Time dependent self-consistent equations

I am facing the following problem. I need to solve numerically a set of coupled equations $$i\frac{d}{dt}f_{n}^{(i)}(t) = \left[U\cdot n(n-1) + \mu\cdot n\right]f_{n}^{(i)}(t) - \sqrt{n+1}\Phi_i^{*}\...
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219 views

Minimal surface finite differences problem - Matlab assemble

I face to the following problem: $$(1+u_x^2)u_{yy} - 2u_xu_yu_{xy} + (1+u_y^2)u_{xx}=0.$$ Problem needs to be discretized and assembled. Does anybody know how to proceed in Matlab?
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535 views

General algorithm to solve systems of symbolic equations

I want to simplify (solve) a system of linear + nonlinear symbolic equations as much as possible. the equations are of random orders, without differentiation. is there a general & well-known ...
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328 views

How to solve a nonlinear diffusion equation?

Consider a thin film with a perpendicular applied magnetic field $H_a$ in $z$-axis. The nonlocal relation between $H_a$, the self-field $H_\text{self}$ (generated by the eddy current $J$) and the ...
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68 views

Interpreting multivariable root-finding results from Matlab's fsolve algorithm

Edit: So I was able to get the same value of r that's given, when coding up the sum of squares of function values directly in the script file, rather than on the Command Window. So, maybe there's a ...
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67 views

Derivative-free ill-conditioned non-linear least squares

I am looking for a package which can solve (non-linear) least squares problems without the use of derivatives (because of an expensive model), but which also deals with ill-conditioning well (such as ...
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51 views

Non-linear differential equation

I have this equation $$y\left(\dot y^2+1\right)=m + \Lambda y^3,$$ where $\Lambda=1.1\cdot 10^{-52} $ (Cosmological constant). I want to get the graph of the solution of this equation (2-parametric ...
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31 views

Estimating the dimension of a solution space in nonlinear least squares

Suppose I have a nonlinear least squares problem, $$ \min_{\mathbf{x}} || \mathbf{f}(\mathbf{x}) ||^2 $$ with $n$ residuals and $m$ parameters, so that $\mathbf{x} \in \mathbb{R}^m$, and $\mathbf{f} \...
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16 views

How to multiply 2 decision variables and a matrix using python

So, basically our agenda is to assign tour guides to tour groups based on this equation and that will be done by these 2 decision variables z(u,g) and y(g,p) where z(u,g) will be 1 if tour guide 'u' ...
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74 views

Solving a nonlinear problem with a very small components with finite element method

In solving nonlinear hyperelastic solid mechanics problems, to converge to the correct solution we need to do step-by-step loading which makes the deformation at each step very small (for my ...
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22 views

Efficient Alternatives to Operator Splitting in NLSE

Lately i've been trying to decide my thesis theme and i've become interested in adaptive finite elements and finite volumes algorithms. However, I need my thesis to fit into a physics related theme. ...
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57 views

A non linear ode with boundary conditions at infinity

I want to solve the non-linear ODE $$\frac{d^2}{dx^2}y=a(y+y^3)$$ With the boundary conditions that $$\lim_{x\to \pm \infty} y(x) =0$$ I am not aware of any analytical method for solving this kind ...
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381 views

Stability of nonlinear partial differential equation

I want to find an expression for the stability of the nonlinear Poisson equation. I know about von Neumann stability analysis which applies to linear equations as far as I know. Any suggestion how to ...
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95 views

nonlinear boundary condition

I have a nonlinear system of algebraic equations with a nonlinear boundary condition of the form \begin{equation} f'''(x,t) - (f'(x,t))^3 - \alpha(t) f'(x,t) = 0, \end{equation} at $x =0$, where $\...
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185 views

Numerical solution of non-linear heat-diffusion PDE using the Crank-Nicholson Method

I am trying to solve numerically the following 1D EBM: $C\frac{\partial T[x,t] }{\partial t} - \frac{\partial }{\partial x}\left ( D(1-x^2)\frac{\partial T[x,t] }{\partial x} \right ) + I[T] = S[x,t](...
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47 views

Order of accuracy of linearised vs non-linear system

Does the order of accuracy of a combination of schemes applied to solve a system of non-linear equations, match those of the same schemes applied to the linearised version of the system? In other ...
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46 views

comparison of stability of two non-linear methods

I have solved a numerical problem using two different sets of non-linear governing equations. I want to get an understanding of the stability of the methods relative to each other. To do so, I solving ...
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64 views

Finding the root of an equation

I have given $i_1$, $i_2$ and $\alpha$ which can be real or integer. How can I find the roots of: $$ (i_1 + i_p )^{\alpha} - i_p^{\alpha} - (i_2 \cdot i_p^{\alpha-1}) = 0 $$
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132 views

system of coupled nonlinear ODEs with complex coefficients

I am interested in numerically solving the following system of coupled ODEs $$\left(i-\frac{1}{2\Omega}f_{m,n}\right) \frac{d a_{m,n}(t)}{dt} =E_{m,n}^{\text{kin}}(t) + \left(\omega_{0}+V_{m,n}-\frac{...
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1answer
306 views

Applying Runge-Kutta to nonlinear system of PDEs

I am applying a 4th order Runge-Kutta code, using the method of lines, to solve the following: \begin{equation} \frac {\partial y_1}{\partial t} = y_2 y_3 - C_1 y_1 \end{equation} \begin{equation} \...
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1answer
25 views

Semi-Definite relaxation of non-linear constraint?

I am implementing an optimization problem using semi-definite approach. One of my constraints is of following form $ trace(A∗X)−(k∗trace(A∗X))+(k∗\sqrt {(trace(B∗X)} )==0$ where k is a constant, A ...
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1answer
108 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 ...

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