Questions tagged [nonlinear-equations]

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

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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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1answer
267 views

Numerically solving a non-linear PDE

I have this non-linear partial differential equation. $$ \frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial x}\right)^2+C\frac{\partial^2 C}{\partial x^2} $$ I want to use the finite ...
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2answers
715 views

Newton method for a nonlinear system of time-independent PDEs

I have taken a course on undergrad scientific computing which discussed nonlinear algebraic equations about half-way in, and PDEs at the very end, but never discussed nonlinear PDEs. However in my ...
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1answer
42 views

Why might the time taken to compute the solution of an ODE system over some interval increase non-linearly with increasing size of interval?

Currently, my problem requires me to solve a system a large system of non-linear ODEs (up to ~5000). So far, I have been using scipy.integrate.odeint as my ...
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1answer
82 views

Solve convection-diffusion equation with a non-linear source term

I would like to solve this equation (which is adapted in my case, a plug flow reactor when a reaction occurs): $ \frac{df}{dt} = D \frac{d²f}{dz²} - u \frac{df}{dz} + r(z,t) $ with $r(z,t)= - k f^{n}...
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39 views

Coupled pdes of the first order

May question is about possible approaches to solve the following system $$ \begin{array}{rcl} \nabla{n}&=&n\,\mathbf{E},\\ \nabla\cdot\mathbf{E}&=&1-n, \end{array} $$ in general with ...
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1answer
111 views

How to use non-dimensional form in open source codes instead of Units

I am using an open source FEM platform, which requires you to convert your equation system to non-dimensional form. So, there are no units specified for the parameters in the problem. If you use ...
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1answer
112 views

NONLINEAR ENERGY MINIMIZATION EXAMPLE

I am learning about FEM methods and nonlinear optimization. I would like to try my nonlinear trust region solver on some simple nonlinear problem. What would be good example to implement for ...
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1answer
851 views

Solving coupled differential equations and Algebraic equation in MATLAB

I want to solve a system of 7 coupled differential equations and 1 algebraic equation in MATLAB with the method of lines. I could do it for each independent equation with some assumptions, but I can'...
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71 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 ...
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1answer
630 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 ...
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557 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 MATLAB ...
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290 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 nwa_1^2AE+nwa_0a_1AB+2pnka_0a_1^...
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1answer
10k 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 ...
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2answers
121 views

diffusivity matrix assembly in nonlinear finite element analysis

I want to solve a diffusion analysis using finite elements. According to fick's law, governing equation is $$\frac{\partial h}{\partial t} = D \frac{\partial^{2} h}{\partial x^{2}}$$ . h is relative ...
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130 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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126 views

Method of Lines Runge-Kutta nonlinear stability and behavior

I have a system of 4 nonlinear 1st-order PDEs. I want to solve them numerically by method of lines, first discretizing space. This leads to the system of $N\times 4$ coupled ODEs. $$ \mathbf{u}_{i} =...
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1answer
115 views

FDM on nonlinear PDEs

I'm working with a 2D Navier Stokes PDE in the unstabilized version - the equation is a linear equation of the type $\frac{∂u}{∂t} = F(u,t)$. In order to perform time discretization with FDM (finite ...
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1answer
22 views

Python: Getting second output variable from minimizing a computationally intensive function on first outputs

I have a function in python that is quite computationally expensive to evaluate, of the form: ...
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1answer
129 views

Numerically solving nonlinear parabolic stochastic PDEs

For a project I'm doing, I have to numerically solve a nonlinear parabolic stochastic partial differential equation, of the form $$ u_t = u_{xx} + f(u)(u_x)^2 + a(u) + b(u)W(t, x), $$ where primes ...
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1answer
165 views

Solving $n$ coupled equations numerically in Matlab

I would like to solve the following equations simultaneously and numerically for all $X, Y, Z, W$ where i = 1:Nw, j = 1:Nl, k = 1:K. $W_\text{net1}$, $W_\text{net2}...
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1answer
189 views

Combining multiple coupled 1st order equations in python

I'm having serious troubles with solving translating 3 coupled differential equations into python. The 3 DE's stem from a 4th order DE used to calculate the bending moment of an underwater pipeline ...
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1answer
105 views

Numerical solution of non-linear first order partial differential equation (HJB)

I am trying to solve a simple optimal control problem using the Hamilton-Jacobi-Bellman equation, numerically in Python. This is proving to be rather difficult as I end up having to solve the ...
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3answers
184 views

Initial conditions for pendulum Jerk equation

I have a very simple problem, but can't seem to understand what I need to do. In simulating a pendulum from it's jerk equation, I'm having a hard time setting initial conditions to get it to work out....
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1answer
971 views

Crank–Nicolson method for nonlinear differential equation

I want to solve the following differential equation from a paper with the boundary condition: The paper used the Crank–Nicolson method for solving it. I think I understand the method after googling ...
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1answer
253 views

PDEPE nonlinear

I would like to use Matlab's pdepe to solve this system: $$ s_t =(sr)_x + s_{ xx } \\ r_t =(\frac{ A }{ B }r^2+s)_x + \frac{ A }{ -K } r_{ xx } $$ where $A$, $B$ ...
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1answer
88 views

Principle of virtual work - extra term needed for deformation dependent loading?

I'm working on a problem in nonlinear elasticity, for which the external forces (loadings) depend on the displacements. Following Klaus-Jürgen Bathe's book "Finite Element Procedures", the virtual ...
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2answers
117 views

Computable alternative to "almost everywhere"

I am working with finite elements for Maxwell's Equations (i.e. with Nedelec's edge elements) and for computation I'm using the FEniCS-project. While implementing the Augmented Lagragian Method, I ...
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2answers
158 views

implicit odes solution using fdm

I am solving a non linear second order implicit initial value problem using finite difference method, but my results do not converge. Please guide me with an example, how we can apply finite ...
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1answer
168 views

Numerical solution of nonlinear thermoconductivity equation

I have to find and plot a numerical solution for the following equation (I have to write a solver): $$u_{t} = (u^2 u_x)_x$$ with the following conditions $u(0,t) =0, u(1,t) = \sqrt{\frac{2c-2}{t}}, u(...
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1answer
391 views

Obtaining extra output argument(s) from the objective function used by fsolve in MATLAB

I have a MATLAB code (see below) that employs 'fsolve' from the optimization toolbox for a root finding problem. The bottleneck is that, within the objective function calculation, there is a ...
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1answer
929 views

How to find order of convergence in finite element method computationally when exact solution is unknown for time dependent problem

Say for heat equation or Burgers equation with nonlinear boundary condition. Exact solution is unknown. So I am taking for small mesh size the discrete solution as exact solution. Then how to write ...
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1answer
2k views

Solving coupled PDE in COMSOL [closed]

I have the system of equations \begin{align} &A \frac{\partial u_1}{\partial t} = 1 - u_1 B \frac{\partial u_2}{\partial y}\\ &\frac{\partial u_2}{\partial t} = \frac{\partial}{\partial y}\...
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1answer
78 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} =& +\frac{i}{\...
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50 views

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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53 views

Discretizing Multi-species Ion Exchange Equations by Finite Volume Method

I'm solving a system of multispecies ion exchange equations (diffusion+drift fluxes) in 1-d spherical domain using finite volume method to obtain the ion concentrations at the next time step. After ...
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1answer
117 views

Discretization of a non-linear ODE using FDM isn't grid indepenent

I am trying to solve the ODE : $\frac{d^2T}{dx^2} = \omega_1 T+\omega_2 T^2$ + using different numerical methods. I have tried the following discretizations so far and none of them seem to be grid ...
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56 views

Proof of kinematic relationship between updated and total Lagrangian 2nd Piola-Kirchhoff stresses

On page 587 of Finite Element Procedures by Bathe the author gives the following kinematic transformations $$ {}^t\tau_{ij} = \frac{{}^t\rho}{{}^o\rho} \; {}^t_ox_{i,r} \; {}^t_oS_{rs} \; {}^t_ox_{j,...
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1answer
267 views

Solve non-linear equation in R

I need to solve the following equation for $x$ in [0, 1]. Assume $0<\alpha<1$ and $0<\lambda$. $$(1 - x)^{\alpha+1} - \lambda (x+1)^{\alpha+1} = -2\lambda (\alpha + 1) x^\alpha$$ Would very ...
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74 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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76 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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65 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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113 views

Non-linearities in modal analysis of flexible beam

I'm trying to analyse the behaviour of a wind turbine blade rigidly connected to a wall during fatigue testing, being excited in it's first mode in two orthogaonal directions simultaneously (the first ...
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418 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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96 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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1answer
199 views

Slight Modification to Backward Euler Stiff ODE Solver

I am trying to implement a stiff ODE solver that uses step doubling with the backward Euler method but I want to make a modification and I am unsure how to incorporate it. I am trying to solve $y'=f(...
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1answer
373 views

FEM, Direct Stiffness Method with a nonlinear displacement constraint in one node

i have a question about a FE problem im working on. I made a finite element model of an linear elastic block of material (double striped block) attached with a rigid connection to the environment (...
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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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47 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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