Questions tagged [nonlinear-equations]

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

Filter by
Sorted by
Tagged with
1
vote
0answers
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 ...
2
votes
0answers
54 views

Nonlinear Sylvester-Like Equation

Maybe you can point me to some results already developed for this. I have to solve for $X$ the following "Sylvester-like" equation: $$ AX - XB = F(X)$$ where $A\in\mathbb{R}^{a\times n}$, $B\in\...
0
votes
1answer
112 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 ...
0
votes
1answer
779 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 ...
5
votes
1answer
165 views

Quantifying the degree of nonlinearity in a heat transfer problem

I’m working with a heat equation of the form. $$\frac{d(\rho(T)c_p(T)T)}{dt}-\nabla\cdot(k(T)\nabla T)=f$$ with temperature dependent density $\rho(T)$, specific heat $c_p(T)$, and thermal ...
4
votes
1answer
593 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 {\...
0
votes
1answer
230 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 $A$, $B$ ...
0
votes
1answer
69 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 ...
1
vote
2answers
495 views

C++ template design pattern for groups (algebra)

Having both programmed my share of c++ and studied some beginners group theory some year ago, I got curious about this... Is there any particularly popular template based (object oriented) design ...
2
votes
0answers
222 views

Pros and cons of optimization vs. variational calculus, re: nonlinear elasticity [closed]

I'm trying to solve a problem of finding the displacements of an elastic material subjected to external forces. Those external forces are themselves a nonlinear function of the material displacements....
0
votes
2answers
115 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 ...
1
vote
0answers
40 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 ...
4
votes
1answer
279 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
2answers
142 views

How does Mathematica compute real and complex solutions to single, non-polynomial equations?

This question on StackOverflow has led me to ask myself what would be involved in solving an equation like this one using tools available to the Python programmer. $$ \frac{0.125567841}{d^{2.25}} = \...
0
votes
2answers
595 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 ...
4
votes
2answers
112 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 ...
2
votes
0answers
148 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 ...
2
votes
0answers
114 views

Precision not improving by decreasing step-size in nonlinear Schrödinger

I tried to simulate soliton propagation by solving the nonlinear Schrödinger equation using the split-step Fourier method. The following is an example of the Matlab code copied from a textbook. ...
1
vote
1answer
141 views

Solving systems of nonlinear ODEs in epidemiology

I have a 9 systems of nonlinear ODEs to solve. I want to determine the endemic equilibrium points. How do I go about it? I tried manual calculation but it becomes cumbersome. Can software be used ...
0
votes
2answers
148 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 ...
1
vote
0answers
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 ...
1
vote
0answers
87 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{\...
2
votes
1answer
182 views

Implicit method for two coupled PDEs

I have two equations (coupled), with the variables $T_1$ and $T_2$ and the constant $T_0$, which are (when written unitless, i.e. without prefactors): $$\partial_t T_1 = 1-T_1^3+T_1+\nabla\left(\frac{...
2
votes
1answer
163 views

Solving this system of equations numerically

In a personal project of mine, I've derived a couple of equations. $$ \sum_{j=1}^Nu_{ij}^*(\theta_1,...,\theta_N,J) - k_{i} = 0 \qquad \forall 1\leq i \leq N$$ $$\sum_{i<j}^N\big(u_{ij}^*(\...
1
vote
2answers
156 views

Improving calculation algorithm for coupled PDEs

I have the following two PDEs: $$\partial_zU=\nabla_r^2U+\varrho U$$ $$\partial_t\varrho=a\vert U\vert^4$$ with $a$ a constant and $$dt=dz\cdot\frac{n}{c}$$ with $n$ the refractive index of a ...
2
votes
1answer
133 views

Newton's method stagnates at small error

I have a system of the form $$A(u)f(u)=b$$ where $A$ is basically a matrix originating from the Finite Element Method. I try to solve it using the Newton method: $$R = A(u_{i}) f(u_{i}) - b $$ $$...
2
votes
0answers
68 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 ...
2
votes
1answer
377 views

Linearization in Finite Difference Method: Why?

I have a very basic question, and I hope some of you might be able to help me: In Fluid Dynamics, a common equation turning up time and time again is the so-called continuity equation: $$\frac{\...
2
votes
0answers
104 views

Backing out a function of parameters from system of nonlinear equations

I have a system of equations that cannot be solved for in closed form: $F_1(x_1,x_2,\beta)=0 ~\&~ F_2(x_1,x_2,\beta)=0 $ I want to solve for functions $x_1=x_1(\beta) ~\&~ x_2=x_2(\beta)$ ...
1
vote
0answers
121 views

Nonlinear 2D thermoconductivity equation(numerical solution) [closed]

I have to write a solver for 2D equation: $$\partial_t u = u^2(\partial_x ^2 u + \partial_y ^2 u)$$ I try to use explicit method: $$\partial_t u = \frac{u_{i,j}^{k+1} - u_{i,j}^k}{\tau}$$ and $$\...
0
votes
0answers
392 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 ...
3
votes
0answers
245 views

How to avoid the Broyden's jacobian approximation becoming poorer with the number of iterations?

I have to solve many times a nonlinear system of the form $$f(x) = b^{(n)}$$ inside a loop. The function $f$ is expensive to compute and I do not have its jacobian, so I have tried the good Broyden's ...
2
votes
1answer
573 views

Stopping criteria in iterative methods for solving nonlinear equations

Is it a good criterion to stop iterative methods for solving non-linear equations, such as Newton-Raphson and good Broyden's methods, when $|x_k-x_{k-1}|<|x_k|\,reltol + abstol$ OR when $|f_k|<...
0
votes
1answer
163 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(...
1
vote
0answers
226 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_{...
0
votes
1answer
320 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 ...
3
votes
1answer
737 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=\...
1
vote
0answers
95 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 ...
10
votes
1answer
363 views

Solving a difficult system of equations numerically

I have a system of $n$ non-linear equations that I want to solve numerically: $$\mathbf{f}(\mathbf{x})=\mathbf{a}$$ $$\mathbf{f}=(f_1,\dots,f_n)\quad\mathbf{x}=(x_1,\dots,x_n)$$ This system has a ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
0
votes
0answers
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 $\...
0
votes
0answers
193 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](...
0
votes
1answer
186 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(...
5
votes
1answer
273 views

Convergence of Jacobi's method for a semilinear elliptic PDE

I have an iterative finite difference scheme for the Poisson equation $$ \nabla^2 u=-\rho $$ It's the Jacobi method, which has the form (for 1D systems) $$ u^{n+1}_{i} = \frac{1}{2}(u^n_{i+1} + u^n_{...
2
votes
0answers
52 views

A test suite of large systems of nonlinear equations

I am looking for a kind of modern test set of large nonlinear problems. The only option I managed to find so far is rather dated: http://folk.uib.no/ssu029/Pdf_file/Testproblems/testprobRheinboldt03....
0
votes
1answer
110 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 ...
0
votes
1answer
321 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} \...
6
votes
1answer
193 views

What would be a good approach to solving this large data non-linear least squares optimisation

Introduction to Problem I'm using a Truncated Signed Distance Function to perform 3D reconstruction from depth images. Essentially I have a large voxel grid where each voxel contains the signed ...