Questions tagged [nonlinear-equations]
Solution of nonlinear systems of equations. The equations might be algebraic or differential equations.
291
questions
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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 ...
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
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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
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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 ...
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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
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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
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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)}$$
inside a loop.
The function $f$ is expensive to compute and I do not have its jacobian, so I have tried the good Broyden's ...
2
votes
1answer
573 views
Stopping criteria in iterative methods for solving nonlinear equations
Is it a good criterion to stop iterative methods for solving non-linear equations, such as Newton-Raphson and good Broyden's methods, when $|x_k-x_{k-1}|<|x_k|\,reltol + abstol$ OR when $|f_k|<...
0
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1answer
163 views
Numerical solution of nonlinear thermoconductivity equation
I have to find and plot a numerical solution for the following equation (I have to write a solver):
$$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_{...
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0answers
289 views
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
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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 ...
