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

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-1
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1answer
49 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 ...
0
votes
0answers
28 views

Shooting method MATLAB upper order non linear ODE [duplicate]

How can I solve a system of nonlinear differential equations using Matlab? I know I need to use the shooting method but how should I do it? I know I have to control the value of f'' so that it ...
4
votes
1answer
100 views

Newton's method goes to zero determinant Jacobian

I am using the Newton's method to solve $3\times3$ systems. For some particular cases, it turns out that at a given iteration, the Jacobian matrix cannot be inverted and that its determinant is very ...
3
votes
0answers
56 views

Poisson equation in frequency domain

I need some help in numerically solving the nonlinear Poisson's equation for electrons in frequency domain. The steady-state equation is: \begin{equation} \nabla.(\epsilon\nabla\varphi) = ...
0
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0answers
42 views

Solving Equations By Reducing To One Variable

To solve an equation while creating a computer algebra system, is it reasonable to take all the variables in a given system and (using equations already given) replace them with one arbitrary ...
1
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0answers
42 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 ...
6
votes
1answer
74 views

Algorithm for solving system of quadratic equations and linear equations

Let $x \in R^N$. From a Spectral Chebyshev collocation method, I have a system of quadratic and linear equations. Denote them, $$ x^T Q_i x + L_i^T x = 0 $$ and $$ A x = 0 $$ Furthermore, I know ...
0
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0answers
42 views

How to model softening of concrete?

I have a question about acquiring the softening part of the compressive stress-stain curve of concrete under uniaxial static loading. I have tried very different options: Concrete Damaged Plasticity ...
0
votes
0answers
64 views

Techniques to solve this complicated ODE

I have to solve the following ordinary differential equation: $$A(\rho, \Phi)\Phi'' + C(\rho, \rho', \Phi)\Phi' - D(\rho, \rho')\Phi^2 - \lambda \rho^4 \Phi^3 = 0 \enspace .$$ Here, prime $(')$ ...
2
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0answers
86 views

Applications of algorithm for solving systems of equations with uncertainty

We have been developing algorithms for detecting "robust" zeros of multidimensional functions $f: X\to\Bbb R^n$ where $X$ is an $m$-dimensional domain in $\Bbb R^m$. More precisely, for a given $f$, ...
2
votes
3answers
159 views

Solving nonlinear differential equations with Newton's method

I have difficulties with this equation $$\frac{d^2 u}{d x^2} + u^2 - x^2 = 0$$ with boundary conditions: $u(0)=u(1)=0$ I do not know how to solve nonlinear differential equations with Newton's ...
1
vote
2answers
77 views

2d Euler manufactured solutions

Where can I find manufactured solutions for the 2d Euler equations, with the complete analytical terms, including the Jacobian of the source term ?
1
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0answers
25 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 ...
1
vote
1answer
67 views

Solving this nonlinear system of equations

Suppose I have this set of equations: $$a = x + z\qquad (1)$$ $$b = y + \frac{z}{2}\qquad (2)$$ $$ z = k_0x\sqrt{y}\qquad (3)$$ Where $a$, $b$ $\in \mathbb{R}$ and $k_0 > 0$. The values of $a$ ...
1
vote
0answers
43 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) - ...
1
vote
2answers
60 views

Solving nonlinear boundary value problem

I have an ODE of the form $$C_1 d_y u + C_2 (d_y u)^n = C_3 y + C_4$$ with boundary conditions $u(0) = 0, d_y u(L) = 0$, where $C_1 \to C_4$ are known constants, and where $0 < n \leq 1$ is a real ...
1
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0answers
38 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 ...
0
votes
1answer
81 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 ...
3
votes
0answers
183 views

How could we solve coupled PDE with finite difference method and Newton-Raphson method?

I'm trying to solve coupled PDE by Crank-Nicolson (CN) and Newton-Raphson method with MATLAB. I have used CN method but not for coupled problem. Please if someone could help let me know to add more ...
6
votes
1answer
116 views

Newton iteration applied to nonlinear PDE

I'm having difficulty understanding how to apply Newton iteration to nonlinear PDEs and then use a fully implicit scheme to time step. For example, I want to solve Burgers equation $$u_{t} + u u_{x} ...
1
vote
1answer
69 views

Convergence problem in iterative method

I am trying to solve two non-linear equations self-consistently in a Gummel loop. Sometimes (every once in a while), I get to a situation when the loop repeats itself with wrong solutions and a ...
0
votes
0answers
38 views

Modeling a dark soliton in 1d

I have modeled, a few times, the linear wave equation, in time and space. It was nothing too complex - more of a homework assignment in Crank-Nicholson or Finite Elements. So I hear about these ...
1
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0answers
55 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: $$ (i_1 + i_p )^{\alpha} - i_p^{\alpha} - (i_2 \cdot i_p^{\alpha-1}) = 0 $$
1
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0answers
100 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?
2
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0answers
304 views

Finite difference scheme for solving nonlinear least-squares problem

I am dealing with following problem: $$ \min_{u,\gamma}\Bigg\{ \frac{1}{1000} \iint_{S_2} {\gamma (x,y)^2 dxdy} + \iint_{S_2} {[u(x,y) - u_0 (x,y)]^2 dxdy} + \iint_{S_2} {[\Delta u(x,y) - \gamma ...
1
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0answers
45 views

Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f(u) = 0, $$ $$ u|_\Gamma = u_0 $$ by Newton’s method when its convergence is global and monotonic. ...
2
votes
2answers
507 views

Solve non-linear set of three equations using scipy

I need to solve a non-linear set of three equations using scipy. However, I do not have any clue on which algorithm is suitable for my problem from a mathematical point of view (stability, ...
1
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0answers
121 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 ...
2
votes
2answers
202 views

Solving a linear equation system with pure Neumann condition

I am trying to solve a linear equation system $\textbf{A}\textbf{x}=\textbf{b}$, e.g. a Poisson equation discretized in strong form, using biCGstab method. Since there are only natural Neumann ...
2
votes
0answers
41 views

Rank deficient Jacobian in discretized periodic solutions to autonomous ODE

I'm trying to numerically find periodic solutions to different systems of autonomous nonlinear ordinary differential equations. I decided to use a finite difference scheme and solve the resulting ...
0
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0answers
63 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) + ...
1
vote
0answers
71 views

Implementing a Hill-Type Muscle Model

I'm interested in implementing the muscle model used in Geijtenbeek and Wang et al's work. Both papers link to the paper by Geyer and Herr, which describes this model: However, the paper on this ...
1
vote
3answers
110 views

Dealing with errors in non-linear least square problem

I am currently working with a optimization problem involving a non-linear least square problem. I have chosen to use lsqnonlin in Matlab. What follows is a ...
3
votes
2answers
489 views

Is “tangent stiffness matrix” the same as “stiffness matrix”?

I'm trying to implement nonzero Displacement Boundary Conditions in VegaFEM on a non-linear model, using the method outlined in §3.6.2 of University of Colorado's intro to FEM (modify $f = Ku$: set ...
0
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0answers
72 views

How to solve a nonlocal diffusion equation?

Consider a thin film with a perpendicular applied magnetic field ${H_{a}}$ (A/m) in z-axis. In fact, By increasing ${H_{a}}$ the magnetic field penetrates gradually the film. The equation for the time ...
0
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0answers
112 views

How to solve a nonlinear diffusion equation?

Consider a thin film with a perpendicular applied magnetic field Ha in z-axis. The nonlocal relation between Ha, the self-field Hself (generated by the eddy current J) and the local magnetic flux ...
4
votes
1answer
74 views

Eikonal Equation solver with different grid densities

The Fast Marching Method, Fast Iterative Method, and Fast Sweeping Method are three ways of solving the Eikonal Equation on a discrete grid, essentially just a wavefront spreading out from initial ...
4
votes
0answers
115 views

numerical analysis of a partial integro-differential equation

I have to numerically solve a nonlinear partial integro-differential equation. This is my equation, $$\frac{\partial y(x,t)}{\partial t}=\int_{-1/2}^{1/2} \frac{\pi\cos u}{\sin\pi u-\sin\pi x} ...
1
vote
1answer
108 views

OpenMP threaded nonlinear solver for complex numbers

Problem: I have translated Jacobian-Free Newton-Krylov solver written by C. T. Kelley to Fortran and now want to parallelize it on shared-memory system with OpenMP. In addition, I want to precondition ...
6
votes
2answers
205 views

Strong coupling of a non-linear multiphysic problem: failure with Newton Raphson method

I am trying to solve a multiphysic problem using finite elements and a Newton Raphson solution scheme. I have two non-linear subsystems that are coupled bi-directionally. The first subsystem includes ...
2
votes
1answer
125 views

solving for unknown inside an expectation

I have a an equation that need to find its root. The function is the following $f(\theta) \equiv E[R(\theta;\eta)]=0$ for some unknown $\theta$ which is deterministic, while the expectation is taken ...
3
votes
0answers
65 views

Finding self-kissing points on a plane curve?

I have a curve in the complex plane given by $$ f(t) = \sum_k r_k\exp(2\pi\mathrm{i}(t+\varphi_k)p_k). $$ Some of the parameters are specially chosen: $r_k>0$, $\sum_k r_k=1$, $p_k\in\mathbb{Z}$, ...
3
votes
2answers
150 views

Bessel EVP and fzero

I am trying to solve the Eigenvalue problem $$ x^2 y''+ x y' + x^2 y = \lambda^2 y\,,\quad x\in(0,1)\,,\quad y(0)=0\,,\quad y'(1)=y(1) $$ The differential equation is the Bessel equation. The solution ...
1
vote
1answer
74 views

methods for a peculiar BVP system

Consider the following system defined on the open interval (-1, 1): $y_1' = c y_3 \\ y_2' = c y_4 \\ y_3' = -f(y_1, y_2)y_2 \\ y_4' = f(y_1, y_2)y_1 $ given $ y_3(-1) = 0 = y_3(1) \\ ...
0
votes
1answer
178 views

Implement Robin Boundary Condition

This is a follow up to this question. I have a nonlinear BVP on $x=0$ to $L$: $$ (T^2)\frac{\partial^2 T}{\partial x^2} + T \left( \frac{\partial T}{\partial x}\right)^2 + Q = 0 $$ to which I apply a ...
4
votes
2answers
203 views

Large-scale nonlinear optimization problem

I want to solve a nonlinear optimization problem of the following form \begin{equation} min( \sum_i d^{x_i}c_{i})\\ 0 \leq x_{i} \leq a\\ \sum_{i} x_{i} \leq b \end{equation} $a$, $b$, $0.95 < d ...
2
votes
1answer
410 views

Solving a Nonlinear BVP using Finite Difference Method

I am trying to write a code to solve a nonlinear BVP using the Finite Difference Method. The BVP is: $(T^2)\frac{\partial^2 T}{\partial x^2} + T \left( \frac{\partial T}{\partial x}\right)^2 + Q = 0$ ...
2
votes
1answer
1k views

Nonlinear ODE to solve Duffing's equation

I am trying to solve the Duffing's equation in MATLAB. $ m\ddot{y}+c\dot{y}+ky+k_{3}y^{3} = f(t) $ where $ f(t) = A \sin{\omega t}$ To do that I wrote a function to be given to the ode45. ...
0
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0answers
34 views

Solving a nonlinear equation with a Markov process and RVs

Assume that we have the following equation and the following assumption. The scope is to solve for some particular variables expressed later. Update $$E_{t}\left[ b(A_{t+1})^{1-\gamma} ...
2
votes
1answer
143 views

Solving a nonlinear equation with random variable

I would like to solve an equation that looks like this UPDATE $E[(R^{1-\gamma})(r_k+\theta-r_z)]=0$ , where $R=\phi r_z+(1-\phi)(r_k+\theta)$ and $\phi\in[0,1]$, $\theta$, is a random variable ...