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

learn more… | top users | synonyms

0
votes
1answer
34 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 ...
1
vote
0answers
68 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)\\ ...
0
votes
0answers
41 views

What does a negative time stepping mean? (Adaptive time stepping)

Summary behind the problem: The following code aims at solving a static elasto-plastic problem. Like a 2D square mesh based on an elasto-plastic constitutive model like Von-Mises or Drucker-Prager ...
6
votes
1answer
62 views

Evolving nonlinear Schrodinger equation with higher-order algorithms?

First I will give the relevant information for my question, and then I'll ask the question. $\large{\textrm{Background}}$ For evolving the nonlinear Schrodinger equation (NLS), one typically uses ...
4
votes
1answer
90 views

Methods of solving non-linear advection-diffusion systems beyond Newton-Raphson?

I'm working on a project where I have two adv-diff coupled domains through their respective source terms (one domain adds mass, the other subtracts mass). For brevity, I'm modeling them in steady ...
2
votes
1answer
72 views

Pseudo Code for non linear power function fit needed

I am struggling finding pseudo Code for a non-linear fit of the following function: $y = a\, x^b$ Package NLS in R does perform well, but utilizing external software is not practicable in my program ...
1
vote
1answer
130 views

How to solve the problem without using symbolic computation

I have the following simple nonlinear equations with two unknowns only: $$\left\{ \begin{array}{c} \int_1^2{\dfrac{ e^{{a_1} x+{a_2} x^3}}{1+x^2}} \, dx=1 \\[13pt] \int_1^2{ x^2 e^{{a_1} x+{a_2} ...
1
vote
2answers
59 views

The second variation of displacement interpolation function in Finite Element Method

I need to calculate the second variation of displacement interpolation function $u = \sum N_a u_a$ in Finite Element Analysis, where $N_a$ are the shape functions and $u_a$ are the nodal values. ...
4
votes
1answer
65 views

What is the preferred method for evolving the Nonlinear Schrödinger Equation?

I am interested in evolving the (cubic) self-focusing nonlinear Schrödinger equation, $$i\frac{\partial \psi}{\partial t}+\frac{1}{2}\frac{\partial^2 \psi}{\partial x^2}+\left|\psi\right|^2\psi=0$$ ...
0
votes
0answers
61 views

slow convergence in displacement using newton method in nonlinear FEM

I have been running a code on crack propagation using phase-field, and viscosity is considered. My code works okay for low viscosity but runs into some converging issue when viscosity becomes high. ...
0
votes
0answers
14 views

multivariate root finding in a finite domain

I am looking for an algorithm that is able to find a root of two-dimensional function in a finite domain. I have the following function: $$G(x,y) = \left[f_{1}(x,y), f_2(x,y) \right]$$ and I want to ...
5
votes
1answer
170 views

Solving a set of linear equations with block structure and weak coupling

I have a standard set of linear equations $Ax=b$ where the Hessian matrix $A$ has the special block structure as shown: $A= \begin{pmatrix} T & U\\ U^T & V \end{pmatrix}$, $x= ...
0
votes
0answers
38 views

Finding the one solution of a nonlinear system of equations, using default values if under-constrained

I'm not sure if this is a math question or a programming question, but it is scientific computation so I hope it fits here. I have a system of equations in which I can symbolically solve any ...
1
vote
0answers
64 views

Solving a nonlinear poisson equation via variational minimization

I am kind of new in finite elements and I am solving simple "Poisson nonlinear" problem. $- \nabla ((1 + u^2) \nabla u) = f$ $u = 0 \ \text{on} \ \Omega $ I am using Newton solver, where I have ...
3
votes
1answer
44 views

Implementation of Backward-Euler scheme, Newton-Raphson iteration scheme to time dependent nonlinear differential equation

I just knew how to do Newton-Raphson iteration in time-independent 1D nonlinear differential equation. Then I applied to time-dependent 1D nonlinear differential equation, and I got confused. Below ...
0
votes
1answer
33 views

How to use time delays in the solvepde function in MATLAB for a system of PDEs?

The solvepde function was introduced in MATLAB R2016a. I am able to solve my system of PDEs if there are no time delays involved. Does anyone know how to include time delays in the solvepde function?
3
votes
1answer
60 views

Solving $\sum_r \frac {\mathrm B_{z_r}(a+m_r+1,b)}{\mathrm B_{z_r}(a+m_r,b)}=K$ for $a$ and $b$

How to numerically find the solution for $a$ and $b$ of this equation $$\sum_r \frac {\mathrm B_{z_r}(a+m_r+1,b)}{\mathrm B_{z_r}(a+m_r,b)}=K$$ where $m_r$ are non-negative integers, $0<K<1$, ...
-1
votes
1answer
77 views

Stability analysis for coupled nonlinear system of partial differential equations

I'm trying to solve a nonlinear partial differential equation \begin{equation} L(u_{xxtt},u_{xx}u_{tt},u_{xt}^2,u_{xt},u_{tt})=0 \end{equation} using finite difference methods. In order to remove the ...
6
votes
1answer
161 views

What does the Von Neumann's stability analysis tell us about non-linear finite difference equations?

I am reading a paper [1] where they solve the following non-linear equation \begin{equation} u_t + u_x + uu_x - u_{xxt} = 0 \end{equation} using finite difference methods. They also analyse the ...
-1
votes
1answer
125 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
136 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
60 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
votes
0answers
43 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
vote
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
104 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
votes
0answers
57 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
66 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
votes
0answers
87 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
186 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
106 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
vote
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
79 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
46 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
63 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
vote
0answers
39 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
100 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
233 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
144 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
71 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
41 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
vote
0answers
58 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
vote
0answers
110 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
votes
0answers
306 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
vote
0answers
46 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
923 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
vote
0answers
149 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
221 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
43 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
votes
0answers
69 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) + ...