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

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4
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2answers
73 views

Finding quick solution to a collection of systems of fairly simple but nonlinear equations

So I have a collection of systems of equations, basically $n$ systems of equations, each composed of $k$ equations: $$\frac{a_1x_{1j}}{a_1x_{1j} + \cdots + a_kx_{kj}} + \log x_{1j} + 1 - B_{1j} = 0$$ ...
4
votes
2answers
50 views

numerical approach for system of non-linear partial-ordinary differential equations

I am interested in the numerical solution of the following system of non-linear partial-differential algebraic equations, where the independent variables are $X$ and $T$, representing non-dimensional ...
0
votes
0answers
23 views

Solving nonlinear coupled PDE using FiPy

I have recently been trying to solve these 6 coupled, nonlinear PDEs of the general form: $ \frac {\partial N_1}{\partial t} = -a(P_2E_1 + P_1E_2) - bN_1 \\ \frac {\partial N_2}{\partial t} = -c ...
3
votes
1answer
60 views

Efficient solution of large systems of non linear algebraic equations

I have quite simple problem of FEM solution of 1D differential non-linear equation. Altough the problem itself is simple, the numerical solution of the arising system of non-linear algebraic equations ...
1
vote
0answers
23 views

Numerical solution of non-linear advection equation other than inviscid burgers

I am solving a non-linear advection equation of the form $u_t + f(u)_x = 0$ where $f(u)$ is a complicated function of $u$. I am solving this equation using a first order fully implicit scheme (...
1
vote
0answers
41 views

Solving system of constrained linear and non-linear equations in MATLAB

Solving system of constrained linear and non-linear equations in MATLAB I'm solving a FEM problem in MATLAB with use of the direct stiffness method. The problem is now formulated as a system of nn ...
0
votes
1answer
38 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 ...
0
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0answers
15 views

Roots of zeros dimensional system of polynomial equations

I have a zeros dimensional system of polynomial equations with as many equations as variables (10 equations and 10 variables) and monomials have degree at most 7. My goal consists in computing all its ...
0
votes
1answer
54 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
70 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)\\ \end{...
0
votes
0answers
45 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
65 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
110 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
80 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
133 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} x^...
1
vote
2answers
61 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
66 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
63 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
174 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= \begin{...
0
votes
0answers
40 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 ...
2
votes
0answers
65 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
51 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
42 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
84 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
169 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 ...
0
votes
1answer
160 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}\...
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
143 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
62 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) = q\left(n_i\...
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 variable?...
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
115 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
59 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
199 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
116 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 ?
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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
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 $a$ ...
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vote
0answers
47 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) - \sqrt{n+1}\Phi_i^{*}\...
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 ...
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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
115 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'...
3
votes
0answers
242 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
149 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
74 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
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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 ...
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vote
0answers
59 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: $$ (i_1 + i_p )^{\alpha} - i_p^{\alpha} - (i_2 \cdot i_p^{\alpha-1}) = 0 $$