In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.
In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.
As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.
Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:
Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.
Homework Statement
y+4y^2=(y^(4)+x)y', IC: y(1)=1
Homework Equations
?
The Attempt at a Solution
Ive tried to figure out a substitution that will make this linear, and i can't seem to figure one... I am unsure of how to approach this?
Hi!
I have to solve the nonlinear equations of motion in the article (16) (17) (18).
I Trasform the system in a system of first order differential equations but i don't have the initial conditions. Is it possible to solve it with the ode45 MATLAB function?
this is the problem:
x2y' = (2y2 - x2)
here's what i have done so far:
dy = (2y2/x2 - 1)dx
(2y2/x2 - 1)dx - dy = 0
i used the substitution y = xv then found an integrating factor and got
dx/x - dv/(2v2 - 2v - 1) = 0
but i am stuck at this point..
i know ln(x) + C is the first part...
Homework Statement
Solve the diferential equation: dx/dt + Rx^2 + G = 0
G constant = 10^18
R constant = 10^-10
Initial conditions x(0) = 10^8Homework Equations
what approach to take?
The Attempt at a Solution
First I try to apply bernoulli, but since in this equation I do not have a term...
1. For each of the following equations, state the order and whether it is nonlinear, linear inhomogeneous, or linear homogeneous; provide reasons.
(a) ut-uxx+1=0
(b) ut-uxx+xu=0
(c) ut-uxxt+uux=0
(d) utt-uxx+x2=0
(e) iut-uxx+u/x=0
(f) ux(1+u2x)-1/2+uy(1+u2y)-1/2=0
(g) ux+eyuy=0
(h)...
Hello.
Can anyone tell me what would be the best software to use for the nonlinear regression? I have an equation with two unknown parameters which I would like to find out. Equation looks something like this: f(x)=a*b*x/(1+b*x), where x and f(x) are known (several points) and I would like to...
Nonlinear Dielectric Heating and more!
Hello all,
Here is an interesting query for some work I am trying to figure out. Everyone knows about dielectric heating, which is the principle on which microwave ovens operate. Without deriving the formula for the power dissipated into an object...
What is the best approach for obtaining the inverse of a system of equations involving nonlinear equations?
Say:
3x^2 - 2y = i
x + y = j
Solving for x and y in terms of i and j?
Note: This is not a homework problem, just a general question.
Can anybody please suggest some references (preferably review articles or lecture notes etc, freely available online) for learning linear and nonlinear sigma models and their applications in particle physics?
Modelling a solid bed cooled by an uprising gas flow gave the following system of 1st order NONLINEAR PDEs
dv/dt + f(t) * dv/dx = h*(u-v)
du/dt = -h*(u-v)
BC
v(x=0,t) = 0
dv/dx(x=1,t)=0
IC
u(x,t=0) = 1
where:
u, v are temperatures function of (x,t) range [0,1]...
Hi all,
I am having a hard time solving a partial second order differential equation with an imaginary part. I basically took a much bigger function with real and imaginary parts and simplified it down to this. I also know the solution to a similar equation (shown in image). Any help would...
Nonlinear 1st order ODE
\frac{dH}{dt}=B-A*(H-Z)^{3/2}
where:
B,A and Z are known values
H=f(t); H is function of t
I've already solve this ODE numerically using a 4th order RK routine. But my question is, it is possible to get an analytical solution for H(t)?
I would like to perform a complicated nonlinear fit with mathematica, which
involves a numerical integral in the fitted model. But i can't get it to work.
If I say something like:
NonlinearRegress[data, NIntegrate[f[x, p1, p2, y], {y, 0, 1}], {p1, p2}, x]
it says the integrand evaluated...
Hi
From Wiki (http://en.wikipedia.org/wiki/Nonlinear_optics): "Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light."
My...
Here is the equation I don't know how to solve:
\begin{aligned}
\left( {\frac{{{{\rm{d}}^2}}}{{{\rm{d}}{t^2}}} + \beta _1^2} \right){u_1} = {g_1}u_2^{}{u_3} \\
\left( {\frac{{{{\rm{d}}^2}}}{{{\rm{d}}{t^2}}} + \beta _2^2} \right){u_2} = {g_2}u_1^{}{u_3} \\
\left(...
Homework Statement
y''+4\left(y'\right)^{2}+8=0
Homework Equations
u=y'?
The Attempt at a Solution
I don't really know where to start, do I use u=y' substituted? So, y''=u*(du/dy)?
That leads to u\frac{du}{dy}+4u^{2}+8=0
I don't think this is correct, since it leads to...
Homework Statement
y'(t) = (y-5t)/(y+t) IVP: y(1)=0
Homework Equations
M(x,y)dx+N(x,y)dy = 0
- or do i use -
y'+p(x)y=q(x)
The Attempt at a Solution
Well I used the first equation (with M and N):
1. first checked that it was exact, by taking the partial of M and N with respect to y and t...
High
I am working on the geometrically nonlinear finite element method. I now that in this method, the internal forces on each node must be equal to external forces. How can I calculate internal forces?
Hi. I'm not even sure if I'm posting this in the best forum!
I'm having a lot of trouble grasping parts of this paper..
Eur Biophys J. 2009 Jun;38(5):637-47. Epub 2009 Mar 4.
A nonlinear model of ionic wave propagation along microtubules.
Specifically, they use a phase space plot that...
From a textbook - The reason why the polarization plays a key role in the description of nonlinear optical phenomena is that a time-varying polarization can act as the source of new components of the electromagnetic field... the wave equation in nonlinear optical media often has the form...
Nonlinear DE (with e^t) ?
Good day forum,
I have this wonderful DE :
dx/dt = [a - f '(t)]x + (b + d(c^t))(x^2) - 1
with,
t \in [s,T]
x(T) = 0
a, b, d & c are constants.
f(t) = g + h(k^t) , where g, h & k are constants (but I think specifying this is of no...
Hi All - I am trying to immerse myself in NLO and purchased Robert W. Boyd's Third Edition on Nonlinear Optics. I'm already struggling just 3 pages into the book.
We are looking at the polarization of a material in a NLO chromophore, so:
P(t)=\epsilon [X(1)E(t)+X(2)E2(t)+X(3)E3(t)...]...
Hello everybody,
could you please direct me how to solve this nonlinear differential equation analytically, so by mathematica or matlab? I really need to solve it for my research project, so please help me
du/dx=d/dx[a*u^(-1/2)*du/dx]-n*u^(3/2)*(u-c)/b
boundry conditions are:
u(0)=b+c...
How to solve this nonlinear PDE? Please help!
Hello Everyone,
I am trying to solve the following nonlinear PDE which is driven from the Hamilton Jacobi Bellman (HJB) equation in ergodic control of a nonlinear dynamical system.
v\nabla_x h - \frac{1}{4}\|\nabla_v h\|^2 + \frac{1}{2} \sigma...
Hi,
I am a physics student who just finished his bachelor thesis in nonlinear dynamics. I am about to start the physics master programme at my home university (in Germany). Within a year, I will have to start writing my master thesis, so I'm trying to figure out in which domain I want to work...
How to detect redundant equation from a system of nonlinear equation?
It means how to find out a system of nonlinear equation is "linear independence"?
One equation from the system can not be represented by the others in the system of nonlinear equation.
Hello guys,
I am not good in advance mathematics.
I have system of nonlinear equation and I want to solve it analytically, but I face some difficulties.
Homework Statement
The system is :
(a-b)y^2-exy^2-2fx=-c
2fx^2-2cx-2fy^2=-d
Homework Equations
where: a,b,c,d,e,and f...
Hello everybody. I have a quick question. I have the following system of nonlinear differential equations:
(di/dt)(x)+(x^2 - dx/dt)(i)+ v(t) =0 _ _ _1
dv/dt = i/C _ _ _2
I know my Initial Conditions: i(0) = 0, di(0)/dt = 0, x(0) = L, dx(0)/dt = 0, v(0)=V
PS- x is displacement, t is time, i...
Homework Statement
y' + Ay2 = B
A & B are constants and y is a function of x
Find the general solution to the differential equation. (Find y(x)).
Homework Equations
The Attempt at a Solution
This differential equation came up when I was trying to solve a problem in...
I derived an equation describing the free surface of an electrified fluid. I am currently seeking traveling wave solutions for this problem, the equation I am looking at is (1-F) f+\frac{1}{90}h^{4}f^{(4)}+\frac{3}{4h}f^{2}-\frac{1}{2}\Bigg( B-\frac{1}{3}\Bigg)...
I have been working on the problem of electrified fluid flow down a channel with a moving pressure distribution. I have derived an equation which describes the free surface of said fluid flow which is a Benjamin-Ono like equation. I have a numerical solution for this equation and it gives the...
First order nonlinear ODE -- Integrating factor + exact differentials, or not?
Hello everyone,
(I apologize if this did not format properly, if not I will attempt to edit it if that functionality is available upon submitting a question).
I recently came across the following nonlinear ODE...
Hello,
Can you give some suggestions to solve the following system of 1st order nonlinear differential equations?
Thank you.
\[
\begin{array}{l}
u'(t) = Au^2 (t) + B(t)u + C(t) \\
u(t) = \left[ {\begin{array}{*{20}c}
{x_1 (t)} \\
{x_2 (t)} \\
\end{array}} \right] \\
A = \left[...
Given the ODE system:
v' = u(u2-1)
u' = v-u
Define w=u2+v2. Compute w'.
Find the largest radius R for which u2+v2<R so that the if the solution curve (u,v) is inside that circle the solution tends to (0,0) as t--> +\infty
Any guidance would be appriciated !
Dear Everyone,
I am working on a physics problem of exciton diffusion involved in organic optoelectronics.
It is in the form of
y''+a*y+b*y^2=0.
Is there a general solution to this equation?
Thanks!
elfine
What conditions are necessary to use the constitutive relations for Maxwell's equations? I am working in a nonlinear media, but am a little confused about whether I can assume isotropy or not.
If I am assuming the media is nonlinear is it necessarily anisotropic? Or, is it possible to have...
I have the following equation to solve:
u_{tt}=12{u_x}^2+12u_{xx}+{u_x}^4+6u_{xx}{u_x}^2+4u_{xxx}u_x+3{u_{xx}}^2+u_{xxxx}
I have been told to look into FDM or FEM. My question, is it possible to code something in MATLAB to solve this and if so what is the best method to use and how do I do...
second order nonlinear ODE -- not autonomous
This equation has arisen from a steady state problem of diffusion with nonlinear reaction:
(dependent var=c; independent variable=x)
c'' = ko + k1 c + k2 c^2 + k3 c x
ko, etc are constants.
I can obtain a solution if I drop the last term...
Hey guys I'm completing an experiment and I'm required to extrapolate data by performing nonlinear curve fitting.
I have 3 sets of data: (x1, y1); (x2, y2); (x3, y3).
I'm required to fit the above mentioned sets of data to the following equation ...
What is meant by "waveform". Working in strogatz nonlinear dynamics, global bifurcati
Homework Statement
Consider the system r' = r(1-r^2), O' = m - sin(O) for m slightly greater than 2. Let x = rcos(O) and y = rsin(O). Sketch the waveforms of x(t) and y(t). (These are typical of what one...
Hi everyone,
I'm having a hard time analyzing the following problem:
b v(x) = -exp(-x) - 1/2 ( g v'(x) )^2 - n x v'(x) + S(g) v''(x)
where:
v' = dv/dx, etc.
0 < b< 1
g > 0
n > 0
S(g) >0 and S'(g) >0
x \in (-inf, inf)
The main goal is to figure out what happens as g...
Homework Statement
I'm working on a problem for my robotics class and could really use some help. I am suppose to be modeling a planar scara manipulator and have managed to come up with two nonlinear differential equations that describe the system; they are shown below. \Theta_{1} and...
In class, my teacher gave the following equations as examples of linear and nonlinear ODE. In the first equation, there are x's in front of some of the y's yet it is linear. In the second equation, there is an x in front of y^2 yet it is nonlinear - why? Also, why is the final equation...
So I thought I would try to figure out the maximum distance of our 7.62x54 round. obviously I used F=ma and for resistance I used Fdrag=(mass of bullet*density of air*v^2)/ballistic constant. the v^2 terms makes the equation nonlinear and not exactly solvable right? also, the angle at which...
Nonlinear ODE's versus "Statistics".
Hi all,
My university offers a course on Probability followed by a more theoretical course on Statistics (not sure how standardized the names are).
They also offer a nonlinear ODE course.
From a physics major's point of view, which subject is harder...