Hello. I'm an electrical engineering undergrad and I'm interested in optical nonlinear effects. I understand that is all about this type of differential equation. I have an undergrad knowledge of calculus, algebra, basic differential equations and electromagnetic theory. What is a good way to understand this topic? What are the requirements? Thanks.
I have just released Gomez - a pure Rust library for solving nonlinear systems of equations.
The goals are:
- Derivative-free methods so that users don't need to care about providing gradient or Jacobian. Methods that are based on Jacobian use finite difference technique that should be fine in practice.
- Application in real-world problems. For example, the library supports specifying variable bounds (useful if the variables have some physical meaning for instance). More advanced constraints are currently out of scope, but contributions for their support are welcome.
- Global convergence. Numerical algorithms are very powerful, but also sensitive to initial guesses. I would like to provide tools for helping to overcome this issue.
- Control over the process. Solvers implement a low-level, iterative interface (similar to argmin) and should provide many settings to tweak them.
- Performance.
Supported algorithms in the initial version:
- Trust region -- This is mostly dogleg method with Levenberg-Marquardt fallback when Jacobian is singular. From simple benchmarks and my experience, it seems to be comparable with GSL hybrids implementation, although I would like to extend the benchmark and fix some todos before making any specific statement.
- Cuckoo search -- This is a first attempt to help with the sensitivity to initial guesses mentioned above. Basically any global optimization algorithm has a potential to be helpful, but this particular algorithm worked quite well for me in the past.
- Nelder-Mead (simplex) -- This was just an experiment, not recommended in general.
My main focus in the future will be on the global convergence topic, I have some interesting articles that I want to try.
Hello everyone,
I am trying to solve a set of coupled non-linear differential equations using ode45 but i am not getting the desired results. By desired results I mean, setting all the initial conditions to be zero and setting torques for both joints to be 0, there should be no change in coordinate or change in velocity of the manipulator in other words if you plot the solution of the ode. It should be a horizontal line parallel to the time axis. But this is not the case when I run the code. Given below are the set of equations that I am trying to solve numerically:
https://preview.redd.it/zhe6jlawgk981.png?width=1033&format=png&auto=webp&s=81f08680b9e08ab50a130ab283b3a5953e7d56f2
And this is the code that i am using to solve the above system :
function xdot = DynOde(t,y)
%% init constants;
m1 = 5;
m2 = 2;
a1 = 0.34;
a2 = 0.34;
g = 9.81;
T1 = 0;
T2 = 0;
x1dot = y(2);
x1ddot = (T1*a2 - 2*a2 - 2*a1*cos(y(3)) - a1*a2*g*m1*cos(y(1)) - a1*a2*g*m2*cos(y(1)) + a1*a2^2*m2*sin(y(3))*y(2)^2 + a1*a2^2*m2*sin(y(3))*y(4)^2 + a1*a2*g*m2*cos(y(3))*cos(y(1) + y(3)) + a1^2*a2*m2*cos(y(3))*sin(y(3))*y(2)^2 + 2*a1*a2^2*m2*sin(y(3))*y(2)*y(4))/(a2*(a1^2*m1 + a1^2*m2 - a1^2*m2*cos(y(3))^2));
x2dot = y(4) ;
x2ddot = (T1*a2 - 2*a2 - 2*a1*cos(y(3)) - a1*a2*g*m1*cos(y(1)) - a1*a2*g*m2*cos(y(1)) + a1*a2^2*m2*sin(y(3))*y(2)^2 + a1*a2^2*m2*sin(y(3))*y(4)^2 + a1*a2*g*m2*cos(y(3))*cos(y(1) + y(3)) + a1^2*a2*m2*cos(y(3))*sin(y(3))*y(2)^2 + 2*a1*a2^2*m2*sin(y(3))*y(2)*y(4))/(a2*(a1^2*m1 + a1^2*m2 - a1^2*m2*cos(y(3))^2));
xdot = [x1dot;x1ddot;x2dot;x2ddot];
end
Please let me know if I have written the correct vector field representation of the two equations in the picture.
Any advice would be of great help.
Thank you.
One who has knowledge of the Schrödinger Equation, will understand that it literally describes the law of attraction, aswell our non-dual nature.
When plotted on a graph in time, it can be observed the equation follows a straigt line on a point in "time". The imaginary "point" that follows the imaginary "line" is what in non-dual teachings / Taoism / Buddhism is called our "true nature".
It is this very equation that proves law of attraction. As at the very vore of the equation, the input desciibes the output. Simpler said: That what you believe, will shape your reality in some form.
Hello. I have this system, which i can't find the optimal steps to solve.
2x^2-3xy+y^2=3
X^2+2xy-2y^2=6
The first equation in the system can be factorised, but it doesnt lead anywhere. Also it's possible to add them, which equals to 3x^2-xy-y^2=9, but i dont see how i could move it from here? Or do i need to take completely different approach? Thanks in advance.
People usually refer to matrices, vectors, etc. as linear algebra. But for example in structural mechanics and fluid dynamics, nonlinear equations are solved using matrices and vectors in an iterative manner. Is it still called linear algebra in this scenario? I haven't heard people using the term "nonlinear algebra".
Title says it all. From what I can tell it doesn't seem very related to how I think about linear or nonlinear functions.
Github - https://github.com/quantum-visualizations/qmsolve
QMsolve seeks to provide an easy solid and easy-to-use solver, capable of solving the Schrödinger equation for one and two particles, and creating descriptive and stunning visualizations of its solutions both in 1D, 2D, and 3D.
Example of the simulation of the eigenstates of a particle confined in two wells
Installation
pip install qmsolve
How the simulator works
The way this simulator works is by discretizing the Hamiltonian with an arbitrary potential, specified as a function of the particle observables. This is achieved with the Hamiltonian constructor.
Then, the Hamiltonian.solve the method efficiently diagonalizes the Hamiltonian and outputs the energies and the eigenstates of the system. Finally, the eigenstates can be plotted with the use of the visualization class.
The visualization.superpositionsmethod features the possibility of interactively visualizing a superposition of the computed eigenstates and studying the time dependence of the resulting wavefunction.
For a quick start, take a look at the examples found in the examples subdirectory.
Writing my PhD in literary studies, trying to make sure my scientific info is accurate.
Obviously, I recognize that the compound structure of the sentence probably simplifies the formalism of quantum mechanics beyond the point of usefulness, but still, is it wrong? Does it miss the relationship between various concepts?
If you can think of a better way to express all of the above in one sentence I won't say no to reading it.
Thanks!
I have the wavefunction in terms of x (radius) and X_0 (some constant maximum radius), as well as the potential term for a spherically symmetric potential for an l=0 system. I have the general form of the expectation energy for such a system, and am trying to find an expression for the expectation energy of for a specific given wavefunction by substituting into the general form. However, the integration limits in the given general form are (0, ∞). My wavefunction and potential function are valid for |x|<=X_0, and 0 elsewhere. I am unsure whether I should change the integration limits to (0,X_0), or to (-X_0,X_0). I have attempted to find the expression using the latter, and it was very messy. Integrating between (0,X_0) makes more sense to me physically, as a negative radius value doesn’t seem right, and that the original expression gives the lower integration limits as 0. But I am unsure due to the wavefunction being defined as valid for the modulus of x being less than the maximum radius. If somebody could tell me which integration limits I should use, and why, it would be much appreciated. Thanks!
