Archive of posts with tag “ode”
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From Picard iteration to Feynman path integral
The Schrödinger equation is an ODE, so we can approach its solution through Picard iteration. This approach leads to a sum over walks on the graph formed by an orthonormal basis as vertices and the Hamiltonian matrix elements as edge weights. This sum is exactly the Feynman path integral if we choose the position basis and take the continuum limit. -
Eigenfunctions of the Laplacian on an annulus with homogeneous Neumann boundary condition
Several features of the eigenfunctions of the Laplacian on an annulus with homogeneous Neumann boundary condition are discussed. The distribution of the eigenvalues is discussed in detail, making use of a phase angle function called . The limiting cases of a disk and a circle are discussed. -
Solving ODE by recursive integration
By recursively integrating according to from , we can get the solution of the ODE
with initial conditions as the limit of the sequence of functions. -
Solving linear homogeneous ODE with constant coefficients
By using power series, we can prove that the problem of solving linear homogeneous ODE with constant coefficients can be reduced to the problem of solving a polynomial with those coefficients. This article illustrates this point in detail, but it uses a very awful notation… -
Simulating a mechanical system using rpg_core.js
Continuing my last work of simulating a mechanical system using RGSS3, I made a new version using rpg_core.js, the game scripting system shipped with RPG Maker MV. This version is live on web! -
Simulating a mechanical system using RGSS3
Hamiltonian mechanics gives us a good way to simulate mechanical systems as long as we can get its Hamiltonian and its initial conditions. I implemented this simulation in RGSS3, the game scripting system shipped with RPG Maker VX Ace. -
The concentration change of gas in reversible reactions
A reversible elementary reaction takes place inside a closed, highly thermally conductive container of constant volume, whose reactants are all gases. Given the reaction equations and the reaction rate constants, a natural question to ask is how the concentration of each gas changes w.r.t. time. In this article, I will answer this question by proposing a general approach to solve it.
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