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Overview. The Hamilton–Jacobi equation is a first-order, non-linear partial differential equation. for a system of particles at coordinates . The function is the system's Hamiltonian giving the system's energy. The solution of the equation is the action functional, , [ 4] called Hamilton's principal function in older textbooks.
A one-way wave equation is a first-order partial differential equation describing one wave traveling in a direction defined by the vector wave velocity. It contrasts with the second-order two-way wave equation describing a standing wavefield resulting from superposition of two waves in opposite directions (using the squared scalar wave velocity ...
In computational physics, the term advection scheme refers to a class of numerical discretization methods for solving hyperbolic partial differential equations. In the so-called upwind schemes typically, the so-called upstream variables are used to calculate the derivatives in a flow field. That is, derivatives are estimated using a set of data ...
Godunov's scheme. In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by Sergei Godunov in 1959, [ 1] for solving partial differential equations. One can think of this method as a conservative finite volume method which solves exact, or approximate Riemann problems at each inter ...
Rate equation. In chemistry, the rate equation (also known as the rate law or empirical differential rate equation) is an empirical differential mathematical expression for the reaction rate of a given reaction in terms of concentrations of chemical species and constant parameters (normally rate coefficients and partial orders of reaction) only ...
Hamilton–Jacobi–Bellman equation. The Hamilton-Jacobi-Bellman ( HJB) equation is a nonlinear partial differential equation that provides necessary and sufficient conditions for optimality of a control with respect to a loss function. [ 1] Its solution is the value function of the optimal control problem which, once known, can be used to ...
The convection–diffusion equation is a relatively simple equation describing flows, or alternatively, describing a stochastically-changing system. Therefore, the same or similar equation arises in many contexts unrelated to flows through space. It is formally identical to the Fokker–Planck equation for the velocity of a particle.
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