Grand Challenge Equations
We call the equations shown below the
grand challenge equations because they are a broad selection of the expressions that relate fundamental quantities in the various computational sciences.
Newton's equation is the fundamental equation of classical mechanics, relating the force acting on a body to the change in its momentum over time. Computational scientists and engineers use it in all calculations involving moving bodies, from civil engineering to astrophysics.
(Sir Isaac Newton, 1642-1727.)
The
Schrödinger equation is the basic equation of quantum mechanics. In time-dependent form, it describes the evolution of atomic-scale motions and is widely used by chemists studying the dynamics of chemical reactions.
(Erwin Schrödinger, 1887-1961.)
The
Navier-Stokes equation is the primary equation of computational fluid dynamics, relating pressure and external forces acting on a fluid to the response of the fluid flow. Forms of this equation are used in computations for aircraft and ship design, weather prediction, and climate modeling.
(Claude Louis Marie Navier, 1785-1836; Sir George Gabriel Stokes, 1819-1903.)
The
Poisson equation is a fundamental equation of mathematical physics, describing the spatial variation of a potential function for given source terms. It is applicable in a wide range of studies, from electrostatics to ocean modeling.
(Denis Simeon Poisson, 1781-1840.)
The
heat equation, also called The
diffusion equation, describes, e.g., temperature distribution in a region as a function of space and time. It is used in studies of reaction-diffusion systems, especially in biology. Such systems may explain, for example, how the leopard gets its spots.
The
Helmholtz equation is used in acoustics and electromagnetic studies. It arises, for example, in the analysis of vibrating membranes, such as the head of a drum.
(Hermann Ludwig Ferdinand von Helmholtz, 1821-1894.)
The
discrete Fourier transform is the equation from which the Fast Fourier Transform (FFT) algorithm is derived. The FFT is one of the most significant numerical techniques of modern applied mathematics, because it helps reduce differential equations to essentially algebraic problems. It describes how any system of data can be broken down into a sum of sine and cosine functions, expressible as a vector-matrix multiplication. FFTs are used in the analysis of wave phenomena, x-ray crystallography, and signal processing.
(Jean Baptiste Joseph Fourier, 1768-1830.)
Maxwell's equations describe the relationship between electric and magnetic fields at any point in space as a function of charge density and electric current at such a point. The wave equation for the propagation of light is derivable from Maxwell's equations, and these are the basic equations of classical electrodynamics. They are used throughout the science of plasma physics, in studies of the earth's magnetic field and its interaction with incoming particles from the sun and the rest of the cosmos, as well as in studies related to the development of fusion power.
(James Clerk Maxwell, 1831-1879.)
The
partition function is a central construct in statistics and statistical mechanics, and it is a bridge between thermodynamics and quantum mechanics because it is formulated as a sum over the energies of states of a macroscopic system at a given temperature. It is widely used in condensed-matter physics and is employed in theoretical studies of high-temperature superconductivity.
The equation for
population dynamics, also called the
logistic equation, is a classical starting point for the dynamics of quasiperiodic and chaotic systems. It describes the fluctuations in populations of animals as functions of their birth rate and the available resources. The addition of terms related to predation leads to
predator-prey models, widely studied in computational biology.
The equation combining the
first and second laws of thermodynamics relates the internal energy, the entropy, and the volume of a system. It is employed in the analysis of engine performance and other engineering studies.
The
radiosity equation is used in the construction of realistically lit scenes in computer graphics, because it can account for light falling on a scene from a light source and then being absorbed, reflected, and diffused throughout the elements of the scene. It must be solved at all wavelengths of input light, and thus results in the computationally intensive nature of computer graphics renderings.
The
rational B-spline is an equation that permits the modeling of free-form curves between two endpoints as a function of a set of intermediate points whose values influence the shape of the curve. It is related to a nineteenth-century curve-drawing technique in which metal weights (called "
ducks") were placed at the data points and a thin, elastic wooden beam (called a "
spline") was passed between the ducks, forming a smooth, aesthetically pleasing curve. The rational B-spline is the quotient of two B-splines, and is used to represent conic curves and quadric surfaces (spheres, cylinders, and cones). The equation is commonly used by engineering designers in the analysis of data sets and in their visualization.
