Numerical methods for ordinary differential equationsNumerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.
Numerical linear algebraNumerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Computers use floating-point arithmetic and cannot exactly represent irrational data, so when a computer algorithm is applied to a matrix of data, it can sometimes increase the difference between a number stored in the computer and the true number that it is an approximation of.
Floridan aquiferThe Floridan aquifer system, composed of the Upper and Lower Floridan aquifers, is a sequence of Paleogene carbonate rock which spans an area of about in the southeastern United States. It underlies the entire state of Florida and parts of Alabama, Georgia, Mississippi, and South Carolina. The Floridan aquifer system is one of the world's most productive aquifers and supplies drinking water for nearly 10 million people.
Longshore driftLongshore drift from longshore current is a geological process that consists of the transportation of sediments (clay, silt, pebbles, sand, shingle) along a coast parallel to the shoreline, which is dependent on the angle of incoming wave direction. Oblique incoming wind squeezes water along the coast, and so generates a water current which moves parallel to the coast. Longshore drift is simply the sediment moved by the longshore current. This current and sediment movement occur within the surf zone.
Numerical integrationIn analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for numerical integration, especially as applied to one-dimensional integrals.
Tidal boreA tidal bore, often simply given as bore in context, is a tidal phenomenon in which the leading edge of the incoming tide forms a wave (or waves) of water that travels up a river or narrow bay, reversing the direction of the river or bay's current. It is a strong tide that pushes up the river, against the current. Bores occur in relatively few locations worldwide, usually in areas with a large tidal range (typically more than between high and low tide) and where incoming tides are funneled into a shallow, narrowing river or lake via a broad bay.
Tidal powerTidal power or tidal energy is harnessed by converting energy from tides into useful forms of power, mainly electricity using various methods. Although not yet widely used, tidal energy has the potential for future electricity generation. Tides are more predictable than the wind and the sun. Among sources of renewable energy, tidal energy has traditionally suffered from relatively high cost and limited availability of sites with sufficiently high tidal ranges or flow velocities, thus constricting its total availability.
Finite differenceA finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted is the operator that maps a function f to the function defined by A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives.
Numerical differentiationIn numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the function. The simplest method is to use finite difference approximations. A simple two-point estimation is to compute the slope of a nearby secant line through the points (x, f(x)) and (x + h, f(x + h)). Choosing a small number h, h represents a small change in x, and it can be either positive or negative.
Probabilistic numericsProbabilistic numerics is an active field of study at the intersection of applied mathematics, statistics, and machine learning centering on the concept of uncertainty in computation. In probabilistic numerics, tasks in numerical analysis such as finding numerical solutions for integration, linear algebra, optimization and simulation and differential equations are seen as problems of statistical, probabilistic, or Bayesian inference.
