finite-difference method
finite-difference method A widely applicable discretization method for the solution of ordinary and partial differential equations. In this approach all derivatives are replaced by approximations that involve solution values only, so in general the differential equation is reduced to a system of nonlinear equations or linear algebraic equations. For example, in the problem y + by′ + cy = d 0←x←1, y(0) = α, y(1) = β,
where b, c, d, α, and β are given constants, the interval [0,1] is first divided into equal subintervals of length h; h is called the stepsize (or mesh or grid size). This gives the mesh points (or grid points) xn, xn = nh, n = 0,1,…,N+1, h = 1/(N+1)
At interior mesh points the derivatives are now replaced by finite-difference approximations, e.g. y′(xn) ≊ (½h)[y(xn+1) – y(xn–1)] y(xn) ≊ (1/h2)[y(xn+1) – 2y(xn) + y(xn–1)]
When combined with the boundary conditions these approximations result in a system of equations for approximations to y(xn), n = 1, 2,…, N. Nonlinear differential equations yield a system of nonlinear equations.
where b, c, d, α, and β are given constants, the interval [0,1] is first divided into equal subintervals of length h; h is called the stepsize (or mesh or grid size). This gives the mesh points (or grid points) xn, xn = nh, n = 0,1,…,N+1, h = 1/(N+1)
At interior mesh points the derivatives are now replaced by finite-difference approximations, e.g. y′(xn) ≊ (½h)[y(xn+1) – y(xn–1)] y(xn) ≊ (1/h2)[y(xn+1) – 2y(xn) + y(xn–1)]
When combined with the boundary conditions these approximations result in a system of equations for approximations to y(xn), n = 1, 2,…, N. Nonlinear differential equations yield a system of nonlinear equations.
More From encyclopedia.com
Chu Shih-chieh , (fl. China, 1280–1303),
mathematics.
Chu Shih-chieh (literary name, Han-ch’ing; appellation, Sung-t’ing) lived in Yen-shan (near modern Peking). Geor… Ishuah , Isuah (Ĭs´yōōə), the same as Ishuah.
Ishuah (Ĭsh´yōōə), in the Bible, son of Asher. It is also spelled Isuah. Y Chromosome , The diploid human genome is packaged within 46 chromosomes, as two pairs of 23 discrete elements, into all cells other than the haploid gametic egg a… dweeb , dweeb / dwēb/ • n. inf. a boring, studious, or socially inept person. DERIVATIVES: dweeb·ish adj. dweeb·y adj.
dweeb •Antibes, Beeb, Delibes, dweeb,… Yuan Shih-kai , Yüan Shih-k'ai (1859-1916), an outstanding Chinese military leader, held the balance of power when the Revolution of 1911 broke out and used it to se… Yosai , Yōsai (Tendai monk): see EISAI.
NEARBY TERMS
finite-difference method