“variational method”的概念、定义、翻译、参考文献-科学参考

variational method

Computer

A technique for the solution of certain classes of ordinary and partial differential equations that involves the use of a variational principle. That is, the solution of the differential equation is expressed as the solution of a minimization problem that involves an integral expression. The equation is then solved by carrying out an approximate minimization. Variational principles arise naturally in many branches of physics and engineering. As an example, the solution of
$\begin{array}{l} y ” + q (x) y = f (x), 0 \leq x \leq 1 \\ y (0) = y (1) = 0 \end{array}$
is also the solution of the problem
$\begin{array}{l} \underset{v ε V}{minimize} \int_{0}^{1} {v' {(x)}^{2} - q (x) v {(x)}^{2} \\ - 2 f (x) v (x)} d x \end{array}$
where V is a class of sufficiently differentiable functions that are zero at x = 0 and x = 1. An approximate minimization can be carried out by minimizing over the subspace of functions
$\sum_{j = 1}^{n} c_{j} ϕ_{j} (x)$
When the trial functions ϕ‎_j(x) are splines, the resulting method is an example of the finite-element method.

单词	variational method
释义	variational method Computer A technique for the solution of certain classes of ordinary and partial differential equations that involves the use of a variational principle. That is, the solution of the differential equation is expressed as the solution of a minimization problem that involves an integral expression. The equation is then solved by carrying out an approximate minimization. Variational principles arise naturally in many branches of physics and engineering. As an example, the solution of $\begin{array}{l} y ” + q (x) y = f (x), 0 \leq x \leq 1 \\ y (0) = y (1) = 0 \end{array}$ is also the solution of the problem $\begin{array}{l} \underset{v ε V}{minimize} \int_{0}^{1} {v' {(x)}^{2} - q (x) v {(x)}^{2} \\ - 2 f (x) v (x)} d x \end{array}$ where V is a class of sufficiently differentiable functions that are zero at x = 0 and x = 1. An approximate minimization can be carried out by minimizing over the subspace of functions $\sum_{j = 1}^{n} c_{j} ϕ_{j} (x)$ When the trial functions ϕ‎_j(x) are splines, the resulting method is an example of the finite-element method.
随便看	rain-wash Rainwater, (Leo) James raised beach raised bog raised cosine window raise(to a power) Rajagopalachariar, Chakravarti (1878–1972) Rajput rake Raketoplan raking Rakosi, Matyas (1892–1971) RAL Raleigh, Sir Walter (1552–1618) RAM r.a.m. Ramachandran plot Ramakrishna movement Raman effect Raman, Sir Chandrasekhara Venkata Ramanuja (1137) Ramanujan, Srinivasa (1887–1920) Ramanujan, Srinivasa Iyengar Ramapithecus Ramaty High-Energy Solar Spectroscopic Imager