By J. N. Reddy
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Extra info for Introduction to the Finite Element Method
Example text
Let wand G be scalar functions defined in a two-dimensional domain Q. 20b) and -f Ja where (V2 G)w ~x dy = ( Vw. 20a), with an appropriate change of variables, is useful in the sequel: i eoax i w-aeo n w-dxdy = - Y Q i awax i aw dxdy= - f. nxwGds -Gdxdy + f. nywGds ay - G dxdy Q Q + r (2. 17b). 3 Functionals An integral expression of the form leu) = f / F(x, u, u') dx, u = u(x), , du dx ut=~ where the integrand F(x, u, u') is a given function of the arguments x, u, and dul dx, is called a junctional.
2 Integral Relations Integration by parts is frequently used in the integral formulation of differential equations. In two-dimensional cases, integration by parts is better known as the gradient and divergence theorems. In this section, we derive some useful identities for future use. w be sufficiently differentiable functions of the coordinate x. Then the following integration-byparts formula holds: INTEGRATION·BY·PARTS FORMULAE. 12) This identity can easily be established. -. -. 12). 12), we obtain b 1 a W d2U fb dw dx2dx=-Ja v dxdx+w(b)v(b)-w(a)v(a) =- ts: du du --dx+w(b)-(b)-w(a)-(o) adxdx dx dx (2.
11) can be satisfied are called eigenvalues, and the associated functions u are called eigenfunctions. The classical (or exact) solution of a differential equation is the function that identically satisfies the differential equation and the specified boundary and/or initial conditions. 2 Integral Relations Integration by parts is frequently used in the integral formulation of differential equations. In two-dimensional cases, integration by parts is better known as the gradient and divergence theorems.
