# Chapter 1 Fundamental Concepts

The strong form of the boundary-value problem, $$(S)$$, is stated as follows: \begin{aligned} (S) \left\{ \begin{array}{rl} Given \ \mathcal{f} : \overline{\Omega} \rightarrow \mathbb{R} \ and \ constants \ q \ and \ h, \ find \ u: \overline{\Omega} \rightarrow \mathbb{R}, \ such \ that \\ u_{,xx}+\mathcal{f}&=0 \ \ \ on \ \Omega\\ u(1)&=q \\ -u_{,x}(0)&=h \end{array} \right.\end{aligned}

Solution  of  $$(S): \ u(x)=q+(1-x)h+\int_x^1 \left\{ \int_0^y f(z)dz \right\} dy$$

The weak or variational form of the boundary-value problem

Defination: $$H^1-functions$$: functions that satisfy $$\int_0^1 (u_{,x})^2 dx < \infty$$
Defination: trial solutions, $$\delta = \{u|u\in H^1, u(1) = q\}$$
Defination: weighting functions, $$\nu = \{w|w\in H^1, w(1) = 0\}$$