91Ó°ÊÓ

The integration-by-parts formula $$ \int_{a}^{b} u \frac{d v}{d x} d x=\left.u v\right|_{a} ^{b}-\int_{a}^{b} v \frac{d u}{d x} d x $$ is known to be valid for functions \(u(x)\) and \(v(x)\) which are continuous and have continuous first derivatives. However, we will assume that \(u, v, d u / d x\), and \(d v / d x\) are continuous only for \(a \leqslant x \leqslant c\) and \(c \leqslant x \leqslant b\); we assume that all quantities may have a jump discontinuity at \(x=c\). *(a) Derive an expression for \(\int_{a}^{b} u d v / d x d x\) in terms of \(\int_{a}^{b} v d u / d x d x\). (b) Show that this reduces to the integration-by-parts formula if \(u\) and \(v\) are continuous across \(x=c\). It is not necessary for \(d u / d x\) and \(d v / d x\) to be continuous at \(x=c !\)

Show that \(e^{x}\) is the sum of an even and an odd function.

Consider the nonhomogeneous heat equation (with a steady heat source): $$ \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}+g(x) . $$ Solve this equation with the initial condition $$ u(x, 0)=f(x) $$ and the boundary conditions $$ u(0, t)=0 \quad \text { and } \quad u(L, t)=0 . $$ Assume that a continuous solution exists (with continuous derivatives). [Hints: Expand the solution as a Fourier sine series (i.e., use the method of eigenfunction expansion). Expand \(g(x)\) as a Fourier sine series. Solve for the Fourier sine series of the solution. Justify all differentiations with respect to \(x\).]

Fourier series can be defined on other intervals besides \(-L \leqslant x \leqslant L\). Suppose that \(g(y)\) is defined for \(a \leqslant y \leqslant b\). Represent \(g(y)\) using periodic trigonometric functions with period \(b-a\). Determine formulas for the coefficients. [Hint: Use the linear transformation $$ \left.y=\frac{a+b}{2}+\frac{b-a}{2 L} x .\right] $$