Chapter 3

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 \(x=c !\)

For the following functions, sketch \(f(x)\), the Fourier series of \(f(x)\), the Fourier sine series of \(f(x)\), and the Fourier cosine series of \(f(x)\). (a) \(f(x)=1\) (b) \(f(x)=1+x\) (c) \(f(x)= \begin{cases}x & x<0 \\ 1+x & x>0\end{cases}\) (d) \(f(x)=e^{x}\) (e) \(f(x)= \begin{cases}2 & x<0 \\ e^{-x} & x>0\end{cases}\)

Suppose that \(f(x)\) and \(d f / d x\) are piecewise smooth. Prove that the Fourier series of \(f(x)\) can be differentiated term by term if the Fourier series of \(f(x)\) is continuous.

For the following functions, sketch the Fourier sine series of \(f(x)\) and determine its Fourier coefficients. (a) \(f(x)=\cos \pi x / L\) (Verify the formula on p. 89 ) (b) \(f(x)= \begin{cases}1 & xL / 2\end{cases}\) (c) \(f(x)= \begin{cases}0 & xL / 2\end{cases}\) (d) \(f(x)= \begin{cases}1 & xL / 2\end{cases}\)

Suppose that \(f(x)\) is continuous [except for a jump discontinuity at \(x=x_{0}\), \(f\left(x_{0}^{-}\right)=\alpha\) and \(\left.f\left(x_{0}^{+}\right)=\beta\right]\) and \(d f / d x\) is piecewise smooth. (a) Determine the Fourier sine series of \(d f / d x\) in terms of the Fourier cosine series coefficients of \(f(x)\). (b) Determine the Fourier cosine series of \(d f / d x\) in terms of the Fourier sine series coefficients of \(f(x)\).

Show that the Fourier series operation is linear; that is, show that the Fourier series of \(c_{1} f(x)+c_{2} g(x)\) is the sum of \(c_{1}\) times the Fourier series of \(f(x)\) and \(c_{2}\) times the Fourier series of \(g(x)\).

Suppose that \(\cosh x \sim \sum_{n=1}^{\infty} b_{n} \sin n \pi x / L\). (a) Determine \(b_{n}\) by correctly differentiating this series twice. (b) Determine \(b\), by integrating this series twice.

Suppose that \(f(x)\) and \(d f / d x\) are piecewise smooth. (a) Prove that the Fouricr sine series of a continuous function \(f(x)\) can only be differentiated term by term if \(f(0)=0\) and \(f(L)=0\). (b) Prove that the Fourier cosine series of a continuous function \(f(x)\) can be differentiated term by term.

Suppose that \(f(x)\) is piecewise smooth. What value does the Fourler series of \(f(x)\) converge to at the end point \(x--L ?\) at \(x-L ?\)

There are some things wrong in the following demonstration. Find the mistakes and correct them. In this problem we attempt to obtain the Fourier cosine coefficients of \(e^{x}\). $$ e^{x}=A_{0}+\sum_{n-1}^{\infty} A_{n} \cos \frac{n \pi x}{L} $$ Differentiating yields $$ e^{x}=-\sum_{n=1}^{\infty} \frac{n \pi}{L} A_{n} \sin \frac{n \pi x}{L}, $$ the Fourier sine series of \(e^{x}\). Differentiating again yields $$ e^{x}=-\sum_{n=1}^{\sim}\left(\frac{n \pi}{L}\right)^{2} A_{n} \cos \frac{n \pi x}{L} \text {. } $$ Since cquations (1) and (2) give Fourier cosine scrics of \(e^{x}\), they must be identical. Thus, $$ \left.\begin{array}{l} A_{0}=0 \\ A_{n}=0 \end{array}\right\\} \quad \text { (obviously wrong!) } $$ By correcting the mistakes you should be able to obtain \(A_{0}\) and \(A_{n}\) without using the typical technique, that is, \(A_{n}=2 / L \int_{0}^{L} e^{x} \cos n \pi x / L d x\).

Prove that the Fourier series of a continuous function \(u(x, t)\) can be differentiated term by term with respect to the parameter \(t\) if \(\partial u / \partial t\) is piecewise smooth.

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

Consider $$ \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}} $$ subject to \(\partial u / \partial x(0, t)=0, \partial u / \partial x(L, t)=0\), and \(u(x, 0)=f(x)\). Solve in the following way. Look for the solution as a Fourier cosine series. Assume that \(u\) and \(\partial u / \partial x\) are continuous and \(\partial^{2} u / \partial x^{2}\) and \(\partial u / \partial t\) are piecewise smooth. Justify all differentiations of infinite series.

(a) Determine formulas for the even extension of any \(f(x)\). Compare to the formula for the even part of \(f(x)\). (b) Do the same for the odd extension of \(f(x)\) and the odd part of \(f(x)\). (c) Calculate and sketch the four functions of parts (a) and (b) if $$ f(x)= \begin{cases}x & x>0 \\ x^{2} & x<0\end{cases} $$ Graphically add the even and odd parts of \(f(x) .\) What occurs? Similarly, add the even and odd extensions. What occurs then?

Consider the heat equation with a known source \(q(x, t)\) : $$ \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}+q(x, t) \quad \text { with } \quad u(0, t)=0 \quad \text { and } \quad u(L, t)=0 \text {. } $$ Assume that \(q(x, t)\) (for each \(t>0\) ) is a piecewise smooth function of \(x\). Also assume that \(u\) and \(\partial u / \partial x\) are continuous functions of \(x\) (for \(t>0\) ) and \(\partial^{2} u / \partial x^{2}\) and \(\partial u / \partial t\) are piecewise smooth. Thus, $$ u(x, t)=\sum_{n=1}^{\infty} b_{n}(t) \sin \frac{n \pi x}{L} $$ What ordinary differential equation does \(b_{n}(t)\) satisfy? Do not solve this differential equation.

If \(f(x)=\left\\{\begin{array}{ll}x^{2} & x<0 \\ e^{-x} & x>0\end{array}\right.\), what are the even and odd parts of \(f(x) ?\)

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 \(x\).]

Given a sketch of \(f(x)\), describe a procedure to sketch the even and odd parts of \(f(x)\).

Solve the following nonhomogeneous problem: \(\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}+e^{-t}+e^{-2 t} \cos \frac{3 \pi x}{L} \quad\) [assume that \(\left.2 \neq k(3 \pi / \mathrm{L})^{2}\right]\) subject to \(\frac{\partial u}{\partial x}(0, t)=0, \frac{\partial u}{\partial x}(L, t)=0, \quad\) and \(\quad u(x, 0)=f(x) .\) Use the following method. Look for the solution as a Fourier cosine series. Justify all differentiations of infinite series (assume appropriate continuity).

(a) Graphically show that the even terms ( \(n\) even) of the Fourier sine series of any function on \(0 \leqslant x \leqslant L\) are odd (antisymmetric) around \(x=L / 2\). (b) Consider a function \(f(x)\) that is odd around \(x=L / 2\). Show that the odd coefficients ( \(n\) odd) of the Fourier sine series of \(f(x)\) on \(0 \leqslant x \leqslant L\) are zero.

Consider $$ \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}} $$ subject to $$ u(0, t)=A(t), \quad u(L, t)=0, \quad \text { and } \quad u(x, 0)=g(x) . $$ Assume that \(u(x, t)\) has a Fourier sine series. Determine a differential equation for the Fourier coefficients (assume appropriate continuity).

Consider a function \(f(x)\) that is even around \(x=L / 2\). Show that the even coefficients ( \(n\) even) of the Fourier sine series of \(f(x)\) on \(0 \leqslant x \leqslant L\) are zero.

(a) Consider a function \(f(x)\) which is even around \(x=L / 2\). Show that the odd coefficients ( \(n\) odd) of the Fourier cosine series of \(f(x)\) on \(0 \leqslant x \leqslant L\) are zero. (b) Explain the result of part (a) by considering a Fourier cosine series of \(f(x)\) on the interval \(0 \leqslant x \leqslant L / 2\).

Consider a function \(f(x)\) that is odd around \(x=L / 2\). Show that the even coefficients ( \(n\) even) of the Fourier cosine series of \(f(x)\) on \(0 \leqslant x \leqslant L\) are zero.

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 $$ \left.y=\frac{a+b}{2}+\frac{b-a}{2 L} x .\right] $$

Consider $$ \int_{0}^{1} \frac{d x}{1+x^{2}} $$ (a) Evaluate explicitly. (b) Use the Taylor scrics of \(1 /\left(1, x^{2}\right)\) (itsclf a geometric series) to obtain an infinite series for the integral. (c) Equate part (a) to part (b) in order to derive a formula for \(\pi\).

For continuous functions: (a) Under what conditions does \(f(x)\) equal its Fourier series for all \(x\), \(-L \leqslant x \leqslant L\) ? (b) Under what conditions does \(f(x)\) equal its Fourier sine series for all \(x\), \(0 \leqslant x \leqslant L\) ? (c) Under what conditions does \(f(x)\) equal its Fourier cosine series for all \(x\), \(0 \leqslant x \leqslant L\) ?