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[Topic in Differential Equations] Final Exam

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Topic in Differential Equations (Course Number: TA10320389), Department of Mathematics, National Taiwan Normal University

Total: 100 points
Date: 17 June 2020
Duration: 3 hr

Problem 1

(a) (15 points) Find the Fourier series for the function

\displaystyle x^2, \quad -\pi<x\leq \pi

and f(x+2\pi)=f(x) for all x.

(b) (5 points) Discuss the convergence of the series you found in (a).

(c) (5 points) Derive

\displaystyle \dfrac{\pi^2}{6}=1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+\cdots.

Problem 2

(15 points) Prove that the eigenvalues of the Sturm-Liouville Problem

\begin{array}{l} y''(x)+\lambda y(x)=0\quad (0<x<\pi);\\\quad y(0)=0,\quad y(\pi)=0\end{array}

are \lambda_n=n^2\quad( n=1,2,3,\ldots) and the associated eigenfunctions are \phi_n(x)=\sin(nx)\quad(n=1,2,3,\ldots).

Problem 3

(10 points) Assume that f has a Fourier sine series

\displaystyle f(x)=\sum^\infty_{n=1}\sin\left(\dfrac{n\pi x}{L}\right),\quad 0\leq x\leq L.

Show formally that

\displaystyle \dfrac{2}{L}\int^L_0[f(x)]^2\text{d}x=\sum^\infty_{n=1}b_n^2.

Problem 4

Solve the following problem

\begin{array}{ll}u_t(x,t)=u_{xx}(x,t)+5,&(0<x<\pi,\quad t>0);\\u(0,t)=1,\quad u(\pi,t)=6,&(t>0),\\u(x,0)=1+\frac{5}{\pi}x+2\sin {3x}&(0<x<\pi).\end{array}

by the following steps. You may regard each part as an individual problem and solve it.

(i) (15 points) Use the method of separation of variables to find a function u(x,t) satisfying

\begin{array}{ll}u_t(x,t)=u_{xx}(x,t),&(0<x<\pi,\quad t>0);\\u(0,t)=0,\quad u(\pi,t)=0,&(t>0).\end{array}

Note: Calculation process is required, though you can use the result of Problem 2 directly.

(ii) (15 points) Find a solution u(x,t) of the following problem

\begin{array}{ll}u_t(x,t)=u_{xx}(x,t),&(0<x<\pi,\quad t>0);\\u(0,t)=1,\quad u(\pi,t)=6,&(t>0)\\u(x,0)=1+\frac{5}{\pi}x+2\sin {3x}&(0<x<\pi).\end{array}

(iii) (15 points) Find a solution u(x,t) of the following problem

\begin{array}{ll}u_t(x,t)=u_{xx}(x,t)+5,&(0<x<\pi,\quad t>0);\\u(0,t)=0,\quad u(\pi,t)=0,&(t>0)\\u(x,0)=0,&(0<x<\pi).\end{array}

(iv) (5 points) Use parts (ii) and (iii) to construct a solution of

\begin{array}{ll}u_t(x,t)=u_{xx}(x,t)+5,&(0<x<\pi,\quad t>0);\\u(0,t)=1,\quad u(\pi,t)=6,&(t>0),\\u(x,0)=1+\frac{5}{\pi}x+2\sin {3x}&(0<x<\pi).\end{array}

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