Differential Equations Partial Differential Equations and Boundary-Value Problems Syllabus

Partial Differential Equations and Boundary-Value Problems syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Partial Differential Equations and Boundary-Value Problems in Differential Equations, not a summary of it.

1. 1

   Fourier Series

   4 topics

   • Orthogonal Functions and Series
   • Fourier Series Expansion
   • Fourier Cosine and Sine Series
   • Convergence of Fourier Series
2. 2

   Sturm-Liouville Theory

   3 topics

   • Sturm-Liouville Boundary-Value Problems
   • Eigenvalues and Eigenfunctions
   • Eigenfunction Expansions
3. 3

   Classification and Separation of Variables

   3 topics

   • Classification of Second-Order PDEs
   • Method of Separation of Variables
   • Superposition and Boundary Conditions
4. 4

   The Heat Equation

   3 topics

   • Heat Equation Derivation
   • Heat Equation with Homogeneous Boundary Conditions
   • Nonhomogeneous Boundary Conditions
5. 5

   The Wave Equation

   3 topics

   • Vibrating String Problem
   • d'Alembert's Solution
   • Normal Modes and Standing Waves
6. 6

   Laplace's Equation and Other Methods

   4 topics

   • Laplace's Equation and Steady-State Problems
   • Boundary-Value Problems in Polar and Cylindrical Coordinates
   • Fourier Transform Method
   • Method of Characteristics for First-Order PDEs

Partial Differential Equations and Boundary-Value Problems flashcards for Differential Equations

24 of 50 cards from the Partial Differential Equations and Boundary-Value Problems deck — real questions with worked answers.

1. What does it mean for a set of functions $\{\phi_{n}(x)\}$ to be orthogonal on an interval $[a,b]$?

   They are orthogonal if the inner product of any two distinct members vanishes: $$\int_{a}^{b}\phi_{m}(x)\phi_{n}(x)\,dx = 0 \quad \text{for } m \neq n.$$

2. How is the square norm $\lVert \phi_{n}\rVert^{2}$ of a function defined on $[a,b]$, and what makes a set orthonormal?

   The square norm is $\lVert \phi_{n}\rVert^{2} = \int_{a}^{b}\phi_{n}^{2}(x)\,dx$. A set is orthonormal if it is orthogonal and each function has unit norm, i.e. $\lVert \phi_{n}\rVert = 1$.

3. For an orthogonal set $\{\phi_{n}\}$ on $[a,b]$, what is the coefficient $c_{n}$ in the generalized Fourier series $f(x)=\sum_{n} c_{n}\phi_{n}(x)$?

   $$c_{n} = \frac{\int_{a}^{b} f(x)\phi_{n}(x)\,dx}{\int_{a}^{b}\phi_{n}^{2}(x)\,dx} = \frac{(f,\phi_{n})}{\lVert\phi_{n}\rVert^{2}}.$$

4. What is meant by orthogonality with respect to a weight function $w(x)$ on $[a,b]$?

   The set is orthogonal with respect to weight $w(x) > 0$ if $$\int_{a}^{b} w(x)\phi_{m}(x)\phi_{n}(x)\,dx = 0 \quad \text{for } m \neq n.$$

5. Write the Fourier series of a function $f(x)$ defined on the interval $(-p,p)$.

   $$f(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n}\cos\frac{n\pi x}{p}+b_{n}\sin\frac{n\pi x}{p}\right).$$

6. Give the Euler formulas for the Fourier coefficients $a_{0}$, $a_{n}$, $b_{n}$ of $f(x)$ on $(-p,p)$.

   $$a_{0}=\frac{1}{p}\int_{-p}^{p}f(x)\,dx,\quad a_{n}=\frac{1}{p}\int_{-p}^{p}f(x)\cos\frac{n\pi x}{p}\,dx,$$ $$b_{n}=\frac{1}{p}\int_{-p}^{p}f(x)\sin\frac{n\pi x}{p}\,dx.$$

7. State the Fourier convergence theorem: to what value does a Fourier series converge at a point of the interval?

   If $f$ and $f'$ are piecewise continuous on $(-p,p)$, the series converges to $f(x)$ at points of continuity, and to the average $\dfrac{f(x^{+})+f(x^{-})}{2}$ at a jump discontinuity.

8. What is a cosine (even) Fourier series, and when is it used?

   Used when $f$ is even on $(-p,p)$: all $b_{n}=0$ and $$f(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}a_{n}\cos\frac{n\pi x}{p},\quad a_{n}=\frac{2}{p}\int_{0}^{p}f(x)\cos\frac{n\pi x}{p}\,dx.$$

9. What is a sine (odd) Fourier series, and when is it used?

   Used when $f$ is odd on $(-p,p)$: $a_{0}=a_{n}=0$ and $$f(x)=\sum_{n=1}^{\infty}b_{n}\sin\frac{n\pi x}{p},\quad b_{n}=\frac{2}{p}\int_{0}^{p}f(x)\sin\frac{n\pi x}{p}\,dx.$$

10. On $(0,L)$, what is a half-range expansion, and what three choices are available?

    A half-range expansion represents $f$ given only on $(0,L)$ by extending it. The three choices are: even reflection (cosine series), odd reflection (sine series), or a period-$L$ extension (full Fourier series).

11. Are $\cos\frac{n\pi x}{p}$ and $\sin\frac{n\pi x}{p}$ orthogonal on $(-p,p)$? State the key integral results.

    Yes. For $m\neq n$, $\int_{-p}^{p}\cos\frac{m\pi x}{p}\cos\frac{n\pi x}{p}\,dx=0$ and similarly for sines; also $\int_{-p}^{p}\cos\frac{m\pi x}{p}\sin\frac{n\pi x}{p}\,dx=0$ for all $m,n$. For $m=n\neq0$ each cosine/sine integral equals $p$.

12. Distinguish pointwise, uniform, and mean-square (L2) convergence of a Fourier series.

    Pointwise: converges at each individual $x$. Uniform: converges at the same rate for all $x$ (requires $f$ continuous and periodic with piecewise-continuous $f'$). Mean-square: $\int_{-p}^{p}\lvert f - S_{N}\rvert^{2}dx \to 0$; guaranteed whenever $f$ is square-integrable.

13. State Parseval's (completeness) relation for a Fourier series on $(-p,p)$.

    $$\frac{1}{p}\int_{-p}^{p}[f(x)]^{2}\,dx=\frac{a_{0}^{2}}{2}+\sum_{n=1}^{\infty}\left(a_{n}^{2}+b_{n}^{2}\right).$$

14. What is the Gibbs phenomenon?

    Near a jump discontinuity, the partial sums of a Fourier series overshoot the function by a fixed amount (about $9\%$ of the jump). The overshoot does not diminish as $N\to\infty$; it merely narrows toward the discontinuity.

15. Write the standard (regular) Sturm-Liouville problem in self-adjoint form.

    $$\frac{d}{dx}\left[r(x)\frac{dy}{dx}\right]+\left(q(x)+\lambda p(x)\right)y=0,\quad a\le x\le b,$$ with separated boundary conditions at $x=a$ and $x=b$.

16. What conditions define a regular Sturm-Liouville problem on $[a,b]$?

    $r(x), r'(x), q(x), p(x)$ are continuous, with $r(x)>0$ and $p(x)>0$ on $[a,b]$, together with separated (homogeneous) boundary conditions of the form $A_{1}y(a)+B_{1}y'(a)=0$ and $A_{2}y(b)+B_{2}y'(b)=0$.

17. List the key properties of the eigenvalues and eigenfunctions of a regular Sturm-Liouville problem.

    Eigenvalues are real and form an increasing infinite sequence $\lambda_{1}<\lambda_{2}<\cdots\to\infty$; each eigenvalue has a one-dimensional eigenspace; eigenfunctions for distinct eigenvalues are orthogonal with respect to weight $p(x)$; the eigenfunctions form a complete set.

18. With respect to what weight are Sturm-Liouville eigenfunctions orthogonal, and state the relation.

    Orthogonal with respect to the weight $p(x)$: $$\int_{a}^{b} p(x)\,y_{m}(x)\,y_{n}(x)\,dx=0 \quad \text{for } \lambda_{m}\neq\lambda_{n}.$$

19. For the eigenvalue problem $y''+\lambda y=0$, $y(0)=0$, $y(L)=0$, give the eigenvalues and eigenfunctions.

    $$\lambda_{n}=\left(\frac{n\pi}{L}\right)^{2},\quad y_{n}(x)=\sin\frac{n\pi x}{L},\quad n=1,2,3,\dots$$

20. For $y''+\lambda y=0$ with Neumann conditions $y'(0)=0$, $y'(L)=0$, give the eigenvalues and eigenfunctions.

    $$\lambda_{n}=\left(\frac{n\pi}{L}\right)^{2},\; n=0,1,2,\dots,\quad y_{n}(x)=\cos\frac{n\pi x}{L}.$$ Note $\lambda_{0}=0$ with $y_{0}=1$ is included.

21. What is a singular Sturm-Liouville problem?

    One where the regularity conditions fail: e.g. $r(x)=0$ at an endpoint, or the interval is infinite, or $p$ or $r$ is unbounded. Boundedness of $y$ at the singular endpoint typically replaces a separated boundary condition.

22. What is an eigenfunction expansion of a function $f$ relative to a Sturm-Liouville system?

    $$f(x)=\sum_{n=1}^{\infty}c_{n}y_{n}(x),\quad c_{n}=\frac{\int_{a}^{b}p(x)f(x)y_{n}(x)\,dx}{\int_{a}^{b}p(x)y_{n}^{2}(x)\,dx}.$$ Fourier sine/cosine series are special cases.

23. Give the general linear second-order PDE in two variables and the discriminant used to classify it.

    For $A u_{xx}+B u_{xy}+C u_{yy}+D u_{x}+E u_{y}+F u=G$, the discriminant is $B^{2}-4AC$.

24. State the classification of second-order linear PDEs by the discriminant $B^{2}-4AC$.

    $B^{2}-4AC<0$: elliptic (e.g. Laplace's equation); $B^{2}-4AC=0$: parabolic (e.g. heat equation); $B^{2}-4AC>0$: hyperbolic (e.g. wave equation).

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