GATE Mathematics Partial Differential Equations Flashcards

24 sample cards from the Partial Differential Equations deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Why is periodicity $\Theta(\theta)=\Theta(\theta+2\pi)$ imposed in polar problems, and what does it force?

   Single-valuedness of $u$ requires period $2\pi$, forcing the separation constant to be $n^{2}$ with integer $n=0,1,2,\dots$ and $\Theta=a\cos n\theta+b\sin n\theta$.

2. Give the general (Fourier) solution of Laplace's equation inside a disk of radius $a$ (bounded at $r=0$).

   $$u(r,\theta)=\frac{a_0}{2}+\sum_{n=1}^{\infty} r^{n}\left(a_n\cos n\theta + b_n\sin n\theta\right)$$ where negative powers and $\ln r$ are dropped for boundedness.

3. For the exterior problem ($r>a$, bounded at infinity), which radial terms are retained?

   Only the decaying terms $r^{-n}$ (and the constant) are kept; the growing $r^{n}$ terms are discarded.

4. State Poisson's integral formula for the value of a harmonic function at the centre of a disk of radius $a$.

   $u(0)=\frac{1}{2\pi}\int_0^{2\pi} u(a,\theta)\,d\theta$, the average of boundary values (a special case of the mean value property).

5. Write the one-dimensional heat (diffusion) equation.

   $$u_t = k\,u_{xx}$$ where $k>0$ is the thermal diffusivity.

6. Separating $u=X(x)T(t)$ in the heat equation, what ODEs result?

   $\frac{T'}{kT}=\frac{X''}{X}=-\lambda$, giving $T'+k\lambda T=0$ and $X''+\lambda X=0$.

7. Why must the separation constant be negative ($-\lambda$ with $\lambda>0$) for a heat problem with fixed-zero ends?

   A positive or zero constant gives non-decaying or unbounded $T(t)$ and no nontrivial $X$ satisfying $X(0)=X(L)=0$; only $-\lambda<0$ yields decaying $T(t)=e^{-k\lambda t}$ and sine eigenfunctions.

8. Give the series solution of $u_t=k u_{xx}$ on $0<x<L$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=f(x)$.

   $$u(x,t)=\sum_{n=1}^{\infty} b_n \sin\!\frac{n\pi x}{L}\, e^{-k\left(\frac{n\pi}{L}\right)^{2} t}, \quad b_n=\frac{2}{L}\int_0^{L} f(x)\sin\frac{n\pi x}{L}\,dx$$

9. What happens to the temperature distribution of the heat equation as $t\to\infty$ (insulated/zero-end case)?

   Each mode decays like $e^{-k(n\pi/L)^{2}t}$, so $u\to$ the steady state (here $u\to0$ for zero-Dirichlet ends); higher modes decay fastest.

10. What is the fundamental solution (heat kernel) of the 1-D heat equation on the infinite line?

    $$u(x,t)=\frac{1}{\sqrt{4\pi k t}}\,\exp\!\left(-\frac{x^{2}}{4kt}\right)$$ representing the response to an initial point source.

11. Does the heat equation describe a forward or reverse smoothing process, and is it time-reversible?

    It smooths/diffuses initial data forward in time and is irreversible (ill-posed backward in time).

12. Write the one-dimensional wave equation.

    $$u_{tt}=c^{2}u_{xx}$$ where $c$ is the wave speed.

13. State d'Alembert's solution of the wave equation on the infinite line with $u(x,0)=f(x)$, $u_t(x,0)=g(x)$.

    $$u(x,t)=\frac{1}{2}\big[f(x-ct)+f(x+ct)\big]+\frac{1}{2c}\int_{x-ct}^{x+ct} g(s)\,ds$$

14. What are the characteristic lines of the 1-D wave equation?

    The lines $x-ct=\text{const}$ and $x+ct=\text{const}$, along which left- and right-moving waves propagate.

15. What is the domain of dependence of the point $(x,t)$ for the wave equation?

    The interval $[x-ct,\,x+ct]$ on the initial line; only initial data there affect $u(x,t)$.

16. Give the separated series solution of $u_{tt}=c^{2}u_{xx}$ on $0<x<L$ with fixed ends $u(0,t)=u(L,t)=0$.

    $$u(x,t)=\sum_{n=1}^{\infty}\sin\frac{n\pi x}{L}\left(A_n\cos\frac{n\pi c t}{L}+B_n\sin\frac{n\pi c t}{L}\right)$$

17. For the vibrating string with fixed ends, what are the natural (angular) frequencies of vibration?

    $\omega_n=\frac{n\pi c}{L}$, $n=1,2,3,\dots$, with $n=1$ the fundamental and higher $n$ the overtones.

18. How are the coefficients $A_n$ and $B_n$ in the wave series determined from $u(x,0)=f(x)$, $u_t(x,0)=g(x)$?

    $A_n=\frac{2}{L}\int_0^{L} f(x)\sin\frac{n\pi x}{L}\,dx$ and $B_n=\frac{2}{n\pi c}\int_0^{L} g(x)\sin\frac{n\pi x}{L}\,dx$.

19. Compare the heat and wave equations in terms of order in time and qualitative behavior.

    Heat: first order in $t$, parabolic, dissipative/smoothing, irreversible. Wave: second order in $t$, hyperbolic, energy-conserving, reversible, finite propagation speed.

20. What is the two-dimensional heat equation, and what steady state does it reduce to?

    $u_t=k(u_{xx}+u_{yy})$. At steady state $u_t=0$, it reduces to Laplace's equation $u_{xx}+u_{yy}=0$.

21. Why does the wave equation preserve sharp features while the heat equation does not?

    The wave equation transports data along characteristics without dissipation (energy conserved), whereas the heat equation's exponential damping of high modes smooths out discontinuities.

22. What general solution form does d'Alembert give for $u_{tt}=c^{2}u_{xx}$ in characteristic coordinates?

    $u(x,t)=F(x-ct)+G(x+ct)$, a sum of a right-moving and a left-moving wave with arbitrary $F,G$.

23. In the canonical reduction, what coordinate change makes the wave equation $u_{\xi\eta}=0$?

    Setting $\xi=x-ct$ and $\eta=x+ct$ transforms $u_{tt}=c^{2}u_{xx}$ into $u_{\xi\eta}=0$, integrated to $u=F(\xi)+G(\eta)$.

24. What boundary/initial data make the wave, heat, and Laplace problems well-posed, respectively?

    Wave (hyperbolic): two Cauchy/initial conditions $u,u_t$ plus boundary data. Heat (parabolic): one initial condition $u(x,0)$ plus boundary data. Laplace (elliptic): boundary conditions only (Dirichlet/Neumann), no time data.

What this deck covers

Partial Differential Equations flashcards FAQ