Level 1 — RecognitionPartial Differential Equations

Partial Differential Equations

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Time limit: 20 minutes
Total marks: 30
Instructions: Answer all questions. For True/False questions, a one-line justification is required for full marks. Use $...$ notation for any mathematics.

Section A — Multiple Choice (1 mark each)

Q1. The PDE $u_{xx} + u_{yy} = 0$ is classified as:

(a) parabolic (b) hyperbolic (c) elliptic (d) mixed

Q2. For a second-order PDE $A u_{xx} + B u_{xy} + C u_{yy} + \dots = 0$ , the discriminant test uses:

(a) $B^2 - 4AC$ (b) $B^2 - AC$ (c) $A^2 - BC$ (d) $4AC - B^2$

Q3. The 1D heat equation $u_t = \alpha u_{xx}$ is:

(a) elliptic (b) parabolic (c) hyperbolic (d) first order

Q4. The 1D wave equation $u_{tt} = c^2 u_{xx}$ is:

(a) elliptic (b) parabolic (c) hyperbolic (d) undefined

Q5. D'Alembert's solution of the wave equation has the general form:

(a) $f(x)g(t)$ (b) $f(x-ct) + g(x+ct)$ (c) $\sum a_n \sin nx$ (d) $e^{-\lambda t}\sin x$

Q6. A half-range cosine series represents a function using:

(a) only sine terms (b) only cosine terms (and constant) (c) both (d) exponentials only

Q7. Which condition is NOT part of the Dirichlet conditions for Fourier convergence?

(a) $f$ is periodic (b) finite number of maxima/minima (c) $f$ is infinitely differentiable (d) finite number of finite discontinuities

Q8. A Neumann boundary condition specifies:

(a) the value of $u$ on the boundary (b) the normal derivative $\partial u/\partial n$ on the boundary (c) $u=0$ everywhere (d) periodicity

Q9. Parseval's theorem relates the integral of $[f(x)]^2$ to:

(a) the derivative of $f$ (b) the sum of squares of Fourier coefficients (c) the Laplace transform (d) the discriminant

Q10. Laplace's equation describes a physical system in:

(a) transient heating (b) steady state (c) wave motion (d) shock formation

Section B — Matching (1 mark each, Q11–Q15)

Match each equation/term in Column X to its correct description in Column Y.

#	Column X		Column Y
Q11	$u_t = \alpha u_{xx}$	A	eigenvalue problem with orthogonal eigenfunctions
Q12	Sturm–Liouville problem	B	diffusion / heat conduction
Q13	$\hat f(k)=\int_{-\infty}^{\infty} f(x)e^{-ikx}dx$	C	steady-state temperature / potential
Q14	$u_{xx}+u_{yy}=0$	D	Fourier transform definition
Q15	Convolution theorem	E	$\widehat{(f*g)} = \hat f\,\hat g$

Section C — True/False with justification (2 marks each: 1 T/F + 1 justification)

Q16. An initial value problem (IVP) specifies conditions at a single value of the time variable, whereas a boundary value problem (BVP) specifies conditions at the ends of the spatial domain. (2)

Q17. Separation of variables assumes a solution of the form $u(x,t)=X(x)\,T(t)$ . (2)

Q18. The discriminant $B^2-4AC = 0$ implies the PDE is hyperbolic. (2)

Q19. Laplace's equation on a disk is naturally solved in polar coordinates, and the radial part leads to Bessel-type/Euler equations. (2)

Q20. Finite difference methods approximate derivatives by replacing them with difference quotients on a grid. (2)

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1. (c) elliptic. Laplace's equation; $A=C=1,B=0 \Rightarrow B^2-4AC=-4<0$ . (1)

Q2. (a) $B^2-4AC$ . Standard discriminant test. (1)

Q3. (b) parabolic. Write $u_t-\alpha u_{xx}=0$ : only $u_{xx}$ present, so $A=-\alpha,B=C=0$ , $B^2-4AC=0$ . (1)

Q4. (c) hyperbolic. $u_{tt}-c^2u_{xx}=0$ : $A=1,C=-c^2,B=0$ , $B^2-4AC=4c^2>0$ . (1)

Q5. (b) $f(x-ct)+g(x+ct)$ . Superposition of right- and left-travelling waves. (1)

Q6. (b) only cosine terms (and constant). Even extension → cosine series. (1)

Q7. (c) $f$ is infinitely differentiable. Dirichlet conditions require only piecewise smoothness, not infinite differentiability. (1)

Q8. (b) the normal derivative on the boundary. Definition of Neumann condition. (1)

Q9. (b) sum of squares of Fourier coefficients. Energy identity. (1)

Q10. (b) steady state. No time dependence; equilibrium. (1)

Section B (1 mark each)

Q11 → B (heat/diffusion). (1)
Q12 → A (eigenvalue problem, orthogonal eigenfunctions). (1)
Q13 → D (Fourier transform definition). (1)
Q14 → C (steady-state potential). (1)
Q15 → E (convolution theorem). (1)

Section C (2 marks each: 1 for correct T/F, 1 for justification)

Q16. TRUE. (1) Justification: IVP gives data at $t=0$ (e.g. initial temperature/displacement); BVP gives data at spatial boundaries $x=0,L$ . Many PDEs are mixed IBVPs. (1)

Q17. TRUE. (1) Justification: We seek product solutions $u=X(x)T(t)$ so that the PDE separates into ODEs in each variable equated to a separation constant. (1)

Q18. FALSE. (1) Justification: $B^2-4AC=0$ is parabolic; $>0$ is hyperbolic, $<0$ is elliptic. (1)

Q19. TRUE. (1) Justification: Separation in $(r,\theta)$ gives $\Theta''+n^2\Theta=0$ and an Euler equation $r^2R''+rR'-n^2R=0$ ; with a Helmholtz-type term it becomes Bessel's equation, hence the Bessel-function connection. (1)

Q20. TRUE. (1) Justification: e.g. $u_x\approx (u_{i+1}-u_{i-1})/(2h)$ , $u_{xx}\approx (u_{i+1}-2u_i+u_{i-1})/h^2$ ; derivatives replaced by grid differences. (1)

[
  {"claim":"Laplace u_xx+u_yy: discriminant B^2-4AC = -4 < 0 (elliptic)",
   "code":"A,B,C=1,0,1; disc=B*B-4*A*C; result=(disc<0)"},
  {"claim":"Wave u_tt - c^2 u_xx: discriminant > 0 (hyperbolic)",
   "code":"c=symbols('c',positive=True); A,B,C=1,0,-c**2; disc=B*B-4*A*C; result=simplify(disc>0)==True or disc.subs(c,2)>0"},
  {"claim":"Heat equation: discriminant = 0 (parabolic)",
   "code":"al=symbols('alpha',positive=True); A,B,C=-al,0,0; disc=B*B-4*A*C; result=(simplify(disc)==0)"},
  {"claim":"D'Alembert form f(x-ct)+g(x+ct) satisfies u_tt=c^2 u_xx",
   "code":"x,t,c=symbols('x t c'); f=Function('f'); g=Function('g'); u=f(x-c*t)+g(x+c*t); result=simplify(diff(u,t,2)-c**2*diff(u,x,2))==0"}
]