# PDE modes, boundaries and initial data

> A sine mode for u_t=κu_xx decays as exp(−κk²t).

Course: [Advanced mathematical methods](https://studybrix.com/en/courses/advanced-mathematical-methods/index.md)
Level: advanced
Prerequisites: Partial derivatives and separation of variables.

## Notes

### Heat equation

A sine mode for u_t=κu_xx decays as exp(−κk²t).

### Wave equation

A fixed-end string supports discrete sine modes with angular frequencies cnπ/L.

### Boundary data

The spatial domain and endpoint conditions select admissible modes.

### Uniqueness

An equation alone is not a complete problem; initial and boundary data matter.

## Practice questions

### 1. Verify that u(x,t)=e^(−4κt)sin(2x) solves u_t=κu_xx.

Sample answer / marking guidance: u_t=−4κe^(−4κt)sin(2x) and u_xx=−4e^(−4κt)sin(2x), so u_t=κu_xx.

Explanation: Credit both derivatives and the matching factor.

### 2. On 0<x<π with zero endpoints, which positive integer n makes sin(nx) have one interior zero?

Answer: 2

Explanation: The n=2 mode has its single interior zero at π/2.

### 3. Which heat mode decays faster for the same positive κ: sin x or sin 3x?

A. Neither decays
B. sin 3x
C. They decay identically
D. sin x

Answer: sin 3x

Explanation: The decay exponents scale with n², so 9κ exceeds κ.

### 4. What initial data are normally required for a second-order-in-time wave equation?

Sample answer / marking guidance: Initial displacement and initial velocity, together with suitable boundary conditions.

Explanation: Credit two time initial conditions and domain boundaries.

### 5. For a string with L=2 and c=4, write the fundamental angular frequency as aπ and find a.

Answer: 2

Explanation: ω₁=cπ/L=2π.

### 6. Does the zero function solve homogeneous heat and wave equations with zero data?

A. No, solutions cannot vanish
B. Yes
C. Only at one point
D. Only if κ is negative

Answer: Yes

Explanation: All derivatives and boundary/initial values are zero.

### 7. Explain why a sine mode is natural for zero displacement at both ends of a finite interval.

Sample answer / marking guidance: sin(nπx/L) vanishes at x=0 and x=L for positive integer n, so it satisfies the endpoint conditions automatically.

Explanation: Credit both endpoint evaluations.

### 8. Why is replacing a boundary-value problem by only its differential equation incomplete?

Sample answer / marking guidance: Different functions can satisfy the same PDE while differing in boundary or initial data. Those constraints determine the intended physical and mathematical problem.

Explanation: Credit nonuniqueness without the missing data.

## Further reading

- https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/
- https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.