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Mathematics · lesson 4 of 5

PDE modes, boundaries and initial data

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

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Learn the ideas

Prerequisites: Partial derivatives and separation of variables.

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.

Worked examples

Example 1

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

Credit both derivatives and the matching factor.

u_t=−4κe^(−4κt)sin(2x) and u_xx=−4e^(−4κt)sin(2x), so u_t=κu_xx.

Example 2

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

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

2

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

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

Sample answer and marking guidance

u_t=−4κe^(−4κt)sin(2x) and u_xx=−4e^(−4κt)sin(2x), so u_t=κu_xx.

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?

Show answer and explanation

2

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?

Show answer and explanation

sin 3x

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 and marking guidance

Initial displacement and initial velocity, together with suitable boundary conditions.

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.

Show answer and explanation

2

ω₁=cπ/L=2π.

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

Show answer and explanation

Yes

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 and marking guidance

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

Credit both endpoint evaluations.

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

Sample answer and 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.

Credit nonuniqueness without the missing data.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.