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Mathematics · Advanced university

PDE modes, boundaries and initial data

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

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Prerequisites

Partial derivatives and separation of variables.

Study 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.

Worked examples

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.

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

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

    1. Neither decays
    2. sin 3x
    3. They decay identically
    4. sin x

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

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

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

    1. No, solutions cannot vanish
    2. Yes
    3. Only at one point
    4. Only if κ is negative

Answers and marking guidance

Answers and marking notes
  1. sin 3x

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

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

    Credit two time initial conditions and domain boundaries.

  3. 2

    ω₁=cπ/L=2π.

  4. Yes

    All derivatives and boundary/initial values are zero.

Further reading and review references

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