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CAL-16

1D Laplacian

Δu = u''. Second derivative as a 1D Laplacian.

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Vector calculusLaplacian

Governing equation

Δu=u′′≈(u+−2u+u−)/h2\Delta u=u''\approx(u_+-2u+u_-)/h^2

u∈[−20,20]u \in \left[-20, 20\right]

where

u_+
u(x+h) (—)
u
u(x) (—)
u_-
u(x−h) (—)
h
Spacing h (—)
\Delta u
1D Laplacian (—)

The figure, part by part

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CAL-16 · wave
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①

Governing law — red

Δu=u′′≈(u+−2u+u−)/h2\Delta u=u''\approx(u_+-2u+u_-)/h^2

u∈[−20,20]u \in \left[-20, 20\right]

u=−20u = -20Δu=4205\Delta u = 4205

②

Domain of the variable — blue

umin⁡=−20,umax⁡=20u_{\min} = -20,\quad u_{\max} = 20

u∈[−20,20]u \in \left[-20, 20\right]

u=−20u = -20u=−20u = -20

③

Response curve — green

Δu=u′′≈(u+−2u+u−)/h2\Delta u=u''\approx(u_+-2u+u_-)/h^2

u∈[−20,20]u \in \left[-20, 20\right]

u=−20u = -20Δu=4205\Delta u = 4205

④

Rate of change — amber

dΔudu\frac{\mathrm{d}\Delta u}{\mathrm{d}u}

u∈[−20,20]u \in \left[-20, 20\right]

u=−19.75u = -19.75Δu′=−200\Delta u^{\prime} = -200

Lecture brief

Historical brief

Newton and Leibniz (1670s), Taylor, the fundamental theorem and the trapezoid rule are how change became a number. The lab differentiates, integrates and linearises in one variable. This sheet (CAL-16 — 1D Laplacian) is the form associated with Laplacian. Working symbols: u+u_+, uu, u−u_-, hh →\rightarrow Δu\Delta u. In 1D, ∇² = d²/dx². Harmonic functions are linear. The 3-point stencil (u_{+} − 2u + u_{-})/h² is the discrete Laplacian.

Purpose

Purpose: compute Δu\Delta u from u+u_+, uu, u−u_-, hh in Calculus via Δu=u′′≈(u+−2u+u−)/h2\Delta u=u''\approx(u_+-2u+u_-)/h^2 Δu = u''. Second derivative as a 1D Laplacian. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given u+=1.100 —u_+ = 1.100\,\mathrm{—}, u=1.000 —u = 1.000\,\mathrm{—}, u−=0.950 —u_- = 0.950\,\mathrm{—}, h=0.100 —h = 0.100\,\mathrm{—}, the governing relation Δu=u′′≈(u+−2u+u−)/h2\Delta u=u''\approx(u_+-2u+u_-)/h^2 yields Δu=5.0000 —\Delta u = 5.0000\,\mathrm{—}. Enter u'' directly, or the 3-point stencil with spacing h. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • 1D Laplacian \Delta u5.0000 —
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Narration of this film

Enter u'' directly, or the 3-point stencil with spacing h.

In 1D, ∇² = d²/dx². Harmonic functions are linear. The 3-point stencil (u_{+} − 2u + u_{-})/h² is the discrete Laplacian.

Reading speed

Watch on YouTube