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Heat Equation Demonstration

Heat Equation Demonstration
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2017-08-22 19:36
VectorFunctionCalculusSlider AnimationGraphing
In collections UQ-Applied Mathematical Analysis
This example dynamically visualizes the solution and evolution of the heat equation using geometric constructions. Key math concepts include partial differential equations, numerical methods, vector fields, and function graphing. The construction summary employs commands like Sequence, Max/Min, Textfield, Vector, OrthogonalLine, If, Polygon, Sum, LaTeX, and CurveCartesian. Ideal for high school students and math enthusiasts studying PDEs and numerical simulations.

GeoGebra Commands & Parameters

  1. Sequence(g(i), i, 0, L, 0.1) → list1
  2. Max(list1) → maxlist1
  3. Min(list1) → minlist1
  4. Textfield(f) → InputBox1
  5. Vector(A, B) → u
  6. Vector(A, C) → v
  7. OrthogonalLine(C, yAxis) → h
  8. OrthogonalLine(B, yAxis) → i
  9. If(y(v) > 0, 0, 1) → condy
  10. If(y(u) < 0, 0, 1) → condx
  11. If(ymin ≟ 0, (ymax * (-0.1)), ymin) → condy2
  12. If(ymax ≟ 0, (ymin * (-0.1)), ymax) → condx2
  13. Textfield(L) → InputBox2
  14. Polygon((xmin1, ymin1), ((-0.1 * L), ymin1), ((-0.1 * L), ymax1), (xmax1, ymax1)) → Range1, j, k, l, m
  15. Polygon((xmin2, ymin2), (L, ymin2), (L, ymax2), (xmax2, ymax2)) → Range2, p, q, r, s
  16. Sequence((2 / L * Integral[(f(x) * sin((((π * x) * i)) / L)), 0, L]), i, 1, n) → An
  17. Sequence(((Element[An, i] * exp(((((-a) * i^(2)) * π^(2)) * y / L^(2)))) * sin(((i * π) * x / L))), i, 1, n) → Anexpsin
  18. Sum(Anexpsin, n) → vv
  19. Textfield(a) → InputBox3
  20. Textfield(n) → InputBox4
  21. If(t < 30, t, 30) → hue
  22. LaTeX(hue, true, true) → text1
  23. CurveCartesian(o, heat(o), o, 0, L) → c

AI Drawing Prompt Reference

Create a dynamic visualization of the heat equation solution. On a coordinate plane, draw a curve representing temperature distribution that evolves with a parameter. Include several input boxes for adjusting the initial function, domain length, thermal diffusivity, and number of series terms. Mark the solution domain with two rectangular regions. Add two vectors indicating directions and a vertical auxiliary line. The curve is formed by summing sinusoidal terms with coefficients from integrals, each decaying exponentially with time. Use color mapping to indicate time progression and display the current time value. The layout should be clear for educational demonstration.
The prompt above was compiled from the original drawing commands and is for reference only.