A Solution To Heat Equation
The Jacobi theta function is the unique solution to the one-dimensional heat equation with periodic boundary conditions at time zero. This is most easily seen by taking z = x to be real, and taking τ = it with t real and positive. Then we can write
which solves the heat equation
That this solution is unique can be seen by noting that at t = 0, the theta function becomes the Dirac comb:
where δ is the Dirac delta function. Thus, general solutions can be specified by convolving the (periodic) boundary condition at t = 0 with the theta function.
Read more about this topic: Theta Function
Famous quotes containing the words solution, heat and/or equation:
“Give a scientist a problem and he will probably provide a solution; historians and sociologists, by contrast, can offer only opinions. Ask a dozen chemists the composition of an organic compound such as methane, and within a short time all twelve will have come up with the same solution of CH4. Ask, however, a dozen economists or sociologists to provide policies to reduce unemployment or the level of crime and twelve widely differing opinions are likely to be offered.”
—Derek Gjertsen, British scientist, author. Science and Philosophy: Past and Present, ch. 3, Penguin (1989)
“When the heat of the summer
Made drowsy the land,
A dragon-fly came
And sat on my hand;”
—Eleanor Farjeon (1881–1965)
“Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.”
—Anna Quindlen (b. 1952)