Find the solution of one dimensional heat equation

 The one-dimensional heat equation is a partial differential equation that describes how the temperature of a rod changes over time. The equation is given by:

∂u/∂t = α ∂^2u/∂x^2

Where u is the temperature of the rod at position x and time t, and α is the thermal diffusivity of the material.

The general solution to the one-dimensional heat equation is:

u(x,t) = F(x - αt) + G(x + αt)

Where F and G are arbitrary functions. To find the specific solution for a given problem, you need to specify the initial temperature of the rod (u(x,0)) and the boundary conditions (temperature at the ends of the rod at all times). You can then use these to determine the functions F and G, and hence the solution to the heat equation.

For example, if the initial temperature of the rod is given by u(x,0) = sin(πx/L), and the boundary conditions are u(0,t) = 0 and u(L,t) = 0 for all t, the solution to the heat equation would be:

u(x,t) = (1 - x/L) sin(π(L - αt)/L) + (x/L) sin(π(L + αt)/L)