Solve an Initial Value Problem for the Heat Equation

Specify the heat equation.

In[1]:=

heqn = D[u[x, t], t] == D[u[x, t], {x, 2}];

Prescribe an initial condition for the equation.

In[2]:=

ic = u[x, 0] == E^(-x^2);

Solve the initial value problem.

In[3]:=

sol = DSolveValue[{heqn, ic }, u[x, t], {x, t}]

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Visualize the diffusion of heat with the passage of time.

In[4]:=

Plot[Evaluate[Table[sol, {t, 0, 4}]], {x, -5, 5}, PlotRange -> All, Filling -> Axis]

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Initial value problem for the heat equation with piecewise initial data.

In[5]:=

ic = u[x, 0] == UnitBox[x];

In[6]:=

sol = DSolveValue[{heqn, ic }, u[x, t], {x, t}]

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Discontinuities in the initial data are smoothed instantly.

In[7]:=

Plot3D[sol, {x, -2, 2}, {t, 0, 1}, PlotRange -> All, PlotPoints -> 250, Mesh -> None]

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