Example: Radially Symmetric Heat Equation: Problem 4.2.10

BesselJZero[0,2]//N

Example problem 4.2.10

a=1;
c=1;
f[r_]=100;
(Subscript[λ, n_]):=((BesselJZero[0,n])/a)

(Subscript[A, n_]):=(2/( (a^2)((BesselJ[1,a (Subscript[λ, n])])^2) ))(∫_0^a)f[r]BesselJ[0,r (Subscript[λ, n])]r\[DifferentialD]r

Let's have Mathematica do the integrals for us. We have to truncate the series.

sol[r_,t_,k_]:=(∑_(n=1)^k)(Subscript[A, n]) BesselJ[0,r (Subscript[λ, n])]Exp[-(c^2) ((((Subscript[λ, n])))^2) t]
solution=sol[r,t,5]

Let's plot!

solut[x_,y_,t_]=solution /.r->(Sqrt[(x^2)+(y^2)])//N

solut[x,y,0]

ContourPlot[solut[x,y,0],{x,-1,1},{y,-1,1},ColorFunction->Hue,PlotPoints->8]

Table[ContourPlot[solut[x,y,t],{x,-1,1},{y,-1,1},ColorFunction->Hue,PlotPoints->8],{t,0,0.01,0.01/20}]