IBVP for the heat equation on the half-line

g[x_,t_,k_]=Exp[-x^2/4/k/t]/Sqrt[4 Pi k t]

Plot[g[x,1,1],{x,-10,10}]

u0[x_]=Exp[- x]Cos[x]

Plot[u0[x],{x,0,10},PlotRange->All]

u[x_,t_,k_]:=NIntegrate[u0[y](g[x-y,t,k]-g[x+y,t,k]),{y,0,100}]

Plot[u[x,2,1],{x,0,10}]