Thank you for the answer, I worked with the initial and boundary conditions mentioned in the answer. Another problem I am facing is that, my initial condition is an incomplete gamma function of x. and at xmin and at all times the function is 0. and also at xmax and all times t the concentration is constants value which is 0.001 M. Whenever I apply this the Mathematica shows error message.
In[5]:= tMin = 0; tMax = 10;
xMin = 0; xMax = 1;
(*Solve the PDE*)
sol = NDSolve[{D[u[x, t], t] == 0.000051*D[u[x, t], {x, 2}],
u[x, tMin] ==
1.1193194537721065` (0.892979511569249` - (
0.3333333333333334` (x^3)^(2/3)
Gamma[0.3333333333333333`, 4.299881111111103`*^17 x^3])/x^2),
u[xMin, t] == 0, u[xMax, t] == 0.001},
u, {t, tMin, tMax}, {x, xMin, xMax}]
(*Plot the solution*)
Plot3D[u[x, t] /. sol, {x, xMin, xMax}, {t, tMin, tMax},
PlotRange -> All, AxesLabel -> Automatic]
During evaluation of In[5]:= Power::infy: Infinite expression 1/0.^2 encountered.
During evaluation of In[5]:= Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered.
During evaluation of In[5]:= NDSolve::ndnum: Encountered non-numerical value for a derivative at t == 0.`.
During evaluation of In[5]:= Power::infy: Infinite expression 1/0.^2 encountered.
During evaluation of In[5]:= Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered.
During evaluation of In[5]:= NDSolve::ndnum: Encountered non-numerical value for a derivative at t == 0.`.
Out[7]= NDSolve[{
\!\(\*SuperscriptBox[\(u\),
TagBox[
RowBox[{"(",
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, t] == 0.000051
\!\(\*SuperscriptBox[\(u\),
TagBox[
RowBox[{"(",
RowBox[{"2", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, t],
u[x, 0] ==
1.11932 (0.89298 - (
0.333333 (x^3)^(2/3) Gamma[0.333333, 4.29988*10^17 x^3])/x^2),
u[0, t] == 0, u[1, t] == 0.001}, u, {t, 0, 10}, {x, 0, 1}]
During evaluation of In[5]:= General::munfl: Exp[-157180.] is too small to represent as a normalized machine number; precision may be lost.
During evaluation of In[5]:= NDSolve::dsvar: 0.000715` cannot be used as a variable.
During evaluation of In[5]:= ReplaceAll::reps: {NDSolve[{(u^(0,1))[0.0000715,0.000715]==0.000051 (u^(2,0))[0.0000715,0.000715],u[0.0000715,0]==0.999529,u[0,0.000715]==0,u[1,0.000715]==0.001},u,{0.000715,0,10},{0.0000715,0,1}]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing.
During evaluation of In[5]:= NDSolve::dsvar: 0.000715` cannot be used as a variable.
During evaluation of In[5]:= ReplaceAll::reps: {NDSolve[{(u^(0,1))[0.0000715,0.000715]==0.000051 (u^(2,0))[0.0000715,0.000715],u[0.0000715,0.]==0.999529,u[0.,0.000715]==0.,u[1.,0.000715]==0.001},u,{0.000715,0.,10.},{0.0000715,0.,1.}]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing.
During evaluation of In[5]:= General::munfl: Exp[-1.57172*10^14] is too small to represent as a normalized machine number; precision may be lost.
During evaluation of In[5]:= NDSolve::dsvar: 0.000715` cannot be used as a variable.
During evaluation of In[5]:= General::stop: Further output of NDSolve::dsvar will be suppressed during this calculation.
During evaluation of In[5]:= ReplaceAll::reps: {NDSolve[{(u^(0,1))[0.0715001,0.000715]==0.000051 (u^(2,0))[0.0715001,0.000715],u[0.0715001,0]==0.999529,u[0,0.000715]==0,u[1,0.000715]==0.001},u,{0.000715,0,10},{0.0715001,0,1}]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing.
During evaluation of In[5]:= General::stop: Further output of ReplaceAll::reps will be suppressed during this calculation.
During evaluation of In[5]:= General::munfl: Exp[-1.25549*10^15] is too small to represent as a normalized machine number; precision may be lost.
During evaluation of In[5]:= General::stop: Further output of General::munfl will be suppressed during this calculation.
