If you want to solve the diffusion equation, ut = uxx , the typical approach would be to use the "method of lines" which means that you first discretize the right-hand side in space, for example by a finite difference method (FDM) or a finite element method (FEM). Both highly successful techniques. With this approach, you transform the PDE into a (large) system of ODEs, 

u' = Au

which you now need to solve using a time-stepping method. If you use an explicit method, you will get a restriction on the time step dt, such that dt/dx2 < C, to keep the time-stepping stable (this is referred to as a CFL condition). That's because the ODE system is stiff. It will severely limit your time step dt, and a better choice is to use an unconditionally stable implicit method, such as the trapezoidal rule. In particular, using the trapezoidal rule in combination with diffusion is often referred to as the Crank-Nicolson method, mentioned above.

There are some variants on how to proceed, but the main idea would still be to use method of lines, in combination with your preferred space discretization (FDM or FEM) and a well chosen time stepping scheme. The space discretizations can easily take care of Neumann conditions. Another issue to consider, is what convergence order is used. For example, if a standard Galerkin (FEM) or FDM of second order is used, it's usually sufficient to also take a second order method in time, such as Crank-Nicolson.

Thanks but I am looking for analytical methods or semi analytical methods.  

Is it that you have a particular problem for which the "front line" methods like separation of variables (SoV) is unsuitable? For example, SoV becomes increasingly difficult if the governing equation and/or boundary conditions are non-homogeneous. In this case, it's often good to move to various spatial integral transforms for which the transform and its inverse are known in advance (unlike Laplace transforms, whose solutions can be very difficult to invert). Chapter 6 of this free e-book

https://www.researchgate.net/profile/Michael_Wendl/publications

gives a good introduction to the use of these transforms for the type of equation you mentioned. This publication also shows a very detailed example of solving a difficult non-homogeneous diffusion equation (starting on pp 169) using such methods.

Apologies, the link above is generic. Here is a direct link to the free e-book.

http://wendl.weebly.com/textbook.html

Dear Dr

Some of my papers

1- A. R. Seadawy, Fractional solitary wave solutions of the

nonlinear higher-order extended KdV equation in a stratified

 shear flow: Part I, Comp. and Math. Appl. \textbf{70} (2015)

 345–352.

2- Khater, A. H., Callebaut D. K., Malfliet, W. and Seadawy A. R., Nonlinear Dispersive

Rayleigh-Taylor Instabilities in Magnetohydro-dynamic Flows,

Physica Scripta, 64 (2001) 533-547.

3- Khater, A. H., Callebaut D.

K. and Seadawy A. R., "Nonlinear Dispersive Kelvin-Helmholtz

Instabilities in Magnetohydrodynamic Flows" Physica Scripta, 67

(2003) 340-349.

4- A.R. Seadawy, Exact solutions of a two-dimensional

nonlinear Schrodinger equation, Appl. Math. Lett. 25 (2012) 687.

5- A.R. Seadawy, Stability analysis for Zakharov-Kuznetsov equation of weakly

nonlinear ion-acoustic waves in a plasma, Computers and

Mathematics with Applications 67 (2014) 172-180.

6- A.R. Seadawy, Stability analysis for two-dimensional ion-acoustic waves in

quantum plasmas, PHYSICS OF PLASMAS 21 (2014) 052107.

7- Helal, M. A. and  Seadawy A. R., Variational method for the

derivative nonlinear Schrodinger equation with computational

applications, Physica Scripta, 80, (2009) 350-360.

8- Helal, M. A. and  Seadawy A. R., Exact

soliton solutions of an D-dimensional nonlinear Schrodinger

equation with damping and diffusive terms, Z. Angew. Math. Phys.

(ZAMP) 62 (2011), 839-847.

9-  Khater, A. H., Callebaut D. K. and Seadawy A. R., General

soliton solutions of an n-dimensional Complex Ginzburg-Landau

equation, Physica Scripta, Vol. 62 (2000) 353-357.

10- Khater, A. H., Helal M. A. and Seadawy A. R.,

General soliton solutions of n-dimensional nonlinear Schrodinger

equation" IL Nuovo Cimento 115B, (2000) 1303-1312.

11-  Khater, A. H., Callebaut D. K., Helal, M. A. and  Seadawy A.

R., Variational Method for the Nonlinear Dynamics of an Elliptic

Magnetic Stagnation Line, The European Physical Journal  D, 39,

(2006) 237-245.

12-  Khater, A. H., Callebaut D. K.,

Helal, M. A. and  Seadawy A. R., General Soliton Solutions for

Nonlinear Dispersive Waves in Convective Type Instabilities,

Physica Scripta, 74, (2006) 384.

13- Seadawy A. R., Three-dimensional nonlinear modified Zakharov–Kuznetsov

equation of ion-acoustic waves in a magnetized plasma, Computers

and Mathematics with Applications 71 (2016) 201-212.

14- M.A. Helal and A.R. Seadawy, Benjamin-Feir-instability in

nonlinear dispersive waves, Computers and Mathematics with

Applications 64 (2012) 3557-3568.

15- Seadawy, A.R., and El-Rashidy, K., Traveling wave solutions for some coupled nonlinear

evolution equations by using the direct algebraic method, Math.

and Comp. model. \textbf{57} (2013) 1371.

16- A. R. Seadawy, New exact solutions for the KdV equation with higher  order nonlinearity

by using the variational method, Comp. and Math. Appl. 62 (2011)

3741-3755.

17- Seadawy A. R, Stability analysis solutions for nonlinear

three-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov

equation in a magnetized electron-positron plasma" Physica A:

Statistical Mechanics and its Applications  Physica A 455 (2016)

44-51.

18- A.R. Seadawy, Nonlinear wave solutions of the three-dimensional

Zakharov–Kuznetsov–Burgers equation in dusty plasma, Physica A

439 (2015) 124–131.

19- A.R. Seadawy, Stability analysis of traveling wave

solutions for generalized coupled nonlinear KdV equations, Appl.

Math. Inf. Sci. 10 (1) (2016) 209–214.