In this paper, approximation techniques based on the shifted Jacobi together with spectral tau technique are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Jacobi polynomials. Using the operational matrix of the fractional derivative, the problem can be reduced to a set of linear algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique and a comparison is made with the existing results to show that the proposed method is easy to implement and produce accurate results.

In this paper, the objective is to derive the variational formulation of the electrochemical machining problem (ECM for short) and to evaluate the numerical solution using the direction Ritz method. This problem is of degenerate problem which has so many difficulties to be solved using other approach.