Table 2 lists the maximum absolute errors (MAEs) for different choices of N, while the best result for the maximum absolute error achieved by using the linearized compact difference method (LCDM [39]) with 80 space steps and (6400) time steps was (1.23times10^{-8}).
For several points, a comparison between the absolute errors of problem (61) which were obtained using the linearized compact difference method (LCDM [39]) and the results obtained by our method is presented in Table 1 at (N=10).
Compact difference method two must solve an algebraic nonlinear equation obtained by Pantazopoulos (1998) at every time step.
We need point out that the fourth-order compact difference method, which is employed to discretize the differential equation, can be replaced by other numerical methods, such as the FE method and the finite volume method.
The combination of the compact difference method and alternating direction implicit method are used for solving two-dimensional fractional Cattaneo equation in [5].
More recently, the authors of [32] and [33] proposed a linearized compact multi-splitting method and compact difference method combined with an extrapolation scheme to solve convection-reaction-diffusion and neutral parabolic PDEs with delay, respectively.
We have also derived fourth-order compact difference methods for the normal derivatives of the solution to the concerned problem.
Using only three grid points at each time level, three-level compact difference methods of order two in time and four in space for the solution of differential equation (1) for uniform mesh were reported by Mohanty and Evans [37], but these methods fail at singular points, and a special technique was needed to solve singular problems.
To do this, Poisson equation is first discretized by fourth-order compact finite difference method.
The proposed compact finite difference method was tested by comparing the frequency response, by applying to an analytical magnetic field, and by applying to measured magnetic field data of the K1200 superconducting cyclotron at the National Superconducting Cyclotron Laboratory.
