On skewed grid point iterative method for solving 2D hyperbolic telegraph fractional differential equation

Abstract

This paper presents the development of a new iterative method for solving the two-dimensional hyperbolic telegraph fractional differential equation (2D-HTFDE) which is central to the mathematical modeling of transmission line satisfying certain relationship between voltage and current waves in specific distance and time. This equation can be obtained from the classical two-dimensional hyperbolic telegraph partial differential equation by replacing the first and second order time derivatives by the Caputo time fractional derivatives of order 2α and α respectively, with 1/2 < α < 1. The iterative scheme, called the fractional skewed grid Crank-Nicolson FSkG(C-N), is derived from finite difference approximations discretized on a skewed grid rotated clockwise 45 0 from the standard grid. The skewed finite difference scheme combined with Crank-Nicolson discretization formula will be shown to be unconditionally stable and convergent by the Fourier analysis. The developed FSkG(C-N) scheme will be compared with the fractional Crank-Nicolson scheme on the standard grid to confirm the effectiveness of the proposed scheme in terms of computational complexities and computing efforts. It will be shown that the new proposed scheme demonstrates more superior capabilities in terms of the number of iterations and CPU timings compared to its counterpart on the standard grid but with the same order of accuracy.

The Solution of 2d Elliptic Equation Using Modified Geometric Mean Method on Skewed Grid With Red-Black Ordering

“…As stated in the previous section, GM is a two-stage iterative method and its iterative process involves of solving two independent systems such as u (1) and u (2) which are calculated at Level 1 and 2, respectively. The square root of the product of the two systems is computed at Level 3.…”

“…The same procedure is implemented in the calculation at Level 2. In Level 3, the square root of multiplication of intermediate values u (1) and u (2) obtained from Level 1 and Level 2 is computed to obtain the updated value of u (k+1) .…”

“…Previous works demostrated that the iteration procedure on skewed grid drastically reduced the computational complexity and speed up the convergence rate of the iterative methods [1,3,12,16,2,10].…”

New group iterative schemes for solving the two-dimensional anomalous fractional sub-diffusion equation

“…Another implicit difference scheme can be constructed by rotating Eq. (2.1) an angle 45 • to the standard mesh [6]. We have the following fractional implicit rotated point (FIRP) iterative scheme,…”

A Modified Accelerated Overrelaxation (Maor) Scheme for Solving the Shifted Gr\"{u}nwald Estimation Fractional Poisson Equation

“…Furthermore, some time-fractional diffusion equations [4,9,43] and space-fractional diffusion equations [33,34,35] have sometimes been solved using various sweeping iterative methods. Another equation includes the two-dimensional time-fractional telegraph equation which has been solved on the rotated iterative method [5] and by utilizing the explicit group iterative method [3].…”