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< alpha < 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 450 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.
Highlights
Fractional calculus is of great importance in mathematical modeling of various phenomena in the field of engineering [1, 2], quantum mechanics [3], hydrology [4], viscoelasticity, [5, 6], bio science [7], control system [8], and other sciences [9,10,11,12,13]
In 2018, Kumar et al [18] proposed a non-differentiable solution for the vehicular fractal traffic flow problem. They solved the problem with the help of the local-fractional homotopy perturbation Sumudu transform scheme, and the results were computationally rigorous for similar kinds of fractional differential equations occurring in natural sciences
We have extended the formulation of the skewed grid iterative method on solving the more complicated 2D-hyperbolic telegraph fractional differential equation (HTFDE) and observed the efficient numerical results
Summary
Fractional calculus is of great importance in mathematical modeling of various phenomena in the field of engineering [1, 2], quantum mechanics [3], hydrology [4], viscoelasticity, [5, 6], bio science [7], control system [8], and other sciences [9,10,11,12,13]. The fractional derivative which simultaneously possesses memory and nonlocal property can describe different nonlinear phenomena more accurately and efficiently in comparison with the integer-order derivative. This makes fractional calculus a powerful tool for modeling the complex dynamical systems [14]. In 2018, Kumar et al [18] proposed a non-differentiable solution for the vehicular fractal traffic flow problem They solved the problem with the help of the local-fractional homotopy perturbation Sumudu transform scheme, and the results were computationally rigorous for similar kinds of fractional differential equations occurring in natural sciences
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