A New Numerical Approach for Solving 1D Fractional Diffusion-Wave Equation

Abstract

Fractional derivative is nonlocal, which is more suitable to simulate physical phenomena and provides more accurate models of physical systems such as earthquake vibration and polymers. The present study suggested a new numerical approach for the fractional diffusion-wave equation (FDWE). The fractional-order derivative is in the Riemann-Liouville (R-L) sense. Discussed the theoretical analysis of stability, consistency, and convergence. The numerical examples demonstrate that the method is more workable and excellently holds the theoretical analysis, showing the scheme’s feasibility.

On some novel bright, dark and optical solitons to the cubic-quintic nonlinear non-paraxial pulse propagation model

“…Such models are broadly utilized to explain more complex physical phenomena in fluid mechanics, plasma waves, optical fiber telecommunication, biophysics, soliton theory, and atmospheric science. For instance, the Jacobi elliptic function method, the hyperbolic function expansion method, the homogeneous balance method, the F-expansion method, G ′ G -expansion method, the Weierstrass elliptic function method, the Hirota bilinear transformation method, the modified simple equation method, the perturbation method, and the variational iteration method has been established for the exploration of exact traveling wave solutions (sKhater, M. M., Mohamed, M. S., Khater et al 2021aKhater et al , 2021bKhater et al , 2021cKhater et al , 2021dAlshahrani et al 2021;Khater and Ghanbari 2021;Ali et al 2021;Akbar et al 2021;Akinyemi et al 2021aAkinyemi et al , 2021bAkinyemi et al , 2021cAhmad et al 2021).…”

The Rayleigh–Stokes Problem for a Heated Generalized Second-Grade Fluid with Fractional Derivative: An Implicit Scheme via Riemann–Liouville Integral

“…e proof is completed via the induction method. □ Theorem 1. e modified implicit difference scheme l 2 is convergent, and the order of convergence is O(τ + τ(Δx) 2 + τ(Δy) 2 ).…”

“…e use of fractional-order calculus in a variety of elds of science and engineering, including geometric phenomena, has sparked a lot of interest in this area [1]. e rst discussion of fractional calculus took place between Leibniz and L'Hospital at the end of the seventeenth century [2].…”

Solution of a Complex Nonlinear Fractional Biochemical Reaction Model

“…Most classical numerical methods used for ordinary differential equations have been successfully modified for fractional differential equations such as implicit Euler scheme [10], spectral collocation methods [22], Adams-Bashforth methods [23], and Runge-Kutta methods [24]. Some of the newly developed numerical methods include a new predictor-corrector formula, Legendre spectral method, discretization of Riemann-Liouville, and a modified Adams-Bashforth method [25][26][27][28].…”