scholarly journals On the Fractional Wave Equation

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 874
Author(s):  
Francesco Iafrate ◽  
Enzo Orsingher
Keyword(s):  
Brownian Motion ◽  
Wave Equation ◽  
Wave Equations ◽  
Stable Process ◽  
Space Time ◽  
Probabilistic Interpretation ◽  
Symmetric Stable Process ◽  
Fractional Wave Equation ◽  
Fractional Wave Equations ◽  
Time Fractional Wave Equation

In this paper we study the time-fractional wave equation of order 1 < ν < 2 and give a probabilistic interpretation of its solution. In the case 0 < ν < 1 , d = 1 , the solution can be interpreted as a time-changed Brownian motion, while for 1 < ν < 2 it coincides with the density of a symmetric stable process of order 2 / ν . We give here an interpretation of the fractional wave equation for d > 1 in terms of laws of stable d−dimensional processes. We give a hint at the case of a fractional wave equation for ν > 2 and also at space-time fractional wave equations.

On a backward problem for nonlinear time fractional wave equations

2021 ◽  
pp. 1-24
Author(s):  
Jia Wei He ◽  
Yong Zhou
Keyword(s):  
Wave Equation ◽  
Regularization Method ◽  
Existence And Uniqueness ◽  
Wave Equations ◽  
Mild Solutions ◽  
Fractional Wave Equation ◽  
Backward Problem ◽  
Fractional Wave Equations ◽  
Time Fractional Wave Equation ◽  
Eigenvalue Expansion

In this paper, we concern with a backward problem for a nonlinear time fractional wave equation in a bounded domain. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we establish some results about the existence and uniqueness of the mild solutions of the proposed problem based on the compact technique. Due to the ill-posedness of backward problem in the sense of Hadamard, a general filter regularization method is utilized to approximate the solution and further we prove the convergence rate for the regularized solutions.

scholarly journals Error Analysis of An Explicit Finite Difference Approximation for the Space Fractional Wave Equations

2012 ◽  
Vol 17 (2) ◽  
pp. 245
Author(s):  
N.H. Sweilam ◽  
T.A. Assiri
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Error Analysis ◽  
Wave Equations ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Fractional Wave Equation ◽  
Fractional Wave Equations ◽  
Explicit Finite Difference ◽  
The Stability

In this paper, the space fractional wave equation (SFWE) is numerically studied, where the fractional derivative is defined in the sense of Caputo. An explicit finite difference approximation (EFDA) for SFWE is presented. The stability and the error analysis of the EFDA are discussed. To demonstrate the effectiveness of the approximated method, some test examples are presented.   

scholarly journals SOLUSI PENDEKATAN PERSAMAAN GELOMBANG FRAKSIONAL NON LINEAR MENGGUNAKAN NEW VERSION OF OPTIMAL HOMOTOPY ASYMPTOTIC METHOD

2020 ◽  
Vol 14 (4) ◽  
pp. 523-534
Author(s):  
Faiqul Fikri ◽  
Eddy Djauhari ◽  
Endang Rusyaman
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Asymptotic Method ◽  
Wave Equations ◽  
Percentage Error ◽  
Approximation Solution ◽  
Fractional Wave Equation ◽  
Non Linear ◽  
Fractional Wave Equations ◽  
Optimal Homotopy Asymptotic Method

Non-linear differential equations with fractional derivative order are mathematical models that are widely used in modeling physical phenomena, one of the applications of these models is non-linear fractional wave equations. Many methods for solving non-linear fractional partial differential equations, one of which is the New Version of Optimal Homotopy Asymptotic Method which is developed by Liaqat Ali in 2016. The author will use this method to solve non-linear fractional wave equations predetermined, so that the convergence of function of the approximation solution non-linear fractional wave equation can be observed and it can be observed that the function of approximation solution of non-linear fractional wave equation solution using the New Version of Optimal Homotopy Asymptotic Method is simple and has a value error using Mean Absolute Percentage Error which is categorized very well

scholarly journals Fractional wave equations with attenuation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Peter Straka ◽  
Mark Meerschaert ◽  
Robert McGough ◽  
Yuzhen Zhou
Keyword(s):  
Wave Equation ◽  
Power Law ◽  
Weak Solutions ◽  
Wave Equations ◽  
Inhomogeneous Media ◽  
Medical Ultrasound ◽  
Sound Waves ◽  
Fractional Wave Equations

AbstractFractional wave equations with attenuation have been proposed by Caputo [5], Szabo [28], Chen and Holm [7], and Kelly et al. [11]. These equations capture the power-law attenuation with frequency observed in many experimental settings when sound waves travel through inhomogeneous media. In particular, these models are useful for medical ultrasound. This paper develops stochastic solutions and weak solutions to the power law wave equation of Kelly et al. [11].

Robustness of Boundary Control of Fractional Wave Equations With Delayed Boundary Measurement Using Fractional Order Controller and the Smith Predictor

2005 ◽  
Author(s):  
Jinsong Liang ◽  
Weiwei Zhang ◽  
YangQuan Chen ◽  
Igor Podlubny
Keyword(s):  
Fractional Order ◽  
Wave Equations ◽  
Small Difference ◽  
Smith Predictor ◽  
Fractional Wave Equation ◽  
Instability Problem ◽  
Boundary Measurement ◽  
Fractional Wave Equations ◽  
Fractional Order Controller ◽  
Better Than

In this paper, we analyze the robustness of the fractional wave equation with a fractional order boundary controller subject to delayed boundary measurement. Conditions are given to guarantee stability when the delay is small. For large delays, the Smith predictor is applied to solve the instability problem and the scheme is proved to be robust against a small difference between the assumed delay and the actual delay. The analysis shows that fractional order controllers are better than integer order controllers in the robustness against delays in the boundary measurement.

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