On the local fractional wave equation in fractal strings

2019 ◽  
Vol 42 (5) ◽  
pp. 1588-1595 ◽  
Author(s):  
Jagdev Singh ◽  
Devendra Kumar ◽  
Dumitru Baleanu ◽  
Sushila Rathore
Keyword(s):  
Wave Equation ◽  
Fractional Wave Equation ◽  
Fractal Strings

scholarly journals Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Ming-Sheng Hu ◽  
Ravi P. Agarwal ◽  
Xiao-Jun Yang
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Series Solution ◽  
Differential Operators ◽  
Fractional Wave Equation ◽  
Local Fractional Calculus ◽  
Vibrating String ◽  
Fractional Differential ◽  
Fractional Differential Operators

We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag-Leffler function.

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.

scholarly journals Approximate Solutions of the Damped Wave Equation and Dissipative Wave Equation in Fractal Strings

2019 ◽  
Vol 3 (2) ◽  
pp. 26 ◽  
Author(s):  
Dumitru Baleanu ◽  
Hassan Kamil Jassim
Keyword(s):  
Wave Equation ◽  
Decomposition Method ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Damped Wave Equation ◽  
Fractal Strings ◽  
Fractional Derivative Operators ◽  
Laplace Decomposition Method ◽  
Dissipative Wave Equation

In this paper, we apply the local fractional Laplace variational iteration method (LFLVIM) and the local fractional Laplace decomposition method (LFLDM) to obtain approximate solutions for solving the damped wave equation and dissipative wave equation within local fractional derivative operators (LFDOs). The efficiency of the considered methods are illustrated by some examples. The results obtained by LFLVIM and LFLDM are compared with the results obtained by LFVIM. The results reveal that the suggested algorithms are very effective and simple, and can be applied for linear and nonlinear problems in sciences and engineering.

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