Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 696597, 6 pages http://dx.doi.org/10.1155/2013/696597

Research Article A Possible Generalization of Acoustic Wave Equation Using the Concept of Perturbed Derivative Order

Abdon Atangana1 and Adem KJlJçman2

1 Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa 2 Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Malaysia

Correspondence should be addressed to Adem Kılıc¸man; [email protected]

Received 18 February 2013; Accepted 18 March 2013

Academic Editor: Guo-Cheng Wu

Copyright © 2013 A. Atangana and A. Kılıc¸man. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The standard version of acoustic wave equation is modified using the concept of the generalized Riemann-Liouville fractional order derivative. Some properties of the generalized Riemann-Liouville fractional derivative approximation are presented. Some theorems are generalized. The modified equation is approximately solved by using the variational iteration method and the Green function technique. The numerical simulation of solution of the modified equation gives a better prediction than the standard one.

1. Introduction A derivation of general linearized wave equations is discussed by Pierce and Goldstein [1, 2]. However, neglecting Acoustics was in the beginning the study of small pressure the nonlinear effects in this equation, may lead to inaccurate waves in air which can be detected by the human ear: sound. prediction of the propagation of acoustic wave through The possibility of acoustics has been extended to higher the medium. Therefore in order to include explicitly the and lower frequencies: ultrasound and infrasound. Structural effect of corresponding relative density fluctuations in the vibrations are now often included in acoustics. Also the mathematicalformulation,oneneedstoinsertitinthe perception of sound is an area of acoustical research. In our partial differential equation that governs the propagation present paper we will limit ourselves to the original definition of the acoustic wave. Recently, the acoustic equation was andtothepropagationinfluidslikeairandwater.Insucha extended to the concept of fractional order derivative in [3]. caseacousticsisapartoffluiddynamics.Amajorproblemof Therefore in this paper, our concern is the modification of fluid dynamics is that the equations of motion are nonlinear. the previous equation by perturbing the order of the first This implies that an exact general solution of these equations derivative by replacing the first order of the derivative with is not available. Acoustics is a first-order approximation in 1+𝜀where 𝜀 is a positive small parameter. Also, when we which nonlinear effects are neglected. The corresponding consider diffusion process in porous medium, if the medium 𝜌󸀠/𝜌 relative density fluctuations 0 are considered very small structure or external field changes with time, in this situation, [1]. The acoustic wave equation governs the propagation of the ordinary integer order and constant-order fractional acoustic waves through a material medium. The form of the diffusion equation model cannot be used to well characterize equation is a second-order partial differential equation. The such phenomenon (see [3–9]). equation describes the evolution of acoustic pressure 𝑃 or particle velocity 𝑢 as a function of position 𝑟 and time 𝑡.A simplified form of the equation that describes acoustic waves 2. Definitions and Approximation in only one spatial dimension is considered in this paper To describe the propagation of acoustic waves through 𝜕2𝑃 1 𝜕2𝑃 a material medium with coordinate and time-dependent − =0. (1) 𝜕𝑥2 𝑐2 𝜕𝑡2 perturbed dimension, one must use Riemann-Liouville 2 Mathematical Problems in Engineering fractional order derivative that was introduced and used in (ii) Division a number of works (see [3, 6–8, 10]). These derivatives are If 𝑢𝑥 and 1/𝑓(𝑥) are differentiable on the open interval definedas(see[3, 6–8]) I then 󸀠 󸀠 1+𝜀𝑡 𝜇𝑡 [− (1 + 𝑢 )𝑓 (𝑥) +𝑢 𝑓 (𝑥)] 𝐷 =𝐷 𝑓 1+𝑢 1 𝑥 𝑥 +,𝑡 𝐷 𝑥 [ ]≅ 𝑓 (𝑥) 𝑓2 (𝑥) 𝑑 𝑛 𝑡 𝑓 (𝜏) (6) =( ) ∫ [ ]𝑑𝜏, 󸀠 󸀠 𝑢󸀠 𝑓 𝑥 𝑑𝑡 𝜇𝜏−𝑛+1 −𝑓 (𝑥) 𝑢𝑥𝑓 (𝑥) 𝑥 ( ) 0 Γ(𝑛−𝜇𝜏 (𝜏)(𝑡−𝜏) ) = − + . 2 2 2 (2) 𝑓 (𝑥) 𝑓 (𝑥) 𝑓 (𝑥) 1+𝜀 𝜇 𝐷 𝑥 =𝐷𝑥 𝑓 +,𝑥 (iii) Multiplication 𝑢 𝑓(𝑥) 𝑔(𝑥) 𝑑 𝑛 𝑥 𝑓 (𝜏) If 𝑥, and are differentiable in the open interval =( ) ∫ [ ]𝑑𝜏. I then 𝑑𝑥 𝜇𝜏−𝑛+1 0 Γ(𝑛−𝜇 (𝜏)(𝑡−𝜏) ) 1+𝑢 󸀠 󸀠 𝜏 𝐷 𝑥 [𝑓 (𝑥) ⋅𝑔(𝑥)]≅𝑔(𝑥) 𝑓 (𝑥) +𝑓(𝑥) 𝑔 (𝑥) Here, Γ is the Euler gamma function; 𝑛={𝜇}+1,where +(𝑔𝑓󸀠 +𝑓𝑔󸀠) (𝑥) 𝑢 {𝜇} is the integer part of 𝜇 for 𝜇≥0,thatis,𝑛−1≤𝜇<𝑛 𝑥 (7) and 𝑛=0for 𝜇<𝑛.Following(2)wehavethat𝜇𝑡 =1+𝜀𝑡 +𝑢󸀠 (𝑓 (𝑥) 𝑔 (𝑥)). and 𝜇𝑥 =1+𝜀𝑥. The integral operator defined previously 𝑥 for fractional exponents 𝜇𝑥and 𝜇𝑡 depending on coordinates (iv) Power and time can be expressed in terms of ordinary derivative If 𝑢𝑥 and 𝑓(𝑥) are differentiable in the open interval I then and integral [11]for|𝜀| ≪.Here 1 𝜀𝑡 and 𝜀𝑥 are considered 󸀠 1+𝑢 𝑛 󸀠 𝑛−1 as the corresponding relative density fluctuations 𝜌 /𝜌0 that 𝐷 𝑥 [(𝑓 (𝑥)) ]≅𝑛𝑓𝑓 vary slightly in time and space, respectively. For this matter, (8) 󸀠 𝑛−1 󸀠 𝑛 generalized Riemann-Liouville fractional derivatives satisfy +𝑢𝑥𝑛𝑓 𝑓 +𝑢𝑥𝑓 ,𝑛≥1. the approximate relations: If 𝑢𝑥 and 𝑓(𝑥) are two times differentiable in the open 1+𝜀 𝜕𝑓 𝜕𝜀 𝐷 𝑡 𝑓≅(1+𝜀) + 𝑡 𝑓, interval I then 𝑡 𝜕𝑡 𝜕𝑡 1+𝑢𝑥 1+𝑢𝑥 (3) 𝐷 [𝐷 [𝑓 (𝑥)]] ≅ (1 + 𝑢𝑥) 1+𝜀 𝜕𝑓 𝜕𝜀 𝐷 𝑥 𝑓≅(1+𝜀 ) + 𝑥 𝑓. 𝑥 𝜕𝑥 𝜕𝑥 2 2 (9) 𝜕 𝑓 𝜕𝑓 𝜕𝑢𝑥 𝜕 𝑢𝑥 𝜕𝑢𝑥 ×[(1+𝑢𝑥) +3 + 𝑓] + 𝑓. Thepreviousrelationsmakeitpossibletodescribethe 𝜕𝑥2 𝜕𝑥 𝜕𝑥 𝜕𝑥2 𝜕𝑥 dynamic system, including the effect of the corresponding relative density fluctuations, by means of partial differential 3.1. Clairaut’s Theorem for the Approximation. Assume that 2 and integral equations. 𝑓(𝑥,, 𝑦) 𝑢𝑥,and𝑢𝑦 are functions for which 𝜕 𝑓/𝜕𝑥𝜕𝑦, 2 2 2 𝜕 𝑓/𝜕𝑦𝜕𝑥, 𝜕 𝜀𝑥/𝜕𝑥𝜕𝑦,and𝜕 𝜀𝑦/𝜕𝑥𝜕𝑦 exist and are contin- 2 1+𝑢 1+𝑢 3. Some Properties of the Approximation uous over a domain 𝐷⊂R then, 𝐷 𝑥 [𝐷 𝑦 [𝑓(𝑥, 𝑦)]] and 1+𝑢 1+𝑢 𝐷 𝑦 [𝐷 𝑥 [𝑓(𝑥, 𝑦)]] exist and are continuous over the Let us examine some properties of the previous derivative 𝑢 =𝑢 operator. domain D. If in addition 𝑥 𝑦 then (i) Addition 1+𝑢 1+𝑢 𝐷 𝑦 [𝐷 𝑥 [𝑓 (𝑥, 𝑦)]] If 𝑢𝑥, 𝑓(𝑥),and𝑔(𝑥) are differentiable in the opened I (10) interval then, 1+𝑢 1+𝑢 =𝐷 𝑥 [𝐷 𝑦 [𝑓 (𝑥, 𝑦)]] . 1+𝑢 1+𝑢 1+𝑢 𝐷 𝑥 [𝑓 (𝑥) +𝑔(𝑥)]≅𝐷 𝑥 [𝑓 (𝑥)]+𝐷 𝑥 [𝑔 (𝑥)]. (4) Proof. If 𝑓(𝑥,, 𝑦) 𝑢𝑥,and𝑢𝑦 are functions for which Proof. We have 2 2 2 2 𝜕 𝑓/𝜕𝑥𝜕𝑦, 𝜕 𝑓/𝜕𝑦𝜕𝑥, 𝜕 𝜀𝑥/𝜕𝑥𝜕𝑦,and𝜕 𝜀𝑦/𝜕𝑥𝜕𝑦 exist and 1+𝑢 2 𝐷 𝑥 [𝑓 (𝑥) +𝑔(𝑥)] are continuous over a domain 𝐷⊂R then 1+𝑢 1+𝑢 𝜕[𝑓(𝑥) +𝑔(𝑥)] 𝐷 𝑦 [𝐷 𝑥 [𝑓 (𝑥, 𝑦)]] ≅(1+𝑢𝑥) 𝜕𝑥 2 2 𝜕𝑢𝑥 𝜕𝑓 𝜕 𝑓 𝜕 𝑢𝑥 𝜕𝑢𝑥 𝜕𝑓 𝜕𝑢 𝜕[𝑓(𝑥)] ≅(1+𝑢𝑦)[ + + 𝑓(𝑥,𝑦)+ ] + 𝑥 [𝑓 (𝑥) +𝑔(𝑥)](1+𝑢 ) (5) 𝜕𝑦 𝜕𝑥 𝜕𝑦𝜕𝑥 𝜕𝑦𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝑥 𝜕𝑥 𝜕𝑢 𝜕𝑓 𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕[𝑔(𝑥)] 𝜕𝑢 +(1+𝑢 ) 𝑦 + 𝑦 𝑥 𝑓(𝑥,𝑦). + 𝑥 [𝑓 (𝑥)]+(1+𝑢 ) + 𝑥 [𝑔 (𝑥)] 𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑥 𝑥 𝜕𝑥 𝜕𝑥 (11) 1+𝑢 1+𝑢 ≅𝐷 𝑥 [𝑓 (𝑥)]+𝐷 𝑥 [𝑓 (𝑥)]. 1+𝑢 1+𝑢 Now interchanging 𝑥 by 𝑦 we obtain 𝐷 𝑥 [𝐷 𝑦 [𝑓(𝑥, 𝑦)]]. Mathematical Problems in Engineering 3

2 2 If 𝑢𝑥 = 𝑢𝑦 then 𝜕 𝑓/𝜕𝑦𝜕𝑥=𝜕𝑓/𝜕𝑥𝜕𝑦 according to Cla- 4. Modification of the Equation 2 2 iraut’s theorem; thus replacing 𝜕 𝑓/𝜕𝑥𝜕𝑦 by 𝜕 𝑓/𝜕𝑦𝜕𝑥 in 1+𝑢 1+𝑢 𝐷 𝑦 [𝐷 𝑥 [𝑓(𝑥, 𝑦)]] In order to include explicitly the possible effect of the corre- ,weobtainthat sponding relative density fluctuations into the mathematical 1+𝑢 1+𝑢 1+𝑢 1+𝑢 𝐷 𝑦 [𝐷 𝑥 [𝑓 (𝑥, 𝑦)]] =𝐷 𝑥 [𝐷 𝑦 [𝑓 (𝑥, 𝑦)]] . (12) formulation, in this paper, we replace the classical version of the derivative of (1) by the modified Riemann-Liouville fractional derivative approximation (3)toobtain

3.2. Chain-Rule for the Approximation. We have 1+𝜀 1+𝜀 1 1+𝜀 1+𝜀 𝐷 𝑥 [𝐷 𝑥 𝑃] = 𝐷 𝑡 [𝐷 𝑡 𝑃] . (19) 𝜕𝜀 𝑐2 1+𝑢𝑥 󸀠 󸀠 𝑥 𝐷 (𝑓∘𝑔)≅ (1+𝑢𝑥)𝑔 (𝑥) 𝑓 [𝑔 (𝑥)]+ (𝑓∘𝑔) 𝜕𝑥 Making use of (3)andrelation(8), the previous equation can 󸀠 󸀠 󸀠 󸀠 be transformed to the following partial differential equation =𝑔 (𝑥) 𝑓 [𝑔 (𝑥)]+𝑢𝑥𝑔 (𝑥) 𝑓 [𝑔 (𝑥)] for 𝜀≪1: 𝜕𝜀 + 𝑥 (𝑓 ∘ 𝑔) . 𝜕2𝑃 𝜕𝑃 𝜕𝜀 𝜕2𝜀 𝜕𝜀 𝜕𝑥 (1 + 𝜀 )[(1+𝜀 ) +3 𝑥 + 𝑥 𝑃] + 𝑥 𝑃 𝑥 𝑥 𝜕𝑥2 𝜕𝑥 𝜕𝑥 𝜕𝑥2 𝜕𝑥 (13) 1 𝜕2𝑃 = {(1 + 𝜀 )[(1+𝜀) 3.3. Rolle’s Theorem for the Approximation. If a real-valued 𝑐2 𝑡 𝑡 𝜕𝑡2 (20) functions 𝑓 and 𝑢𝑥 are continuous on a closed interval [a, b], (𝑎, 𝑏) 𝑓(𝑎) = 𝑓(𝑏) differentiable on the open interval ,and , 𝜕𝑃 𝜕𝜀 𝜕2𝜀 𝜕𝜀 then there exist a 𝑐 in the open interval (𝑎, 𝑏) and a small +3 𝑡 + 𝑡 𝑃] + 𝑡 𝑃} . 𝜕𝑡 𝜕𝑡 𝜕𝑡2 𝜕𝑡 parameter 𝜇 such that 1+𝑢 2 𝐷 𝑥 𝑓 (𝑐) =𝜇𝑓(𝑐) . (14) Omitting the terms of 𝜀 in the previous equation, we obtain Proof. Following Rolle’s theorem, there exists a c in the open the following: (𝑎, 𝑏) 𝑓󸀠(𝑐) = 0 𝑐 interval such that .Forthis we have that 𝜕2𝑃 𝜕𝑃 𝜕𝜀 𝜕2𝜀 (1 + 2𝜀 ) +3 𝑥 + 𝑥 𝑃 1+𝑢𝑥 󸀠 󸀠 𝑥 2 2 𝐷 𝑓 (𝑐) =(1+𝑢𝑥 (𝑐))𝑓 (𝑐) +𝑢𝑥 (𝑐) 𝑓 (𝑐) 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑥 (21) 󸀠 1 𝜕2𝑃 𝜕𝑃 𝜕𝜀 𝜕2𝜀 =𝑢𝑥 (𝑐) 𝑓 (𝑐) (15) = {(1 + 2𝜀 ) +3 𝑡 + 𝑡 𝑃} . 𝑐2 𝑡 𝜕𝑡2 𝜕𝑡 𝜕𝑡 𝜕𝑡2 =𝜇𝑓(𝑐) . Now since the small parameters representing the pertur- If 𝑔(𝑥), 𝑓(𝑥),and𝑢𝑥 are differentiable in an open interval bation additions to unity are small, the right- and left-hand I, then there exist 2>𝛼>1and 𝛽>0such that sides of (21) can be divided by 1+2𝜀𝑥.Inthiscase,weobtain 󵄨 1+𝑢 1+𝑢 󵄨 󵄨 󸀠 󸀠 󵄨 󵄨𝐷 𝑥 𝑓 (𝑥) −𝐷 𝑥 𝑔 (𝑥)󵄨 ≤𝛼󵄨𝑓 (𝑥) −𝑔 (𝑥)󵄨 󵄨 󵄨 󵄨 󵄨 𝜕2𝑃 𝜕2𝑃 󵄨 󵄨 𝑐 (𝑥,) 𝑡 = +𝐹(𝑥,) 𝑡 +𝐵(𝑥,) 𝑡 𝑃. (22) +𝛽󵄨𝑓 (𝑥) −𝑔(𝑥)󵄨 , ∀𝑥𝜖𝐼. 𝜕𝑡2 𝜕𝑥2 (16) Here, We have, 1 Proof. Let 𝑥∈𝐼;then 𝑐 (𝑥,) 𝑡 = (1 + 2𝜀 −2𝜀 ), 𝑐2 𝑡 𝑥 󵄨 1+𝑢 1+𝑢 󵄨 󵄨𝐷 𝑥 𝑓 (𝑥) −𝐷 𝑥 𝑔 (𝑥)󵄨 󵄨 󵄨 𝜕𝑃 𝜕𝜀 1 𝜕𝑃 𝜕𝜀 𝐹 (𝑥,) 𝑡 =3( 𝑥 − 𝑡 ), 󵄨 2 (23) = 󵄨(1 + 𝑢 (𝑥))𝑓󸀠 (𝑥) +𝑢󸀠 (𝑥) 𝑓 (𝑥) 𝜕𝑥 𝜕𝑥 𝑐 𝜕𝑡 𝜕𝑡 󵄨 𝑥 𝑥 2 2 󸀠 󸀠 󵄨 𝜕 𝜀𝑥 1 𝜕 𝜀𝑡 −(1+𝑢 (𝑥))𝑔 (𝑥) −𝑢 (𝑥) 𝑔 (𝑥)󵄨 𝐵 (𝑥,) 𝑡 =( − ). 𝑥 𝑥 󵄨 𝜕𝑥2 𝑐2 𝜕𝑡2 󵄨 󵄨 󵄨 󸀠 󸀠 󵄨 󵄨 󸀠 󵄨 󵄨 󵄨 = 󵄨1+𝑢 (𝑥)󵄨 󵄨𝑓 (𝑥) −𝑔 (𝑥)󵄨 + 󵄨𝑢 (𝑥)󵄨 󵄨𝑓 (𝑥) −𝑔(𝑥)󵄨 . 󵄨 𝑥 󵄨 󵄨 󵄨 󵄨 𝑥 󵄨 󵄨 󵄨 It is easy to observe that (22)differsfrom(1)inthree (17) properties. 󸀠 First the velocity of the sound in this case depends on time But 𝑢𝑥(𝑥) is very small such that |1+𝑢𝑥(𝑥)| < 2 and |𝑢𝑥(𝑥)| > 0;itfollowsthat and coordinates due to the effect of the corresponding relative 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 density fluctuations. Secondly the force 󵄨1+𝑢 (𝑥)󵄨 󵄨𝑓󸀠 (𝑥) −𝑔󸀠 (𝑥)󵄨 + 󵄨𝑢󸀠 (𝑥)󵄨 󵄨𝑓 (𝑥) −𝑔(𝑥)󵄨 󵄨 𝑥 󵄨 󵄨 󵄨 󵄨 𝑥 󵄨 󵄨 󵄨 𝜕𝑃 𝜕𝜀 1 𝜕𝑃 𝜕𝜀 󵄨 󵄨 (18) 𝐹 (𝑥,) 𝑡 =3( 𝑥 − 𝑡 ) 󵄨 󸀠 󸀠 󵄨 󵄨 󵄨 2 (24) ≤𝛼󵄨𝑓 (𝑥) −𝑔 (𝑥)󵄨 +𝛽󵄨𝑓 (𝑥) −𝑔(𝑥)󵄨 . 𝜕𝑥 𝜕𝑥 𝑐 𝜕𝑡 𝜕𝑡 It is important to observe that if 𝑢=0,werecoverthe appears due to the coordinate and time dependence of properties of normal derivatives. the corresponding relative density fluctuations, and such 4 Mathematical Problems in Engineering force was considered in [12]. Third there is a derivative- The correction functional for5 ( ) can be approximately free term that depends on both time and space, and it is expressed for this matter as follows: proportional to 𝑃 and characterizes, depending on the coef- 𝑃 (𝑥,) 𝑡 =𝑃 (𝑥,) 𝑡 ficient sign, the retardation or enhancement propagation of 𝑛+1 𝑛 acoustic waves through a material medium. Therefore, even 𝑥 𝜕𝑚𝑃 (𝜁,) 𝑡 1 weak memory, which is taken into account by generalized −∫ 𝜆 (𝜁) [ − (𝑃̃(𝜁,)) 𝑡 −𝐻̃(𝜁,) 𝑡 ]𝑑𝜁, 𝜕𝜁𝑚 𝑐2 2𝑡 Riemann-Liouville fractional derivatives and presents the 0 characteristics of a corresponding relative density fluctua- (28) tions, transforms constant-coefficient velocity to varying- where 𝜆 is a general Lagrange multiplier [21, 22], which can be coefficient velocity. Moreover, this memory is responsible recognized optimally by means of variation assumption [21, for a force with which the corresponding relative density (𝑃(𝜁,̃ 𝑡)) 𝐻(𝑟,̃ 𝜏) fluctuations act on a propagation of acoustic waves through a 22]; here 2𝑡,and are considered as constrained material medium. This force appears only if the propagating variations. Making the previous functional stationary, we acoustic wave has memory depending on coordinates and obtain time; that it “remembers” its trajectories and time. Those 𝛿𝑃𝑛+1 (𝑥,) 𝑡 =𝛿𝑃𝑛 (𝑥,) 𝑡 terms in (22) that involve the corresponding relative density fluctuations additions𝐹 ( and 𝐵)tothetimeandspace 𝑥 𝜕𝑚𝑃 (𝜁,) 𝑡 (29) −𝛿∫ 𝜆 (𝜁) [ 𝑚 ]𝑑𝜁. dimensions are very small. Also the exact analytical solution 0 𝜕𝜁 of this modified equation is not easy to be determined. Therefore, this equation can be solved approximately by Capitulates the next Lagrange multipliers, giving up to the changing the function 𝑃(𝑟, 𝑡) to 𝑃0(𝑟, 𝑡), which satisfies1 ( ). following Lagrange multipliers 𝜆=−1for the case where 𝑚= 1 and 𝜆 = 𝑥−𝜁for 𝑚=2. For these matters 𝑚=2,weobtain 5. Solutions of the Modified Equation the following iteration formula: 𝑥 The modified equation can be reformulated as follows: 𝑃𝑛+1 (𝑥,) 𝑡 =𝑃𝑛 (𝑥,) 𝑡 − ∫ (𝑥−𝜁) 0 2 2 1 𝜕 𝑃 𝜕 𝑃 2 2 − + =𝐻(𝜀 ,𝜀,𝑃 (𝑟,) 𝑡 ) . (25) 𝜕 𝑃𝑛 (𝜁,) 𝑡 1 𝜕 𝑃𝑛 (𝜁,) 𝑡 𝑐2 𝜕𝑡2 𝜕𝑥2 𝑥 𝑡 0 ×[ − −𝐻(𝜁,) 𝑡 ]𝑑𝜁. 𝜕𝜁2 𝑐2 𝜕𝑡2

To solve (25) we need to give explicitly 𝜀𝑥 and 𝜀𝑡.For (30) example if one consider these functions to be It is worth noting that if the zeroth component 𝑃0(𝑥, 𝑡) is defined, then the remaining components 𝑛≥1can be 𝜀𝑥 =𝜀𝑜𝑥 sin 𝜃𝑥𝑥,𝑡 𝜀 =𝜀𝑜𝑡 sin 𝜃𝑡𝑡, (26) completely determined such that each term is determined where 𝜀𝑜𝑥 and 𝜀𝑜𝑡 are very significantly small such that |𝜀| ≪. 1 by using the previous terms, and the series solutions are 𝑃(𝑟, 𝑡) Since these parameters are very small then the solution of (25) thus entirely determined. Finally, the solution is canbesoughtintheform𝑃=𝑃1 +𝑃0 where 𝑃0 is the solution approximated by the truncated series 𝑃 ≪𝑃 𝜀 𝜀 𝜃 of (1)and 1 0 is proportional to 𝑥 and 𝑡.Here 𝑥 and 𝑁−1 𝜃𝑡 are frequencies characterizing variation in corresponding 𝑃𝑁 (𝑥,) 𝑡 = ∑ 𝑃𝑛 (𝑥,) 𝑡 , relative density fluctuations. 𝑛=0 (31) To solve (25)togetherwith(26)wemakeuseoftwo lim 𝑃𝑁 (𝑟,) 𝑡 =𝑃1 (𝑥,) 𝑡 . techniques including the Green function techniques and the 𝑁→∞ variational iterative decomposition technique. Here we will start with the variational iteration techniques. Here we choose the first term to be zero meaning 𝑃0(𝑥, 𝑡) = 𝑃(0, 𝑡)=0 and the second term can be determined 5.1. Variational Iteration Method. Variational iteration met- as hod has been favourably applied to various kinds of nonlinear 𝑥 problems. The main property of the method is in its flexibility 𝑃1 (𝑥,) 𝑡 =−∫ (𝑥−𝜁) [−𝐻 (𝜁,) 𝑡 ]𝑑𝜁. (32) andabilitytosolvenonlinearequationsaccuratelyandcon- 0 veniently. Very recently it was recognized that the variational Our next concern is to define 𝐻(𝑥, 𝑡); that is, we first iteration method [11, 13–20] can be an effective procedure need to provide the solution of (1)whichisfoundinthe for solution of various nonlinear problems without usual literatures [23]. The following solutions are obtained by restrictive assumptions. In this paper we will make use of this separation of variables in different coordinate systems. They iterative decomposition technique to solve the modified wave are phasor solutions, meaning that they have an implicit time- (26)togetherwith(25). To solve (25)bymeansofvariational dependence factor of exp(𝑖𝜔𝑡) where 𝜔=2𝜋𝑓is the angular iteration method, we put (25)intheformof frequency. The explicit time dependence is given by(33) 1 𝜔 (𝑃 (𝑥,) 𝑡 ) − (𝑃 (𝑥,) 𝑡 ) −𝐻(𝑥,) 𝑡 =0. 𝑃 (𝑥, 𝑘,) 𝑡 = [𝑝 (𝑘,) 𝑥 [𝑖𝜔] 𝑡 ], 𝑘= . 1 2𝑥 𝑐2 1 2𝑡 (27) Real exp 𝑐 (33) Mathematical Problems in Engineering 5

100 100

80 80 60

60 40

20 40 0 0 0.5 1 1.5 2 2.5 3 20 Figure 2: Topographic map of solution of acoustic wave equation for 𝑐=4and 𝑓 = 0.001. 0 0 0.5 1 1.5 2 2.5 3 100 Figure 1: Topographic map of the solution of the modified acoustic equation for 𝑐=4, 𝑓 = 0.001. 80

60

Introducing the previous expression in 𝐻(𝑥, 𝑡),weobtain the second expression. In this matter two components of 40 the decomposition series were obtained of which 𝑃(𝑥, 𝑡) was evaluatedtohavethefollowingexpansion: 20

𝑃1 (𝑥,) 𝑡 =𝑃1 (𝑥,) 𝑡 +𝑃0 (𝑥,) 𝑡 +⋅⋅⋅ (34) 0 0 0.5 1 1.5 2 2.5 3 The next figures show the graphical representation of theapproximatedsolutionofthemodifiedacousticwave Figure 3: Density plot of the solution of the modified acoustic wave equation and the exact solution of the standard version equation. of acoustic wave equation as function of space and time (contour plot and density plot of both solution). A contour plot gives essentially a topographic map of a solution and the andisgivenlaterinthecaseoftheclosedformsfortheGreen density plot shows the values of the function at a regular erray function for the infinite one-dimensional domain [23]: of points and lighter region of the contour plot is higher. 󵄨 󵄨 From the next Figures 1, 2, 3,and4,respectively,onecan 𝐺(𝑥,𝑡|𝑥0,𝑡0)=2𝜋𝑐𝑢((𝑡−𝑡0)−(󵄨𝑥−𝑥0󵄨)) , (36) see that there are more details with the solution of the mod- ified acoustic wave equation than in the standard solution, where meaning that the details left out by neglecting the small effect of the correspondent relative density fluctuations are very 𝑢 (𝑥) =0 if 𝑥<0,( 𝑢 𝑥) =1 if 𝑥>0. (37) importantwhenoneneedstoobservethepropagationofthe acoustic wave through the material medium.The approximate Following the Green function technique, the general solu- solution of (25) has been depicted in Figures 1 and 3 and the tion of the modified acoustic wave equation is given later as exact solution in Figures 2 and 4. 𝑡 𝑥 𝑃1 (𝑥,) 𝑡 = ∫ ∫ 𝐺(𝑥1,𝑡1 |𝑥0,𝑡0)𝐻(𝑥1,𝑡1)𝑑𝑡1 𝑑𝑥1, (38) 5.2. Green Function Method. To solve (25)togetherwith(26) 0 0 by means of Green function technique, one needs first to where 𝐻(𝑥1,𝑡1) remains the same as defined earlier in construct a suitable green function. Section 5.1. IfGisthegreenfunctiontobeconstructed,thenGmust satisfy the following equation: 6. Conclusion 𝜕2𝐺(𝑥,𝑡|𝑥 ,𝑡 ) 1 𝜕2𝐺(𝑥,𝑡|𝑥 ,𝑡 ) 0 0 − 0 0 𝜕𝑥2 𝑐2 𝜕𝑡2 In this paper, an acoustic wave equation was extended to (35) the concept of the modified Riemann-Liouville fractional =−4𝜋𝛿(𝑥−𝑥0)𝛿(𝑡−𝑡0). order derivative. We presented in detail some properties of the generalized Riemann-Liouville fractional order derivative Weareluckyenough,becausetheGreenfunctiontobe approximation. We presented the analysis of the generalized constructedhereisthegreenfunctionofthewaveequation equation. We highlighted the three differences between the 6 Mathematical Problems in Engineering

100 [9] A. Atangana and J. F. Botha, “Generalized groundwater flow equation using the concept of variable order derivative,” Bound- 80 ary Value Problems,vol.2013,p.53,2013. [10] H. G. Sun, W. Chen, and Y. Q. Chen, “Variable order fractional 60 differential operators in anomalous diffusion modeling,” Phys- ica A,vol.388,no.21,pp.4586–4592,2009. 40 [11] G. C. Wu, “New trends in the variational iteration method,” Communications in Fractional Calculus,vol.2,no.2,pp.59–75, 20 2011. [12]L.Ya.Kobelev,L.Ya.Kobelev,andL.Yu.Klimontovicht” 0 “Anomalous diffusion with time and coordinate-dependent 0 0.5 1 1.5 2 2.5 3 memory,” Dokiady Physics,vol.48,no.6,pp.264–268,2003. [13] G. C. Wu and D. Baleanu, “Variational iteration method for Figure 4: Density plot of the solution of the acoustic wave equation. the Burgers’ flow with fractional derivatives-New Lagrange multipliers,” Applied Mathematical Modelling,vol.37,no.9,pp. 6183–6190, 2013. generalized equation and the standard one. 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[8]R.L.Magin,O.Abdullah,D.Baleanu,andX.J.Zhou,“Anoma- lous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation,” Journal of Magnetic Resonance,vol.190,no.2,pp.255–270,2008. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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