On the solution of an acoustic wave equation with variable-order derivative loss operator
© Atangana; licensee Springer 2013
Received: 17 February 2013
Accepted: 24 May 2013
Published: 12 June 2013
When modeling sound propagation, the use of fractional derivatives leads to models that better describe observations of attenuation and dispersion. The wave equation for viscous losses involving integer-order derivatives only leads to an attenuation which is proportional to the square of the frequency. This does not always reflect reality. The acoustic wave equation with loss operator is generalized to the concept of variable-order derivatives in this work. The generalized equation is solved via the Crank-Nicholson scheme. The stability and the convergence of this case are examined in detail.
Fractional derivatives are well suited to describe wave propagation in complex media. When introduced in classical wave equations, they allow the modeling of attenuation and dispersion that better describes sound propagation in biological tissues. Fractional derivatives have been used for modeling heat transfer or diffusion [1, 2], seismic data  and sound wave propagation, [4–6] only to name a few. When modeling sound propagation, the use of fractional derivatives leads to models that better describe observations of attenuation and dispersion . The wave equation for viscous losses involving integer-order derivatives only leads to an attenuation which is proportional to the square of the frequency. This does not always reflect reality. For instance, for biological tissues  and marine sediments , the frequency dependency of attenuation and dispersion is more complicated. Different forms of the wave equation have been proposed to reflect this complexity [4, 7, 10–12].
Several simulators take a modified nonlinear wave equation as a starting point by replacing the traditional loss operator by fractional derivatives [7, 13, 14] or a convolution in time [15, 16]. Their justification for modifying the standard wave equations is the ability of fractional derivatives to lead to a dispersion equation that better describes attenuation and dispersion. A wave equation based on fractional constitutive equations gives an alternative to modeling absorption and dispersion in complex media like biological tissues. However, in the case where the medium, through which these sounds are propagating, is variable or heterogeneous, neither the wave equation described by integer order nor that described by constant fractional order are suited for describing the phenomena. To solve the above problems, the variable-order (VO) fractional acoustic wave equation models are suggested for use in this work. This present work is therefore devoted to the discussion underpinning the description of the extension of an acoustic wave equation to the concept of the variational-order derivative and the solution of the generalized equation using the Crank-Nicholson scheme.
2 A possible modification of an acoustic wave equation
For the readers that are not acquainted with the concept of the variational-order derivative, we start this section by presenting the basic definition of this derivative.
2.1 Variable-order differential operator
The above derivative is called the Caputo variational-order differential operator, in addition, the derivative of the constant is zero.
2.2 Statement of the problem
where and y are constants that characterize the medium. Such attenuation can be described by wave equations with particular loss operators.
The above proposed loss operator is equivalent to the loss operator of the viscoelastic equation for . This operator is interesting as it turns out to be based on an underlying fractional Kelvin-Voigt model despite the impression Wismer  gives that it was found by inspection just like the previous operators. Although the wave equations given here may successfully be applied for wave propagation simulation, most of them are nevertheless derived through ad hoc procedures which are not directly linked to more basic physical principles.
The above equation is then called the ‘variable-order derivative acoustic wave equation with loss operator’. This new equation does not have obviously an exact analytical solution. In particular, the equation cannot be solved analytically. It is therefore important to examine its solution numerically. The aim of the next section is then devoted to the discussion underpinning the numerical solution of the modified equation using the Crank-Nicholson scheme.
3 Numerical solution of a variable-order derivative acoustic wave equation with loss operator
Numerical methods yield approximate solutions to the governing equation through the discretization of space and time. Within the discredited problem domain, the variable internal properties, boundaries and stresses of the system are approximated. Deterministic, distributed-parameter, numerical models can relax the rigid idealized conditions of analytical models or lumped-parameter models, and they can therefore be more realistic and flexible for simulating fields conditions. The finite difference schemes for constant-order time or space fractional diffusion equations have been widely studied [21, 22]. For constant-order time fractional diffusion equations, the implicit difference approximation scheme was proposed in . The weighted average finite difference methods were introduced in . The matrix approach for fractional diffusion equations was proposed in , and Hanert proposed a flexible numerical scheme for the discretization of the space-time fractional diffusion equation . Lately, the numerical schemes for a VO space fractional advection-dispersion equation was considered by the author of . An investigation of the explicit scheme for a VO nonlinear space fractional diffusion equation was done in . Before performing the numerical methods, we assume that equation (2.3) has a unique and sufficiently smooth solution. To establish the numerical schemes for the above equation, we let , , , , , , h is the step and τ is the time size, M and N are grid points.
3.1 Crank-Nicholson scheme 
4 Stability analysis of the Crank-Nicholson scheme
In this section, we analyze the stability conditions of the Crank-Nicholson scheme for the generalized acoustic wave equation.
To achieve this, we make use of the recurrence technique on the natural number k.
which completes the proof.
5 Convergence analysis of the Crank-Nicholson scheme
This can be achieved via the recurrence technique on the natural number k.
which completes the proof.
The interested can find the solvability of the Crank-Nicholson scheme in the work done by . Therefore the details of the proof will not be presented in this paper.
6 Numerical simulation
The wave equation for viscous losses involving integer-order derivatives only leads to an attenuation which is proportional to the square of the frequency. This does not always reflect reality. The acoustic wave equation with loss operator was generalized using some approaches of variational calculus. Since the modified equation is difficult to solve analytically, we make use of the numerical scheme to solve this new equation. The numerical used in solving this new equation is the Crank-Nicholson scheme. The convergence and the stability of this scheme in this case were presented in details.
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