A Homotopy-Analysis Approach for Nonlinear Wave-Like Equations with Variable Coefficients

and Applied Analysis 3 Example 1. We consider the initial-boundary value problem u tt + u xx = −2u xt , x > 0, t > 0,

Note that the proposed method can be applied for equations like  (, )   −   =  (,   ,   ,   ) +  (, ) , (4) with the same type of initial-boundary conditions.Problems like (1)-( 3) model many problems in classical and quantum mechanics, solitons, and matter physics [1,2].If  is a function of  only and (, ) = const we obtain Klein-Gordon or sine-Gordon type equations.These models can describe some nonlinear phenomena; for example, wave-like equation can describe earthquake stresses [3], coupling currents in a flat multistrand two-layer superconducting cable [4], and nonhomogeneous elastic waves in soils [5].Typical examples of the wave-like equations with variable coefficients are Euler-Tricomi equation [6] or Chaplygin equation [7] given by   −   = 0,   −  ()   = 0, (5) which are useful in the study of transonic flow, where  = (, ) is the stream function of a plane-parallel steady-state gas flow, () is positive at subsonic and negative at supersonic speed, and  is the angle of inclination of the velocity vector.Some Chaplygin type of equation of the special form where  = () is the speed of sound, has also applications in the study of transonic flow [8].Note that we use the term wave-like equation to describe the partial differential equations with the terms   and   ; that is, the term "wave-like" may not correspond to the real physical waves, in general.
Recently, there has been a growing interest for obtaining the explicit solutions to wave-like and heat-like models by analytic techniques.Wazwaz [9] used the tanh method to obtain the exact solution of the sine-Gordon equation.Kaya [10] applied the modified decomposition method to solve the sine-Gordon equation.Aslanov [11] used homotopy perturbation method to solve Klein-Gordon type of equations with unbounded right-hand side.El-Sayed [12] and Wazwaz and Gorguis [13] used Adomian decomposition method for solving wave-like and heat-like problems.
The homotopy analysis method [14][15][16][17][18] is developed to search the accurate asymptotic solutions of nonlinear problems.Liao [17] proved that the homotopy analysis method (HAM) contains some other nonperturbation techniques, such as Adomian's decomposition method and Lyapunov's artificial small parameter.Hayat and Sajid [19] and Abbasbandy [20] pointed out that the homotopy perturbation method is only a special case of the HAM.Aslanov [21] used homotopy perturbation method to solve wave-like equations with initial-boundary conditions.Rajaraman [22] and Alomari at al. [23] applied HAM for solving nonlinear equations with initial conditions.Özis ¸and Agırseven [24] used homotopy perturbation method for solving heat-like and wave-like equations with variable coefficients.
Here we will further extend the applications of HAM to obtain an approximate series solution for the nonlinear wavelike equations with variable coefficients and with initialboundary conditions.The difficulty in the use of standard HAM is that the choice of the linear operator  in standard form (like  =   or  =   ) cannot control the boundary conditions (2)- (3).
Unlike the various approximation techniques, which are usually valid for problems with (only) initial conditions, our technique is applicable for a wide range of initial-boundary problems of types (1)-(3).The central idea here is that the problem, has a unique solution (see, e.g., [32]) and therefore there exists an inverse of the operator  :   −  2   .This operator can control all initial-boundary conditions in each step of HAM.
According to definition (12), the governing equation can be deduced from the zero-order deformation equation (9).Define the vector Differentiating equation ( 9)  times with respect to embedding parameter  and then setting  = 0, we have the so-called th-order deformation equation where

Applications
To demonstrate the advantages of our approach first we consider the wave-like linear equation with constant coefficients.
The traditional methods (with  =   ⋅ ⋅ ⋅ ) do not work for this kind of problems.For example, the operator  =   can not control the condition (0, ) = 0 in every iteration step, the same for the operator  =   ; that is, we need an operator that can control all initial/boundary conditions.Clearly the most appropriate operator should be the wave operator.We take () =  2 (, , )/ 2 −  2 (, , )/ 2 .Equation ( 1) suggests that we define the nonlinear operator as Using the above definition, we construct the zeroth-order deformation equation and the th order deformation equation with the initial/boundary conditions   (, 0) = 0, (  )  (, 0) = 0, and   (0, ) = 0. Now it follows from the theory of wave equations that [32] the solution of the equation   −  = 0 with the same initialboundary conditions is According to (21) we now successively obtain and it follows from that Example 2. We consider the initial-boundary value problem The exact solution is First we rewrite the equation as The solution of the equation   −   = 0 with the same initial-boundary conditions will be taken as an initial approximation: For  1 we have and therefore and continuing in this way, we obtain  2 =  3 = ⋅ ⋅ ⋅ = 0 and  exact =  0 +  1 .
The above example demonstrates the importance of the proposed technique with the use of wave operator.In fact, all traditional approaches with some auxiliary operator  =   do not work, since the (exact) solution is not analytical function on the whole region.For example, the approaches used in [23,24] can not be applied to solve this problem (note that the approaches used in [23,24] are effective and simple for the problems with analytical solutions and/or for the problems with initial conditions).
The exact solution for Example 3 is shown in Figure 1 and the approximate solution is shown in Figure 2. A higher accuracy level can be attained by evaluating some more terms.
The exact solution is  =  +  2 .We take  =   −   , and consequently The absolute errors between the exact and the two-term approximation of the series solution for some values of (, ) ∈ [0, 1] × [0, 1] for ℎ = −1 are shown in Table 2.The exact solution for Example 4 is shown in the Figure 3 and the approximate solution is shown in Figure 4. Example 5. Consider the nonlinear initial-boundary value problem whose exact solution is First we rewrite the equation as The solution of the equation   −   = 0 with the same initial-boundary conditions is For  1 we have

Discussion
The main goal of this work was to propose a reliable method for solving wave-like equations with variable coefficients.The proposed equations may not be solved by the method of separation of variables or by HAM or ADM in standard form.The main difficulty in the use of previous methods is related to the choice of an auxiliary linear operator .Traditional methods work effectively in case of analytical solutions in the whole region, that is, in fact, when some initial/boundary condition is supposed by another one or in case of some special type of equations (homogeneous, etc.).The proposed method was applied directly without any need for restrictive assumptions, and this gives it a wider applicability.This method is capable of greatly reducing the volume of computational work compared to standard approaches while still maintaining high accuracy of the approximate solution.A higher accuracy level can be attained even by evaluating some two-three terms in the series solution.
The approach was tested by employing the method to obtain solutions for several problems.The results obtained in all cases demonstrate the efficiency of this approach.

Figure 1 :
Figure 1: The exact solution for Example 3.

Figure 2 :
Figure 2: The approximate solution for Example 3.

Table 1 :
Maximum errors for Example 3.

Table 2 :
Maximum errors for Example 4.