Splitting the one-Dimensional Wave Equation, Part II: Additional Data are Given by an End Displacement Measurement

In this research, an unknown space-dependent force function in the wave equation is studied. This is a natural continuation of [1] and chapter 2 of [2] and [3], where the finite d ifference method (FDM)/boundary element method (BEM) , with the separation of variables method, were considered. Additional data are given by the one end displacement measurement . Moreover, it is a continuation of [3], with exchanging the boundary condition , where are extra data, by the initial condition . This is an ill-posed inverse force p roblem for linear hyperbolic equation. Therefore, in order to stabilize the solution, a zeroth-order Tikhonov regularization method is provided. To assess the accuracy, the minimum error between exact and numerical solutions for the force is computed for various regularization parameters. Numerical results are presented and a good agreement was obtained for the exact and noisy data.


Introduction
An unknown force function in the wave equation can be experienced in many engineering applications dealing with wave, wind, seismic, explosion, or noise excitations [2,4]. It can be found in physical problems as well; for instance, the vibrations of a spring or membrane, acoustic scattering, etc. The objective of this research is to provide the numerical solution for an inverse force problem for the nonhomogeneous hyperbolic equation, by considering the initial condition with boundary condition. Furthermore, in case of using Dirichlet boundary condition, the Neumann boundary conditions were taken as extra data. It is observed that we could also control the mixed data instead of the Dirichlet data. In a previous study [3], we used the finite difference method (FDM) to numerically discretize the wave equation with the method of separating the variables. Therefore, in order to extend the range of applicability, a different boundary condition has been applied in this study.
Similarly, as in [1][2][3], the resulting system of linear equations is ill-conditioned. Nearly, we obtained the same results (see Table 1, Table 2 and Figure 3 in [1]) and, for that reason, these tables and figures are omitted here. Consequently, we seek the Tikhonov regularization to regularize the solution.
This paper is organized as follows; Section 2 presents the mathematical formulation. Section 3 describes the numerical results and discussion. Section 4 includes the conclusions reached by this work.

Mathematical Formulations
The required equation for a vibrating bounded structure , acted upon by a force is given by the wave equation [1][2][3]5], as follows: 2 where , and represent the displacement, the initial displacement and velocity, respectively. The above equation is a direct well-posed problem if has been given, otherwise the problem becomes an inverse linear problem. Furthermore, in order to determine the pair solution ( ) we need to have extra conditions. For instance, (3) is a Dirichlet boundary condition, then an additional condition can be , namely: Also, we tried different conditions instead of the Dirichlet boundary condition (3) by using mixed boundary conditions. and in this case the additional condition has also changed to By splitting equation (1) into [1][2][3]6], where satisfies the well-posed direct problem, we obtain Numerically, FDM has been used for solving (7)-(9) and to get [2,3,5]. In order to get we will consider a different condition, i.e. using (5), namely the boundary condition (9) is changed to The formula for the solution using FDM [2,7,[5][6][7][8]  For testing the stability, we add noisy data to and respectively, as follows ̅̅̅̅̅ From a Gaussian normal distribution, can be determined where the mean zero and standard deviation are given by [1][2][3][4][5][6][7] o where represents the percentage of noise. The noisy data (26) also cause noise in o [1][2][3], as follows ̅̅̅̅̅ Finally, we apply the condition (24)|(25) with o replaced by o in a least-squares penalised sense by minimizing the Tikhonov functional, which is in general a zeroth-order Tikhonov regularization solution (for more details, see [1][2][3]) to deal with stability, Note that, from equations (26)-(29), "|" means "or".

Numerical Results and Discussion
In order to see how far the changing of boundary condition affects the accuracy of solution and to see the difference between boundary element method (BEM) and finite difference method (FDM), using the same example in [1][2][3], we have with the additional condition If the boundary condition is changed to Then, in this case, the over-determination can be as follows: Figure 1 shows the numerical results for obtained from (7)-(9) and from (7), (8)  In Figure-2(a), we note that the numerical solution of from (24) was received for fixed when considering the exact data without noise (i.e. ) and then comparing with analytical values for the sine Fourier series coefficients √ ∫ (see [1][2][3]). Also, when is given by equation (30), one can see that a good approach was obtained between the numerical and exact values for . However, the method in [1] was changed to FDM and an additional condition to condition Still, no different shape of the obtained figure can be seen; the reason is that the same exact values of and the same boundary conditions and were used. Furthermore, Figure 2(b) shows the ̅̅̅̅̅ obtained from (25), in comparison with the exact sine Fourier series coefficients √ ∫ ( ) , when for given by (30), there is a good agreement between the exact and numerical one, see Figure 3. Therefore, the method was changed to FDM instead of BEM, and there were different shapes of the plots obtained. We can still compare Figure-2 with Figures-(4 and 13) in [1]. In Figure-4, but in here . Also, in Figure-  where Dirichlet boundary condition (9) is applied, while Figure 3(b) represents these solutions when the mixed boundary condition (10) is used. These figures are obtained before adding noise to the exact data (i.e. ). It can be seen that accurate numerical solutions are achieved.
(a) (b)  Figure 4 shows the unstable numerical solution for with various . In Figure 4(a), oscillations become highly unbounded, as increases for when fixing . In this case we selected . In addition, and the various are shown in Figure 4( . In order to choose the best the error was calculated as | | √∑ [4,6,7]. From Figure-5, it can be seen that the minimum error occurs around and , when it reaches the best approach for as illustrated in Figure 6. Figure 5(a) and Figure 5(b) shows the minimum error around and , respectively. Meantime, a good approached value for numerical solution can be seen in Figure-6(a) and Figure-6

Conclusions
In this paper, we apply the one end data of Dirichlet or mixed boundary condition as additional data. Splitting the wave equation in two parts, first part was direct problem part when we solved it by (FDM), and second part was inverse problem part, in this part using separation of variables [3]. The problem is ill-posed, since adding a small noise in extra data causes an unstable force. In order to deal with this unsuitability, we employed the Tikhonov regularization method with minimum errors for selecting a good parameter for regularization. For future work, we plan to use as additional information.