 Journal of Applied Mathematics and Physics, 2013, 1, 54-57 Published Online November 2013 (http://www.scirp.org/journal/jamp) http://dx.doi.org/10.4236/jamp.2013.15008 Open Access JAMP A Semi-Analytical Method for the Solution of Helmholtz Equation M. Tadi Department of Mechanical Engineering, University of Colorado at Denver, Denver, USA Email: mohsen.tadi@ucdenver.edu Received October 2, 2013; revised October 30, 2013; accepted November 5, 2013 Copyright © 2013 M. Tadi. This is an open access article distributed under the Creative Commons Attribution License, which per-mits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT This note is concerned with a semi-analytical method for the solution of 2-D Helmholtz equation in unit square. The method uses orthogonal functions to project the problem down to finite dimensional space. After the projection, the problem simplifies to that of obtaining solutions for second order constant coefficient differential equations which can be done analytically. Numerical results indicate that the method is particularly useful for very high wave numbers. Keywords: Helmholtz Equation; Elliptic Systems; High Wavenumber 1. Introduction In this note we consider a numerical method for the solu-tion of 2-D Helmholtz equation. The goal is provide a solution method that is more suitable for Helmholtz equation at high frequencies and can be applied to the Helmholtz equation in 3-D. Helmholtz equation appears very naturally in the study of wave propagation , after assuming a harmonic field. It is well-known that the numerical solution of the Helmholtz equation is a challenging problem for high frequencies. Higher order finite difference schemes have been developed for Helmholtz equation with homogene-ous domain [2-4]. Additional methods such as boundary element , discontinuous Galerkin method , multi- level multi-integral algorithm , iterative methods , and methods based on parallel computing  have also been developed for the numerical solution of Helmholtz equation. Recent results also include an iterative method based on ADI , a finite-element based semi-analytic method , and a method based on discrete singular convolution . The method presented in this note is based on or-thogonal functions. In addition to good accuracy for high wave numbers, it has the following specific new fea- tures. ● The actual calculation is performed analytically in 1-D only. ● It can also be applied to 3-D domains. Section 2 introduces the basic principles of the method. Section 3 presents the numerical results and Section 4 is devoted to the concluding remarks. 2. Mathematical Formulations Consider a 2-D Helmholtz equation given by 20,,0,1 ,xx yyuuku xy  (1)  0,, 1,,,0, ,1,wesnuyfyuyfyuxfx uxfx where, for simplicity, the domain is assumed to be a unit square. A set of Dirichlet boundary conditions are also given. In the present method, the solution to the above system is obtained as a linear combination of two similar problems. This is only to satisfy nonzero boundary con-ditions (Figure 1). The two problems can now be similarly treated. Con-sider the first problem, where the boundary conditions at 0x and 1x are set equal to zero. Consider an ex-pansion of the solution in the form  1,sinijuxyf yjx (2) where, the zero boundary conditions are satisfied auto-matically. A similar formulation can be used for the Helmholtz equation in 3-D (Appendix). Multiplying both sides of Equation (2) by , integrating both sides fover the domain sin jx0:1 , and using the orthogonal- M. TADI 55 Figure 1. The decomposition of the elliptic problem to two elliptic problems with apporipriate boundary conditions. ity condition  101ifsinsind 20otherwiseijixjx x it is possible to obtain a relationship for the functions jfy given by  102,sin dj.fyuxyjxx (3) Differentiating the above equation with respect to twice and using Equation (1) leads to y  2112200d2sin d2sin ddjxx .fyujxxkujxy x (4) Integrating the first integral on the right-hand side by parts twice leads to  2112011200d2sin cosdsind2sind .jxfujxjujxyju jxxku jxx  0 After applying the boundary conditions, the above equation simplifies to  2222d0.djfy kjfyy j (5) The above equation is now a simple constant coeffi-cient second-order differential equation. The appropriate boundary conditions are given by   101002 ,0sind,12 ,1sind.jjfuxjx xfuxjx x (6) The second-order differential equation can be solved. There are three separate cases. Case 1: , in this case the solution is given by 22kj  10cos0cossin ,sinjjjjfffyf yy (7) where, 22.kj1 Case 2: 22kj, in this case the solution is given in terms of exponential function. One needs to consider two separate cases. ● 22smalljk , or . In this case the solution is given by 22ejk 1e01e 0eeeee ejj jj,yyjff fffy  (8) where, 22jk. 22jk For this case, in order to avoid un- bounded functions, one needs to approximate e0. For this case, the solution is given by   11e01ee.yyjj jjfy fff  (9) Case 3: 22kj This is the singular case. For these cases, it is possible to treat the difaccording to the following. For , the differential equation leads to ferential equation 22kj 22d0.djjfy fyy (10) It is possible to obtain a solution in terms of the regu-lar perturbation according to 012233jj jjjfyf yfyf yf y  (11) The boundary conditions are imposed at the zero-th order. For higher-order terms, zero boundary conditions can be imposed. 0001 0jjjfy ffyf . (12) Additional terms can be added as needed. This com-pletes the solution for the first problem where only two of the boundary conditions are accounted for. The second problem can also be treated in a similar way. 1The case where sin 0 is the degenerate case . We are not treating that case here. Open Access JAMP M. TADI 56 3. Numerical Experiments In this section, we use a numerical example to investigate the applicability of the proposed method. The proposed method is particularly useful for very high values of the wave number. For these values none of the existing nu-merical methods can be applied within the available computer capacity. It is possible to use a problem for which analytical solutions exist . An exact solution for the problem is given by  ,coscos sinuxy kxy (13) where,  is the angle of the incoming wave. Using the exact solution one can provide the boundary conditions for the numerical method. The domain is divided into equal intervals in both x and . Figure 2 presents the numerical results for . The figure shows the error as a function of the number of orthogonal functions. There is little dependency to the mesh size ne. The error is the L2 norm of the difference between the numerical result and the exact solution divide by the number of nodes. y2000kFigures 3-5 show the same error reduction for higher wave numbers. Figure 3 presents the reduction in the error for . The number of intervals is equal to . For higher wave numbers, the same reduction in error is observed. For higher values of k, the method 15000k1600en Figure 2. Reduction in the error as a function of the num-ber of orthogonal functions. Other parameters are , 2000k4, , and the three maximum number of orthogonal functions are . e800n600,800,1600orthn Figure 3. Reduction in the error as a function of the num-ber of orthogonal functions. Other parameters are , 15000k4, , and the three maximum number of orthogonal functions are . e1600n3600,orthn4200,4800 Figure 4． Reduction in the error as a function of the number of orthogonal functions. Other parameters are 25000k, 4, e1600n, and the three maximum number of orthogonal functions are 6400,orthn7200,8400,10800 Figure 5． Reduction in the error as a function of the number of orthogonal functions. Other parameters are 100000k, 4, , and the three maximum number of orthogonal functions are e1600n3620022200,orthn30000,8400, simply requires the inclusion of more orthogonal func-tions in the expansion given in Equation (2). Figure 5 presents the result for and a similar reduction in the error is obtained. 510kThe present method is particularly useful for the Helmholtz equation at higher frequency for which exist-ing methods require a large amount of memory. The ac-tual calculation for the present method is done analyti-cally in one-dimension. 4. Conclusion In this note, we presented a numerical method for the solution of 2-D Helmholtz equation. The method can be applied to 3-D domains. It is based on orthogonal func-tions. Apart from the projection of the problem onto the space of orthogonal functions, the solution is obtained analytically. Numerical results for a number of cases with very high wave numbers were presented. REFERENCES  G. E. 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Li, “The Trefftz Method for the Helmholtz Equation with Degeneracy,” Applied Numerical Mathematics, Vol. 58, No. 2, 2008, pp. 131-159. http://dx.doi.org/10.1016/j.apnum.2006.11.004 Appendix In 3-D one can assume a form given by  11,,sin sin,ijijuxyzf yixjzd (14) here, after using the orthogonality conditions, the func-tions are given by )( yfij   11004,,sinsindij .fyuxyzixjzx z (15) Also, for the 3-D Helmholtz equation, one needs to split the problem into three similar problems.