Abstract
In the paper, a novel boundary consistent method, modified from the Fourier sine series method, is proposed to solve a non-homogeneous wave equation with non-homogeneous boundary conditions. Besides the usual Fourier sine series, we supplement two extra terms in the solution to consider the consistency of the wave equation at the boundaries. We point out a “mistake” in the conventional Fourier sine series method without considering the boundary consistency conditions, which might cause large boundary errors of the Fourier sine series solution. Numerical examples confirm the improvement of the accuracy of the boundary consistent method, which not only overcomes the boundary errors but also improves the accuracy in the whole domain about four orders, upon comparing with the conventional Fourier sine series method.
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References
D.L. Young, J.W. Ruan, Method of fundamental solutions for scattering problems of electromagnetic waves. Comput. Model. Eng. Sci. 7, 223–232 (2005)
L. Godinho, A. Tadeu, P. Amado Mendes, Wave propagation around thin structures using the MFS. Comput. Mater. Contin 5, 117–128 (2007)
D.L. Young, M.H. Gu, C.M. Fan, The time-marching method of fundamental solutions for wave equations. Eng. Anal. Bound. Elem. 33, 1411–1425 (2009)
M.H. Gu, D.L. Young, C.M. Fan, The method of fundamental solutions for one-dimensional wave equations. Comput. Mater. Contin. 11(11), 185–208 (2009)
E. Marsch, Relativistic wave equation for a massive charged particle with arbitrary spin. Eur. Phys. J. Plus 132(4), 188 (2017)
C.-S. Liu, The \(g\)-analytic function theory and wave equation. J. Math. Res. 7, 85–98 (2015)
C.-S. Liu, A global domain/boundary integral equation method for the inverse wave source and backward wave problems. Inv. Prob. Sci. Eng. 25, 506–531 (2017)
P.V. O’neil, Beginning Partial Differential Equations (Wiley, New York, 1999)
T. Myint-U, L. Debnath, Partial Differential Equations for Scientists and Engineers, 3rd edn. (Prentice-Hall, New Jersey, 1987)
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Liu, CS., Wang, F. Overcoming the near boundary error in the solution of non-homogeneous wave equation by a boundary consistent method. Eur. Phys. J. Plus 135, 5 (2020). https://doi.org/10.1140/epjp/s13360-019-00032-z
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DOI: https://doi.org/10.1140/epjp/s13360-019-00032-z