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Hassan Kamil Jassim
Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq.
DOI: https://doi.org/10.17656/jzs.10421
Abstract
In this paper, we proposed local fractional Laplace decomposition method to solve the
Helmholtz and coupled Helmholtz equations on Cantor sets within local fractional
operator. The approximate solutions are obtained by using the local fractional Laplace
decomposition method, which is the coupling method of local fractional Laplace
transform and Adomian decomposition method. Illustrative examples are included to
demonstrate the high accuracy and fast convergence of this new method.
Key Words: Helmholtz equation; coupled Helmholtz equations, Local fractional operator.
References
[1] Kreb, R and Roach, G. F. "Transmission problems for the Helmholtz equation", Journal of Mathematical
Physics, Vol. 19, No. 6, pp. 1433–1437, 1978.
[2] Kleinman, R. E. and Roach, G. F. "Boundary integral equations for the three-dimensional Helmholtz
equation", SIAM Review, Vol. 16, pp. 214–236, 1974.
[3] Karageorghis, A. "The method of fundamental solutions for the calculation of the eigenvalues of the
Helmholtz equation", Applied Mathematics Letters, Vol. 14, No. 7, pp. 837–842, 2001.
[4] Fu, L. S. and Mura, T. "Volume integrals of ellipsoids associated with the inhomogeneous Helmholtz
equation", Wave Motion, Vol. 4, No. 2, pp. 141–149, 1982.
[5] Samuel, M. S. and A. Thomas, A. "On fractional Helmholtz equations", Fractional Calculus and Applied
Analysis, Vol. 13, No. 3, pp. 295–308, 2010.
[6] Hao, Y., Srivastava, H. M., Jafari, H. and Yang, X. J. "Helmholtz and diffusion equations associated with local
fractional derivative operators involving the Cantorian and Cantor-type cylindrical coordinates", Advances in
Mathematical Physics, Article ID 754248, pp. 1-5, 2013.
[7] Wang, X. J., Zhao, Y., Cattani, C. and Yang, X. J. "Local Fractional Variational Iteration Method for
Inhomogeneous Helmholtz Equation within Local Fractional Derivative Operator", Mathematical Problems in
Engineering, Article ID 913202, pp. 1-7, 2014.
[8] Yang, A.M., Chen, Z.S., Srivastava, H. M. and Yang, X. J. "Application of the local fractional series
expansion method and the variational iteration method to the Helmholtz equation involving local fractional
derivative operators,” Abstract and Applied Analysis, Article ID 259125, pp. 1-6, 2013.
[9] Yang, Y. J. and Hua, L. Q. "Variational Iteration Transform Method for Fractional Differential Equations with
Local Fractional Derivative", Abstract and Applied Analysis, Article ID 760957, pp.1-9, 2014.
[10] Jafari, H. and Jassim, H. K. "Numerical Solutions of Telegraph and Laplace Equations on Cantor Sets Using
Local Fractional Laplace Decomposition Method", International Journal of Advances in Applied Mathematics
and Mechanics, Vol. 2, No. 3, pp. 144-151, 2015.
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