A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

• Autorzy: Eid Doha , Ali Bhrawy , Mohammed Abdelkawy
• Języki publikacji: EN

Abstrakty

EN

In this paper, we propose an efficient spectral collocation algorithm to solve numerically wave type equations subject to initial, boundary and non-local conservation conditions. The shifted Jacobi pseudospectral approximation is investigated for the discretization of the spatial variable of such equations. It possesses spectral accuracy in the spatial variable. The shifted Jacobi-Gauss-Lobatto (SJ-GL) quadrature rule is established for treating the non-local conservation conditions, and then the problem with its initial and non-local boundary conditions are reduced to a system of second-order ordinary differential equations in temporal variable. This system is solved by two-stage forth-order A-stable implicit RK scheme. Five numerical examples with comparisons are given. The computational results demonstrate that the proposed algorithm is more accurate than finite difference method, method of lines and spline collocation approach

Słowa kluczowe

EN

non-local boundary conditions; integral conservation condition; collocation method; shifted Jacobi-Gauss-Lobatto quadrature; system of differential equations

• Czasopismo: Open Physics
• Rocznik: 2014
• Tom: 12
• Numer: 9
• Strony: 637-653

Daty

wydano

2014-09-01

online

2014-07-31

Twórcy

autor

Eid Doha

• Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt,

autor

Ali Bhrawy,

autor

Mohammed Abdelkawy

• Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt,

Bibliografia

• [1] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer-Verlag, New York, 2006)
• [2] E. H. Doha, A.H. Bhrawy, D. Baleanu, R.M. Hafez, Appl. Numer. Math. 77, 43 (2014) http://dx.doi.org/10.1016/j.apnum.2013.11.003[Crossref]
• [3] M. Tatari, M. Haghighi, Appl. Math. Model. 38, 1351 (2014) http://dx.doi.org/10.1016/j.apm.2013.08.008[Crossref]
• [4] A. H. Bhrawy, Appl. Math. Comput. 222, 255 (2013) http://dx.doi.org/10.1016/j.amc.2013.07.056[Crossref]
• [5] H. Wang, Comput. Phys. Commun. 181, 325 (2010) http://dx.doi.org/10.1016/j.cpc.2009.10.007[Crossref]
• [6] S. A. Khuri, A. Sayfy, Appl. Math. Comput. 216, 1047 (2010) http://dx.doi.org/10.1016/j.amc.2010.01.122[Crossref]
• [7] A. H. Bhrawy, M.A. Alghamdi, Boundary Value Problems 2012, 62 (2012) http://dx.doi.org/10.1186/1687-2770-2012-62[Crossref]
• [8] E. H. Doha, A.H. Bhrawy, R.M. Hafez, M.A. Abdelkawy, Appl. Math. Inf. Sci. 8, 535 (2014) http://dx.doi.org/10.12785/amis/080211[Crossref]
• [9] A. Saadatmandi, Appl. Math. Model. 38 1365 (2014) http://dx.doi.org/10.1016/j.apm.2013.08.007
• [10] E. H. Doha, A.H. Bhrawy, M.A. Abdelkawy, R.M. Hafez, Centr. Eur. J. Phys. 12, 111 (2014) http://dx.doi.org/10.2478/s11534-014-0429-z[Crossref]
• [11] X. Ma, C. Huang, Appl. Math. Model. 38, 1434 (2014) http://dx.doi.org/10.1016/j.apm.2013.08.013[Crossref]
• [12] E. H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Centr. Eur. J. Phys. 11, 1494 (2013) http://dx.doi.org/10.2478/s11534-013-0264-7[Crossref]
• [13] R. E. Ewing, T. Lin, Adv. Water Resour. 14, 89 (1991) http://dx.doi.org/10.1016/0309-1708(91)90055-S[Crossref]
• [14] L. Formaggia, F. Nobile, A. Quarteroni, A. Veneziani, Comput. Visual. Sci. 2, 75 (1999) http://dx.doi.org/10.1007/s007910050030[Crossref]
• [15] D. Bahuguna, R.K. Shukla, Nonlinear Dynam. Syst. Theor. 5, 345 (2005)
• [16] J. Dabas, D. Bahuguna, Math. Comput. Model. 50, 123 (2009) http://dx.doi.org/10.1016/j.mcm.2009.03.002[Crossref]
• [17] P. Shi, SIAM J. Numer. Anal. 24, 46 (1993) http://dx.doi.org/10.1137/0524004[Crossref]
• [18] J. Lee, R. Sakthivel, Pramana 76, 819 (2011) http://dx.doi.org/10.1007/s12043-011-0105-4[Crossref]
• [19] L. Girgis, A. Biswas, Appl. Math. Comput. 216, 2226 (2010) http://dx.doi.org/10.1016/j.amc.2010.03.056[Crossref]
• [20] A. Biswas, A.H. Kara, Appl. Math. Comput. 217, 4289 (2010) http://dx.doi.org/10.1016/j.amc.2010.09.054[Crossref]
• [21] R. Sakthivela, C. Chuna, A-R. Bae, Int. J. Comput. Math. 87, 2601 (2010) http://dx.doi.org/10.1080/00207160802691660[Crossref]
• [22] M. Dehghan, M. Tatari, Numer. Meth. Part. D. E. 24, 924 (2008) http://dx.doi.org/10.1002/num.20297[Crossref]
• [23] C. Chun, R. Sakthivel, Comput. Phys. Commun. 181, 1021 (2010) http://dx.doi.org/10.1016/j.cpc.2010.02.007[Crossref]
• [24] M. Dehghan, A. Saadatmandi, Chaos Soliton. Fract. 41, 1448 (2009) http://dx.doi.org/10.1016/j.chaos.2008.06.009[Crossref]
• [25] M. Dehghan, Numer. Meth. Part. D. E. 21, 24 (2005) http://dx.doi.org/10.1002/num.20019[Crossref]
• [26] F. Shakeri, M. Dehghan, Comput. Math. Appl. 56, 2175 (2008) http://dx.doi.org/10.1016/j.camwa.2008.03.055[Crossref]
• [27] W-T. Ang, Appl. Numer. Math. 56, 1054 (2006) http://dx.doi.org/10.1016/j.apnum.2005.09.006[Crossref]
• [28] M. Tatari, M. Dehghan, Appl. Math. Model. 33, 1729 (2009) http://dx.doi.org/10.1016/j.apm.2008.03.006[Crossref]
• [29] S. A. Khuri, A. Sayfy, Appl. Math. Comput. 218, 9187 (2012) http://dx.doi.org/10.1016/j.amc.2012.02.075[Crossref]
• [30] S. A. Khuri, A. Sayfy, Appl. Math. Comput. 217, 3993 (2010) http://dx.doi.org/10.1016/j.amc.2010.10.005[Crossref]
• [31] A. H. Bhrawy, A.S. Alofi, Commun. Nonlinear Sci. Numer. Simul. 17, 62 (2012) http://dx.doi.org/10.1016/j.cnsns.2011.04.025[Crossref]
• [32] S. A. Khuri, A. Sayfy, Appl. Math. Model. 38, 2901 (2014) http://dx.doi.org/10.1016/j.apm.2013.11.016[Crossref]
• [33] E. H. Doha, A.H. Bhrawy, R.M. Hafez, Math. Comput. Model. 53, 1820 (2011) http://dx.doi.org/10.1016/j.mcm.2011.01.002[Crossref]
• [34] D. Gottlieb, C.-W. Shu, SIAM Rev. 29, 644 (1997) http://dx.doi.org/10.1137/S0036144596301390[Crossref]
• [35] D. Tchiotsop, D. Wolf, V. Louis-Dorr, R. Husson, Ecg data compression using Jacobi polynomials, in: Proceedings of the 29th Annual International Conference of the IEEE EMBS, 1863 (2007)
• [36] E. H. Doha, A.H. Bhrawy, Numer. Meth. Part. D. E. 25, 712 (2009) http://dx.doi.org/10.1002/num.20369[Crossref]
• [37] M. El-Kady, J. Korean Math. Soc. 49, 99 (2012) http://dx.doi.org/10.4134/JKMS.2012.49.1.099[Crossref]
• [38] G. Szegö,, Colloquium Publications, XXIII, American Mathematical Society, ISBN 978-0-8218-1023-1, MR 0372517G (1939)
• [39] E. H. Doha, A.H. Bhrawy, Comput. Math. Appl. 64, 558 (2012) http://dx.doi.org/10.1016/j.camwa.2011.12.050[Crossref]
• [40] Y. Luke, The Special Functions and Their Approximations (Academic Press, New York, 1969)
• [41] J. C. Butcher, The numerical analysis of ordinary differential equations, Runge-Kutta and general linear methods (Wiley, New York, 1987)
• [42] E. H. Doha, Arab J. Math. 4, 31 (1983)
• [43] B. S. Attili, K. Furati, M.I. Syam, Appl. Math. Comput. 178, 229 (2006) http://dx.doi.org/10.1016/j.amc.2005.11.044[Crossref]
• [44] J. Cooper, J. Butcher, IMA J. Numer. Anal. 3, 127 (1983) http://dx.doi.org/10.1093/imanum/3.2.127[Crossref]
• [45] B. Ehle, Z. Picel, Math. Comput. 29, 501 (1975) http://dx.doi.org/10.1090/S0025-5718-1975-0375737-7[Crossref]
• [46] M. Serbin, Math. Comput. 35, 1231 (1980) http://dx.doi.org/10.1090/S0025-5718-1980-0583500-9[Crossref]
• [47] M. Dehghan, A. Shokri, J. Comput. Appl. Math. 230, 400 (2009) http://dx.doi.org/10.1016/j.cam.2008.12.011[Crossref]
• [48] S. M. El-Sayed, Chaos Soliton Fract. 18, 1025 (2003) http://dx.doi.org/10.1016/S0960-0779(02)00647-1[Crossref]
• [49] E. Y. Deeba, S.A. Khuri, J. Comput. Phys. 124, 442 (1996) http://dx.doi.org/10.1006/jcph.1996.0071[Crossref]
• [50] D. B. Duncan, SIAM J. Numer. Anal. 34, 1742 (1997) http://dx.doi.org/10.1137/S0036142993243106[Crossref]
• [51] M. J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, New York, 1991) http://dx.doi.org/10.1017/CBO9780511623998[Crossref]
• [52] J. Q. Mo, W.J. Zhang, M. He, Acta Phys. Sin. 56, 1847 (2007)

Identyfikatory

DOI

10.2478/s11534-014-0493-4