Authors:
Pankaj,Gurpreet Singh,DOI NO:
https://doi.org/10.26782/jmcms.spl.11/2024.05.00008Keywords:
Telegraph Equations,Laplace Transform,Variational iteration method,Abstract
This study adopts a novel approach that integrates the Laplace transform and the variational iteration method to tackle the (1+1)-Dimensional Telegraph equations, representing the current or voltage flow in electrical circuits. The methodology commences with transforming the telegraph equation into a modified format using Laplace transformation. Following this, the variational iteration method is utilized to derive both numerical and approximate analytical solutions. The paper incorporates practical examples to demonstrate the effectiveness of the proposed approach, supplemented with graphical illustrations depicting the outcomes achieved through the suggested techniques.Refference:
I. A. S. Arife and A. Yildirim. : ‘New modified variational iteration transform method (MVITM) for solving eighth-order boundary value problems in one step’. World Applied Sciences Journal. Vol. 13(10), pp. 2186–2190 (2011).
II. A. Saadatmandi, M. Dehghan. : ‘Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method’. Numerical Methods Partial Differ. Equation. Vol. 26(1), pp. 239-252 (2010). 10.1002/num.20442
III. B. Blbl, M. Sezer. : ‘A Taylor matrix method for the solution of a two-dimensional linear hyperbolic equation’. Applied Mathematics Letter. Vol. 24(10), pp. 1716-1720 (2011). 10.1016/j.aml.2011.04.026
IV. E. Hesameddini and H. Latifizadeh. : ‘Reconstruction of variational iteration algorithms using the Laplace transform’. International Journal of Nonlinear Sciences and Numerical Simulation. Vol. 10(11-12), pp. 1377-1382 (2009). 10.1515/IJNSNS.2009.10.11-12.1377
V. F. Gao, C. Chi. : ‘Unconditionally stable difference schemes for a one space-dimensional linear hyperbolic equation’. Appl. Math. Comput., Vol. 187(2), pp. 1272-1276 (2007). 10.1016/j.amc.2006.09.057
VI. G. C. Wu. : ‘Laplace transform overcoming principle drawbacks in application of the variational iteration method to fractional heat equations’. Thermal Science. Vol. 16(4), pp. 1257-1261, (2012). 10.2298/TSCI1204257W
VII. H. Y. Martinez, J. F. Gomez-Aguilar. : ‘Laplace variational iteration method for modified fractional derivatives with non-singular kernel’. Journal of Applied and Computational Mechanics. Vol. 6(3), pp. 684-698 (2020). 10.22055/JACM.2019.31099.1827
VIII. I. Singh, S. Kumar. : ‘Wavelet methods for solving three- dimensional partial differential equations’. Mathematical Sciences. Vol. 11, pp. 145-154 (2017). 10.1007/s40096-017-0220-6
IX. I. Singh, S. Kumar. : ‘Numerical solution of two- dimensional telegraph equations using Haar wavelets’. Journal of Mathematical Extension. Vol. 11(4), pp. 1-26 (2017). 10.1007/s40096-017-0220-6
X. J. H. He. : ‘Variational iteration method-a kind of non-linear analytical technique: some examples’. International Journal of Non-Linear Mechanics. Vol. 34(4), pp. 699-708 (1999). https://tarjomefa.com/wp-content/uploads/2018/06/9165-English-TarjomeFa.pdf
XI. K. D. Sharma, R. Kumar, M. K. Kakkar, S. Ghangas. : ‘Three dimensional waves propagation in thermo-viscoelastic medium with two temperature and void’. In IOP Conference Series: Materials Science and Engineering Vol. 1033(1), pp. 012059-73 (2021). 10.1088/1757-899X/1033/1/012059
XII. M. Aneja, M. Gaur, T. Bose, P. K. Gantayat, R. Bala. : ‘Computer-based numerical analysis of bioconvective heat and mass transfer across a nonlinear stretching sheet with hybrid nanofluids’. In International Conference on Frontiers of Intelligent Computing: Theory and Applications. pp. 677-686 (2023). 10.1007/978-981-99-6702-5_55
XIII. M. Dehghan, A. Ghesmati. : ‘Solution of the second-order one dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method’. Eng. Anal. Bound. Elem., Vol. 34(1), pp. 51-59 (2010). 10.1016/j.enganabound.2009.07.002
XIV. M. Dehghan, and M. Lakestani. : ‘The use of chebyshev cardinal functions for solution of the second‐order one‐dimensional telegraph equation’. Numerical Methods for Partial Differential Equations. Vol. 25(4), pp. 931-938 (2009). 10.1002/num.20382
XV. M. Javidi, N. Nyamoradi. : ‘Numerical solution of telegraph equation by using LT inversion technique’. International Journal of Advanced Mathematical Sciences. Vol. 1(2), pp. 64-77 (2013). 10.14419/ijams.v1i2.780
XVI. M. Lakestani, B. N. Saray. : ‘Numerical solution of telegraph equation using interpolating scaling functions’. Comput. Math. Appl., Vol. 60(7), pp. 1964-1972 (2010). 10.1016/j.camwa.2010.07.030
XVII. R. C. Mittal, R. Bhatia. : ‘Numerical solution of second order one dimensional hyperbolic telegraph equation by cubic B-spline collocation method’. Appl. Math. Comput., Vol. 220, pp. 496-506 (2013). 10.1016/j.amc.2013.05.081
XVIII. R. Jiwari R, S. Pandit S, R.C. Mittal. : ‘A differential quadrature algorithm for the numerical solution of the second-order one dimensional hyperbolic telegraph equation’. International Journal of Nonlinear Science. Vol. 13(3), pp. 259-266 (2012).
XIX. S. Yüzbaşı, M. Karaçayır. : ‘A Galerkin-type method to solve one-dimensional telegraph equation using collocation points in initial and boundary conditions’. International Journal of Computational Methods. Vol. 25(15), 1850031 (2018). 10.1142/S0219876218500317
XX. T. M. Elzaki, E. M. A. Hilal. : ‘Analytical Solution for Telegraph Equation by Modified of Sumudu Transform “Elzaki Transform”. Mathematical Theory and Modeling. Vol. 2(4), pp. 104-111 (2012). https://core.ac.uk/download/pdf/234678972.pdf
XXI. T. M. Elzaki. : ‘Solution of nonlinear partial differential equations by new Laplace variational iteration method’. Differential Equations: Theory and Current Research. (2018).
XXII. T. S. Jang. : ‘A new solution procedure for the nonlinear telegraph equation’. Communications in Nonlinear Science and Numerical Simulation. Vol. 29(1-3), pp. 307-326 (2013). 10.1016/j.cnsns.2015.05.004
XXIII. Z. Hammouch and T. Mekkaoui. : ‘A Laplace-variational iteration method for solving the homogeneous Smoluchowski coagulation equation’. Applied Mathematical Sciences. Vol. 6(18), pp. 879-886 (2012). https://www.m-hikari.com/ams/ams-2012/ams-17-20-2012/hammouchAMS17-20-2012.pdf