Laplace Transform Analytical Restructure

Affiliation(s)

Department of Mathematics, Faculty of Basic Education, PAAET, Shaamyia, Kuwait.

M. S. Software Engg, SITE, V. I. T. University, Vellore, India.

Department of Mathematics, Faculty of Basic Education, PAAET, Shaamyia, Kuwait.

M. S. Software Engg, SITE, V. I. T. University, Vellore, India.

ABSTRACT

In this paper, the Laplace transform definition is implemented without resorting to Adomian decomposition nor Homotopy perturbation methods. We show that the said transform can be simply calculated by differentiation of the original function. Various analytic consequent results are given. The simplicity and efficacy of the method are illustrated through many examples with shown Maple graphs, and transform tables are provided. Finally, a new infinite series representation related to Laplace transforms of trigonometric functions is proposed.

In this paper, the Laplace transform definition is implemented without resorting to Adomian decomposition nor Homotopy perturbation methods. We show that the said transform can be simply calculated by differentiation of the original function. Various analytic consequent results are given. The simplicity and efficacy of the method are illustrated through many examples with shown Maple graphs, and transform tables are provided. Finally, a new infinite series representation related to Laplace transforms of trigonometric functions is proposed.

Cite this paper

F. Belgacem and R. Silambarasan, "Laplace Transform Analytical Restructure,"*Applied Mathematics*, Vol. 4 No. 6, 2013, pp. 919-932. doi: 10.4236/am.2013.46128.

F. Belgacem and R. Silambarasan, "Laplace Transform Analytical Restructure,"

References

[1] L. Debnath and D. Bhatta, “Integral Transforms and Their Applications,” 2nd Edition, C. R. C. Press, London, 2007.

[2] P. P. G. Dyke, “An Introduction to Laplace Transform and Fourier Series,” Springer-Verlag, London, 2004.

[3] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, “Tables of Integral transform,” Vol. 1, McGrawHill, New York, Toronto, London, 1954.

[4] M. Rahman, “Integral Equations and Their Applications,” WIT Press, Boston, 2007.

[5] J. L. Schiff, “Laplace Transform Theory and Applications,” Springer, Auckland, 2005.

[6] M. R. Spiegel, “Theory and Problems of Laplace Transforms,” Schaums Outline Series, McGraw-Hill, New York, 1965.

[7] W. T. Thomson, “Laplace Transformation Theory and Engineering Applications,” Prentice-Hall Engg Design Series, Printice-Hall Inc., New York, 1950.

[8] D. V. Widder, “The Laplace Transform,” Oxford University Press, London, 1946.

[9] F. B. M. Belgacem, A. A. Karaballi and S. L. Kalla, “Analytical Investigations of the Sumudu Transform and Applications to Integral Production Equations,” Mathematical Problems in Engineering (MPE), Vol. 2003, No. 3, 2003, pp. 103-118.

[10] F. B. M. Belgacem and A. A. Karaballi, “Sumudu Transform Fundamental Properties Investigations and Applications,” Journal of Applied Mathematics and Stochastic Analysis (JAMSA), 2005, Article ID: 91083.

[11] F. B. M. Belgacem, “Introducing and Analysing Deeper Sumudu Properties,” Nonlinear Studies Journal (NSJ), Vol. 13, No. 1, 2006, pp. 23-41.

[12] F. B. M. Belgacem, “Applications of Sumudu Transform to Indefinite Periodic Parabolic Equations,” In: Proceedings of the 6th International Conference on Mathematical Problems & Aerospace Sciences (ICNPAA 06), Chap. 6, Cambridge Scientific Publishers, Cambridge, 2007, pp 51-60.

[13] F. B. M. Belgacem, “Sumudu Applications to Maxwell’s Equations,” PIERS Online, Vol. 5, No. 4, 2009, pp 355360. doi:10.2529/PIERS090120050621

[14] F. B. M. Belgacem, “Sumudu Transform Applications to Bessel’s Functions and Equations,” Applied Mathematical Sciences, Vol. 4, No. 74, 2010, pp. 3665-3686.

[15] M. G. M. Hussain and F. B. M. Belgacem, “Transient Solutions of Maxwell’s Equations Based on Sumudu Transform,” Journal of Progress in Electromagnetics Research (PIER), Vol. 74, 2007, pp. 273-289. doi:10.2528/PIER07050904

[16] M. A. Rana, A. M. Siddiqui, Q. K. Ghori and R. Qamar, “Application of He’s Homotopy Perturbation Method to Sumudu Transform,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, No. 2, 2007, pp. 185-190.

[17] F. B. M. Belgacem and R. Silambarasan, “Theory of the Natural Transform,” Mathematics in Engineering, Science and Aerospace (MESA) Journal, Vol. 3, No. 1, 2012, pp. 99-124.

[18] F. B. M. Belgacem and R. Silambarasan, “Maxwell’s Equations Solutions by Means of the Natural Transform,” Mathematics in Engineering, Science and Aerospace (MESA) Journal, Vol. 3, No. 3, 2012, pp. 313-323.

[19] F. B. M. Belgacem and R. Silambarasan, “The Generalized n-th Order Maxwell’s Equations,” PIERS Proceedings, Moscow, 19-23 August 2012, pp. 500-503.

[20] F. B. M. Belgacem and R. Silambarasan, “Advances in the Natural Transform,” AIP Conference Proceedings, Vol. 1493, 2012, pp. 106-110. doi:10.1063/1.4765477

[21] R. Silambarasan and F. B. M. Belgacem, “Applications of the Natural Transform to Maxwell’s Equations,” PIERS Proceedings, Suzhou, 12-16 September 2011, pp. 899902.

[22] S. Abbasbandy, “Application of He’s Homotopy Perturbation Method for Laplace Transform,” Journal of Chaos, Solitons and Fractals, Vol. 30, No. 5, 2006, pp. 12061212. doi:10.1016/j.chaos.2005.08.178

[23] E. Babolian, J. Biazar and A. R. Vahidi, “A New Computational Method for Laplace Transform by Decomposition Method,” Journal of Applied Mathematics and Computation, Vol. 150, No. 3, 2004, pp. 841-846. doi:10.1016/S0096-3003(03)00312-6

[24] C. J. Efthimiou, “Trigonometric Series via Laplace Transform,” 2007. http://arxiv.org/abs/0707.3590v1

[25] S. Saitoh, “Theory of Reproducing Kernels: Applications to Approximate Solutions of Bounded Linear Operator Equations on Hilbert Spaces,” American Mathematical Society Translations, Vol. 230, No. 2, 2010, pp. 107-134.

[1] L. Debnath and D. Bhatta, “Integral Transforms and Their Applications,” 2nd Edition, C. R. C. Press, London, 2007.

[2] P. P. G. Dyke, “An Introduction to Laplace Transform and Fourier Series,” Springer-Verlag, London, 2004.

[3] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, “Tables of Integral transform,” Vol. 1, McGrawHill, New York, Toronto, London, 1954.

[4] M. Rahman, “Integral Equations and Their Applications,” WIT Press, Boston, 2007.

[5] J. L. Schiff, “Laplace Transform Theory and Applications,” Springer, Auckland, 2005.

[6] M. R. Spiegel, “Theory and Problems of Laplace Transforms,” Schaums Outline Series, McGraw-Hill, New York, 1965.

[7] W. T. Thomson, “Laplace Transformation Theory and Engineering Applications,” Prentice-Hall Engg Design Series, Printice-Hall Inc., New York, 1950.

[8] D. V. Widder, “The Laplace Transform,” Oxford University Press, London, 1946.

[9] F. B. M. Belgacem, A. A. Karaballi and S. L. Kalla, “Analytical Investigations of the Sumudu Transform and Applications to Integral Production Equations,” Mathematical Problems in Engineering (MPE), Vol. 2003, No. 3, 2003, pp. 103-118.

[10] F. B. M. Belgacem and A. A. Karaballi, “Sumudu Transform Fundamental Properties Investigations and Applications,” Journal of Applied Mathematics and Stochastic Analysis (JAMSA), 2005, Article ID: 91083.

[11] F. B. M. Belgacem, “Introducing and Analysing Deeper Sumudu Properties,” Nonlinear Studies Journal (NSJ), Vol. 13, No. 1, 2006, pp. 23-41.

[12] F. B. M. Belgacem, “Applications of Sumudu Transform to Indefinite Periodic Parabolic Equations,” In: Proceedings of the 6th International Conference on Mathematical Problems & Aerospace Sciences (ICNPAA 06), Chap. 6, Cambridge Scientific Publishers, Cambridge, 2007, pp 51-60.

[13] F. B. M. Belgacem, “Sumudu Applications to Maxwell’s Equations,” PIERS Online, Vol. 5, No. 4, 2009, pp 355360. doi:10.2529/PIERS090120050621

[14] F. B. M. Belgacem, “Sumudu Transform Applications to Bessel’s Functions and Equations,” Applied Mathematical Sciences, Vol. 4, No. 74, 2010, pp. 3665-3686.

[15] M. G. M. Hussain and F. B. M. Belgacem, “Transient Solutions of Maxwell’s Equations Based on Sumudu Transform,” Journal of Progress in Electromagnetics Research (PIER), Vol. 74, 2007, pp. 273-289. doi:10.2528/PIER07050904

[16] M. A. Rana, A. M. Siddiqui, Q. K. Ghori and R. Qamar, “Application of He’s Homotopy Perturbation Method to Sumudu Transform,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, No. 2, 2007, pp. 185-190.

[17] F. B. M. Belgacem and R. Silambarasan, “Theory of the Natural Transform,” Mathematics in Engineering, Science and Aerospace (MESA) Journal, Vol. 3, No. 1, 2012, pp. 99-124.

[18] F. B. M. Belgacem and R. Silambarasan, “Maxwell’s Equations Solutions by Means of the Natural Transform,” Mathematics in Engineering, Science and Aerospace (MESA) Journal, Vol. 3, No. 3, 2012, pp. 313-323.

[19] F. B. M. Belgacem and R. Silambarasan, “The Generalized n-th Order Maxwell’s Equations,” PIERS Proceedings, Moscow, 19-23 August 2012, pp. 500-503.

[20] F. B. M. Belgacem and R. Silambarasan, “Advances in the Natural Transform,” AIP Conference Proceedings, Vol. 1493, 2012, pp. 106-110. doi:10.1063/1.4765477

[21] R. Silambarasan and F. B. M. Belgacem, “Applications of the Natural Transform to Maxwell’s Equations,” PIERS Proceedings, Suzhou, 12-16 September 2011, pp. 899902.

[22] S. Abbasbandy, “Application of He’s Homotopy Perturbation Method for Laplace Transform,” Journal of Chaos, Solitons and Fractals, Vol. 30, No. 5, 2006, pp. 12061212. doi:10.1016/j.chaos.2005.08.178

[23] E. Babolian, J. Biazar and A. R. Vahidi, “A New Computational Method for Laplace Transform by Decomposition Method,” Journal of Applied Mathematics and Computation, Vol. 150, No. 3, 2004, pp. 841-846. doi:10.1016/S0096-3003(03)00312-6

[24] C. J. Efthimiou, “Trigonometric Series via Laplace Transform,” 2007. http://arxiv.org/abs/0707.3590v1

[25] S. Saitoh, “Theory of Reproducing Kernels: Applications to Approximate Solutions of Bounded Linear Operator Equations on Hilbert Spaces,” American Mathematical Society Translations, Vol. 230, No. 2, 2010, pp. 107-134.