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Space Time Fractional Telegraph Equation and its Application by Using Adomian Decomposition Method

Year 2018, Issue: 22, 73 - 81, 26.03.2018

Abstract

The telegraph equations are a pair of linear differential equations which describe the voltage and
current on an electrical transmission line with distance and time. In this paper the authors give a brief
overview of fractional calculus and extend its application to space-time fractional telegraph equation by using
Adomian decomposition method. The time- space derivates are considered as Caputo fractional derivate. The
solutions are obtained in the series form. 

References

  • [1] A. A. Kilbas, H. M. Srivastava & J. J. Trujillo, (2006), Theory and Applications of Fractional Differential Equations: Elsevier, Amsterdam, The Netherlands.
  • [2] W. Tomasi, (2004), Electronic Communication Systems, Prentice Hall, New Jersey.
  • [3] R. C. Cascaval, E. C. Eckstein, L. Frota & J. A. Goldstein , (2002), Fractional Telegraph Equations: J Math. Anal. Appl., 276, 145-159.
  • [4] E. Orsingher & L. Beghin, (2004), Time Fractional Telegraph Equation and Telegraph Process with Brownian Time: Prob. Theory Relat. Fields, 128, 141-160.
  • [5] J. Chen, F. Liu & V. Anh, (2008), Analytical Solution for the Time-Fractional Telegraph Equation by the method of Separating Variables: J. Math. Anal. Appl., 338, 1364-1377.
  • [6] S. Momani, (2005), Analytic and Approximate solutions of the space and time fractional telegraph equations: Appl. Math. Comput., 170, 1126-1134.
  • [7] G. Adomian, (1986), Non-linear Stochastic Operator Equations: Academic Press, San Diego.
  • [8] G. Adomian, (1994), Solving Frontier Problems of Physics: The Decomposition Method: Kluwer Acad. Pub., Boston.
  • [9] G. Adomian, (1988), A Review of the Decomposition Method in Applied Mathematics: J. Math. Anal. Appl. 135, 501-544.
  • [10] A. M. Wazwaz, (2008), A Study on Linear and Nonlinear Schrodinger equations by the Variational Iteration method: Chaos, Solitons and Fractals 37, 1136-1142.
  • [11] X. G. Luo, (2005), A two step Adomian decomposition method: Appl. Math Comput. 170(1), 570-583.
  • [12] B. Q. Zhang, X. G. Luo and Q. B. Wu, (2006), The restrictions and improvement of the Adomian decomposition method: Appl. Math. Comput. 177, 99-104.
  • [13] M. Caputo, (1969), Elasticita e Dissipazione: Zanichelli, Bologa, Italy.
  • [14] H. Weyl, (1917), Vierteljahrsschr. d. Naturf. Ges.; Zurich, 62, 296–302.
  • [15] E. A. Ibijola, B.J. Adegboyegun and O.Y. Halid, (2008), On Adomian decomposition method (ADM) for numerical solutions of ordinary differential equations: Advances in Natural and Applied Sciences, 3(3) 165-169.
  • [16] A. Atangana, (2015), On the stability and convergence of the time-fractional variable order telegraph equation: Journal of Computational Physics 293, 104-114.
  • [17] M. A. E. Herzallah, (2010), On abstract fractional telegraph equation: J. Comput. Nonlinear Dyn. 5, 5pp.
  • [18] P. Zjaung, F. Liv, (2006), Implicit difference approximation for the time fractional diffusion equation: J. Appl. Math Comput., 22, 87-99.
  • [19] M. Azreg-Ainov, (2009), A developed new algorithm for evaluating Adomian polynomials: CMES, 42(1) 1-18.
  • [20] E. Babolian, A. R. Vahidi and G. H. Asadi Cordshooli, (2005), Solving differential equations by decomposition method: Applied Mathematics and Computation, 167, 1150-115
Year 2018, Issue: 22, 73 - 81, 26.03.2018

Abstract

References

  • [1] A. A. Kilbas, H. M. Srivastava & J. J. Trujillo, (2006), Theory and Applications of Fractional Differential Equations: Elsevier, Amsterdam, The Netherlands.
  • [2] W. Tomasi, (2004), Electronic Communication Systems, Prentice Hall, New Jersey.
  • [3] R. C. Cascaval, E. C. Eckstein, L. Frota & J. A. Goldstein , (2002), Fractional Telegraph Equations: J Math. Anal. Appl., 276, 145-159.
  • [4] E. Orsingher & L. Beghin, (2004), Time Fractional Telegraph Equation and Telegraph Process with Brownian Time: Prob. Theory Relat. Fields, 128, 141-160.
  • [5] J. Chen, F. Liu & V. Anh, (2008), Analytical Solution for the Time-Fractional Telegraph Equation by the method of Separating Variables: J. Math. Anal. Appl., 338, 1364-1377.
  • [6] S. Momani, (2005), Analytic and Approximate solutions of the space and time fractional telegraph equations: Appl. Math. Comput., 170, 1126-1134.
  • [7] G. Adomian, (1986), Non-linear Stochastic Operator Equations: Academic Press, San Diego.
  • [8] G. Adomian, (1994), Solving Frontier Problems of Physics: The Decomposition Method: Kluwer Acad. Pub., Boston.
  • [9] G. Adomian, (1988), A Review of the Decomposition Method in Applied Mathematics: J. Math. Anal. Appl. 135, 501-544.
  • [10] A. M. Wazwaz, (2008), A Study on Linear and Nonlinear Schrodinger equations by the Variational Iteration method: Chaos, Solitons and Fractals 37, 1136-1142.
  • [11] X. G. Luo, (2005), A two step Adomian decomposition method: Appl. Math Comput. 170(1), 570-583.
  • [12] B. Q. Zhang, X. G. Luo and Q. B. Wu, (2006), The restrictions and improvement of the Adomian decomposition method: Appl. Math. Comput. 177, 99-104.
  • [13] M. Caputo, (1969), Elasticita e Dissipazione: Zanichelli, Bologa, Italy.
  • [14] H. Weyl, (1917), Vierteljahrsschr. d. Naturf. Ges.; Zurich, 62, 296–302.
  • [15] E. A. Ibijola, B.J. Adegboyegun and O.Y. Halid, (2008), On Adomian decomposition method (ADM) for numerical solutions of ordinary differential equations: Advances in Natural and Applied Sciences, 3(3) 165-169.
  • [16] A. Atangana, (2015), On the stability and convergence of the time-fractional variable order telegraph equation: Journal of Computational Physics 293, 104-114.
  • [17] M. A. E. Herzallah, (2010), On abstract fractional telegraph equation: J. Comput. Nonlinear Dyn. 5, 5pp.
  • [18] P. Zjaung, F. Liv, (2006), Implicit difference approximation for the time fractional diffusion equation: J. Appl. Math Comput., 22, 87-99.
  • [19] M. Azreg-Ainov, (2009), A developed new algorithm for evaluating Adomian polynomials: CMES, 42(1) 1-18.
  • [20] E. Babolian, A. R. Vahidi and G. H. Asadi Cordshooli, (2005), Solving differential equations by decomposition method: Applied Mathematics and Computation, 167, 1150-115
There are 20 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Research Article
Authors

Manzoor Ahmad This is me

Altaf Ahmad Bhat

Renu Jain This is me

Publication Date March 26, 2018
Submission Date January 28, 2018
Published in Issue Year 2018 Issue: 22

Cite

APA Ahmad, M., Bhat, A. A., & Jain, R. (2018). Space Time Fractional Telegraph Equation and its Application by Using Adomian Decomposition Method. Journal of New Theory(22), 73-81.
AMA Ahmad M, Bhat AA, Jain R. Space Time Fractional Telegraph Equation and its Application by Using Adomian Decomposition Method. JNT. March 2018;(22):73-81.
Chicago Ahmad, Manzoor, Altaf Ahmad Bhat, and Renu Jain. “Space Time Fractional Telegraph Equation and Its Application by Using Adomian Decomposition Method”. Journal of New Theory, no. 22 (March 2018): 73-81.
EndNote Ahmad M, Bhat AA, Jain R (March 1, 2018) Space Time Fractional Telegraph Equation and its Application by Using Adomian Decomposition Method. Journal of New Theory 22 73–81.
IEEE M. Ahmad, A. A. Bhat, and R. Jain, “Space Time Fractional Telegraph Equation and its Application by Using Adomian Decomposition Method”, JNT, no. 22, pp. 73–81, March 2018.
ISNAD Ahmad, Manzoor et al. “Space Time Fractional Telegraph Equation and Its Application by Using Adomian Decomposition Method”. Journal of New Theory 22 (March 2018), 73-81.
JAMA Ahmad M, Bhat AA, Jain R. Space Time Fractional Telegraph Equation and its Application by Using Adomian Decomposition Method. JNT. 2018;:73–81.
MLA Ahmad, Manzoor et al. “Space Time Fractional Telegraph Equation and Its Application by Using Adomian Decomposition Method”. Journal of New Theory, no. 22, 2018, pp. 73-81.
Vancouver Ahmad M, Bhat AA, Jain R. Space Time Fractional Telegraph Equation and its Application by Using Adomian Decomposition Method. JNT. 2018(22):73-81.


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