## Exact analytical solutions of fractional order telegraph equations via triple Laplace transform

1\. | Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber, Pakhtunkhwa, Pakistan  
---|---  
2\. | Department of Mathematics, Sun Yat-Sen University, Guangzhou, China  
3\. | Department of Mathematics, Çankaya University, 06790 Etimesgut, Ankara, Turkey  
  
In this paper, we study initial/boundary value problems for $ 1+1 $ dimensional and $ 1+2 $ dimensional fractional order telegraph equations. We develop the technique of double and triple Laplace transforms and obtain exact analytical solutions of these problems. The techniques we develop are new and are not limited to only telegraph equations but can be used for exact solutions of large class of linear fractional order partial differential equations

**Keywords:** Fractional telegraph equations,  initial/ boundary value problems,  double and triple Laplace transform,  Exact solutions. 

**Mathematics Subject Classification:** Primary:26A33;Secondary:34A08, 35R11. 

**Citation:** Rahmat Ali Khan, Yongjin Li, Fahd Jarad. Exact analytical solutions of fractional order telegraph equations via triple Laplace transform. _Discrete & Continuous Dynamical Systems - S_, doi: 10.3934/dcdss.2020427

##### **References:**

[1] |  M. A. Abdou, Adomian decomposition method for solving telegraph equation in charged particle transport, _J. Quant. Spectosc. Ra._ , **95** (2005), 407-414. doi: 10.1016/j.jqsrt.2004.08.045.  
---|---  
[2] |  A. M. O. Anwar, F Jarad, D Baleanu and F Ayaz, Fractional Caputo heat equation within the double Laplace transform, _Romanian J. Phys._ , **58** (2013), 15-22.  
[3] |  J. Biazar and M. Eslami, Analytic solution for telegraph equation by differential transform method, _Phys. Lett. A_ , **374** (2010), 2904-2906. doi: 10.1016/j.physleta.2010.05.012.  
[4] |  L. Debnath, Recent applications of fractional calculus to science and engineering, _Int. J. Math. Math. Sci._ , **(2003)** (2003), 3413–3442. doi: 10.1155/S0161171203301486.  
[5] |  M. Dehghan and A. Shokri, A numerical method for solving the hyperbolic telegraph equation, _Numer. Meth. Part. D. E._ , **24** (2008), 1080-1093. doi: 10.1002/num.20306.  
[6] |  M. Dehghan and M. Lakestani, The use of Chebyshev cardinal functions for solution of the secondorder one-dimensional telegraph equation, _Numer. Meth. for Part. D. E._ , **25** (2009), 931-938. doi: 10.1002/num.20382.  
[7] |  M. Dehghan and A. Ghesmati, Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation method, _Eng. Anal. Bound. Elem._ , **34** (2010), 51-59. doi: 10.1016/j.enganabound.2009.07.002.  
[8] |  M.Dehghan and A. Mohebbi, High order implicit collocation method for the solution of two-dimensional linear hyperbolic equation, _Numer. Meth. for Part. D. E._ , **25** (2009), 232-243. doi: 10.1002/num.20341.  
[9] |  M. Dehghan and A. Ghesmati, Combination of meshless local weak and strong forms to solve the two dimensional hyperbolic telegraph equation, _Eng. Anal. Bound. Elem._ , **34** (2010), 324-336. doi: 10.1016/j.enganabound.2009.10.010.  
[10] |  M. Dehghan and A. Shokri, A meshless method for numerical solution of a linear hyperbolic equation with variable coefficients in two space dimensions, _Numer. Meth. Part. D. E._ , **25** (2009), 494-506. doi: 10.1002/num.20357.  
[11] |  H. Ding and Y. Zhang, A new fourth-order compact finite difference scheme for the two-dimensional second-order hyperbolic equation, _J. Comput. Appl. Math._ , **230** (2009), 626-632. doi: 10.1016/j.cam.2009.01.001.  
[12] |  H. J. Haubold, A. M. Mathai and R. K. Saxena, Mittag-Leffler Functions and Their Applications, _J. Appl. Math._ , **2011** (2011), Art. ID 298628, 51 pp. doi: 10.1155/2011/298628.  
[13] |  R. Hilfer, _Applications of Fractional Calculus in Physics_ , Word Scientific, Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.  
[14] |  F. Jarad and K. Tas, Application of Sumudu and double Sumudu transforms to Caputo-Fractional differential equations, _J. Comput. Anal. Appl._ , **14** (2012), 475-483.  
[15] |  T. Khan, K. Shah, A. Khan and R. A. Khan, Solution of fractional order heat equation via triple Laplace transform in two dimensions, _Math Meth Appl Sci._ , **41** (2018), 818-825. doi: 10.1002/mma.4646.  
[16] |  A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, _Theory and Application of Fractional Differential Equations_ , North Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.  
[17] |  M. Lakestani and B. N. Saray, Numerical solution of telegraph equation using interpolating scalling function, _Comp. Math. Appl._ , **60** (2010), 1964-1972 doi: 10.1016/j.camwa.2010.07.030.  
[18] |  R. L. Magin, _Fractional Calculus in Bioengineering_ , Begell House Publishers, Redding, 2006.  
[19] |  R. K. Mohanty and M. K. Jain, An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation, _Numer. Meth. for Part. D. E._ , **17** (2001), 684-688. doi: 10.1002/num.1034.  
[20] |  R. K. Mohanty, An operator splitting method for an unconditionally stable difference scheme for a linear hyperbolic equation with variable coefficients in two space dimensions, _Appl. Math. Comput._ , **152** (2004), 799-806. doi: 10.1016/S0096-3003(03)00595-2.  
[21] |  R. K. Mohanty, M. K. Jain and K. George, On the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients, _J. Comp. Appl. Math._ , **72** (1996), 421-431. doi: 10.1016/0377-0427(96)00011-8.  
[22] |  R. K. Mohanty, New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations, _Int. J. Comp. Math._ , **86** (2009), 2061-2071 doi: 10.1080/00207160801965271.  
[23] |  R. K. Mohanty, M. K. Jain and U. Arora, An unconditionally stable ADI method for the linear hyperbolic equation in three space dimensions, _Int. J. Comp. Math._ , **79** (2002), 133-142. doi: 10.1080/00207160211918.  
[24] |  A. Mohebbi and M. Dehghan, High order compact solution of the one-space-dimensional linear hyperbolic equation, _Numer. Meth. Part. D. E._ , **24** (2008), 1222-1235. doi: 10.1002/num.20313.  
[25] |  I. Podlubny, _Fractional Differential Equations_ , Mathematics in Science and Engineering, Academic Press, New York, 1999.   
[26] |  A. Saadatmandi and M. Dehghan, Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method, _Numer. Meth. for Part. D. E._ , **26** (2010), 239- 252. doi: 10.1002/num.20442.  
[27] |  S. G. Samko, A. A. Kilbas and O. I. Marichev, _Fractional Integrals and Derivatives: Theory and Applications_ , Gordon and Breach, Yverdon, (1993).  
[28] |  C. Shu, Q. Yao and K. S. Yeo, Block-marching in time with DQ discretization: An efficient method for time-dependent problems, _Comput. Methods Appl. Mech. Engrg_ , **191** (2002), 4587-4597. doi: 10.1016/S0045-7825(02)00387-0.  
  
##### **References:**

[1] |  M. A. Abdou, Adomian decomposition method for solving telegraph equation in charged particle transport, _J. Quant. Spectosc. Ra._ , **95** (2005), 407-414. doi: 10.1016/j.jqsrt.2004.08.045.  
---|---  
[2] |  A. M. O. Anwar, F Jarad, D Baleanu and F Ayaz, Fractional Caputo heat equation within the double Laplace transform, _Romanian J. Phys._ , **58** (2013), 15-22.  
[3] |  J. Biazar and M. Eslami, Analytic solution for telegraph equation by differential transform method, _Phys. Lett. A_ , **374** (2010), 2904-2906. doi: 10.1016/j.physleta.2010.05.012.  
[4] |  L. Debnath, Recent applications of fractional calculus to science and engineering, _Int. J. Math. Math. Sci._ , **(2003)** (2003), 3413–3442. doi: 10.1155/S0161171203301486.  
[5] |  M. Dehghan and A. Shokri, A numerical method for solving the hyperbolic telegraph equation, _Numer. Meth. Part. D. E._ , **24** (2008), 1080-1093. doi: 10.1002/num.20306.  
[6] |  M. Dehghan and M. Lakestani, The use of Chebyshev cardinal functions for solution of the secondorder one-dimensional telegraph equation, _Numer. Meth. for Part. D. E._ , **25** (2009), 931-938. doi: 10.1002/num.20382.  
[7] |  M. Dehghan and A. Ghesmati, Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation method, _Eng. Anal. Bound. Elem._ , **34** (2010), 51-59. doi: 10.1016/j.enganabound.2009.07.002.  
[8] |  M.Dehghan and A. Mohebbi, High order implicit collocation method for the solution of two-dimensional linear hyperbolic equation, _Numer. Meth. for Part. D. E._ , **25** (2009), 232-243. doi: 10.1002/num.20341.  
[9] |  M. Dehghan and A. Ghesmati, Combination of meshless local weak and strong forms to solve the two dimensional hyperbolic telegraph equation, _Eng. Anal. Bound. Elem._ , **34** (2010), 324-336. doi: 10.1016/j.enganabound.2009.10.010.  
[10] |  M. Dehghan and A. Shokri, A meshless method for numerical solution of a linear hyperbolic equation with variable coefficients in two space dimensions, _Numer. Meth. Part. D. E._ , **25** (2009), 494-506. doi: 10.1002/num.20357.  
[11] |  H. Ding and Y. Zhang, A new fourth-order compact finite difference scheme for the two-dimensional second-order hyperbolic equation, _J. Comput. Appl. Math._ , **230** (2009), 626-632. doi: 10.1016/j.cam.2009.01.001.  
[12] |  H. J. Haubold, A. M. Mathai and R. K. Saxena, Mittag-Leffler Functions and Their Applications, _J. Appl. Math._ , **2011** (2011), Art. ID 298628, 51 pp. doi: 10.1155/2011/298628.  
[13] |  R. Hilfer, _Applications of Fractional Calculus in Physics_ , Word Scientific, Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.  
[14] |  F. Jarad and K. Tas, Application of Sumudu and double Sumudu transforms to Caputo-Fractional differential equations, _J. Comput. Anal. Appl._ , **14** (2012), 475-483.  
[15] |  T. Khan, K. Shah, A. Khan and R. A. Khan, Solution of fractional order heat equation via triple Laplace transform in two dimensions, _Math Meth Appl Sci._ , **41** (2018), 818-825. doi: 10.1002/mma.4646.  
[16] |  A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, _Theory and Application of Fractional Differential Equations_ , North Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.  
[17] |  M. Lakestani and B. N. Saray, Numerical solution of telegraph equation using interpolating scalling function, _Comp. Math. Appl._ , **60** (2010), 1964-1972 doi: 10.1016/j.camwa.2010.07.030.  
[18] |  R. L. Magin, _Fractional Calculus in Bioengineering_ , Begell House Publishers, Redding, 2006.  
[19] |  R. K. Mohanty and M. K. Jain, An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation, _Numer. Meth. for Part. D. E._ , **17** (2001), 684-688. doi: 10.1002/num.1034.  
[20] |  R. K. Mohanty, An operator splitting method for an unconditionally stable difference scheme for a linear hyperbolic equation with variable coefficients in two space dimensions, _Appl. Math. Comput._ , **152** (2004), 799-806. doi: 10.1016/S0096-3003(03)00595-2.  
[21] |  R. K. Mohanty, M. K. Jain and K. George, On the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients, _J. Comp. Appl. Math._ , **72** (1996), 421-431. doi: 10.1016/0377-0427(96)00011-8.  
[22] |  R. K. Mohanty, New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations, _Int. J. Comp. Math._ , **86** (2009), 2061-2071 doi: 10.1080/00207160801965271.  
[23] |  R. K. Mohanty, M. K. Jain and U. Arora, An unconditionally stable ADI method for the linear hyperbolic equation in three space dimensions, _Int. J. Comp. Math._ , **79** (2002), 133-142. doi: 10.1080/00207160211918.  
[24] |  A. Mohebbi and M. Dehghan, High order compact solution of the one-space-dimensional linear hyperbolic equation, _Numer. Meth. Part. D. E._ , **24** (2008), 1222-1235. doi: 10.1002/num.20313.  
[25] |  I. Podlubny, _Fractional Differential Equations_ , Mathematics in Science and Engineering, Academic Press, New York, 1999.   
[26] |  A. Saadatmandi and M. Dehghan, Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method, _Numer. Meth. for Part. D. E._ , **26** (2010), 239- 252. doi: 10.1002/num.20442.  
[27] |  S. G. Samko, A. A. Kilbas and O. I. Marichev, _Fractional Integrals and Derivatives: Theory and Applications_ , Gordon and Breach, Yverdon, (1993).  
[28] |  C. Shu, Q. Yao and K. S. Yeo, Block-marching in time with DQ discretization: An efficient method for time-dependent problems, _Comput. Methods Appl. Mech. Engrg_ , **191** (2002), 4587-4597. doi: 10.1016/S0045-7825(02)00387-0.  
  
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