Study of Telegraph Equation via He-Fractional Laplace Homotopy Perturbation Technique

Authors

Ali Moazzzam University of Agriculture Faisalabad, Pakistan
Ayza Anjum University of Agriculture, Faisalabad, Pakistan
Nimra Saleem University of Agriculture Faisalabad, Pakistan
Emad A. Kuffi Al-Qadisiyah University, College of Engineering, Iraq.

DOI:

https://doi.org/10.30526/36.3.3239

Keywords:

Fractional Laplace method, He’s Polynomials, Homotopy Perturbation, Approximated solutions, Telegraph Equation.

Abstract

A new technique to study the telegraph equation, mostly familiar as damped wave equation is introduced in this study. This phenomenon is mostly rising in electromagnetic influences and production of electric signals. The proposed technique called as He-Fractional Laplace technique with help of Homotopy perturbation is utilized to found the exact and nearly approximated results of differential model and numerical example of telegraph equation or damped wave equation in this article. The most unique term of this technique is that, there is no worry to find the next iteration by integration in recurrence relation. As fractional Laplace integral transformation has some limitations in non-linear terms, to get the result of nonlinear term in this differential mode, He polynomials via homotopy techniques of iteration is proposed to find the result of the computation assignment. The obtained result by this proposed technique directed that this technique is quite ease to apply and convergent rapidly to exact solutions. Numerous examples are described to determine the stability and accuracy of the proposed technique with the graphical explanation.

References

Gong Q.; Liu C.; Xu Y.; Ma C.; Zhou J.; Jiang R.; Zhou C. Nonlinear vibration control with nanocapacitive sensor for electrostatically actuated nanobeam. Journal of Low Frequency Noise, Vibration and Active Control, 2018, 37(2), 235-252. DOI: https://doi.org/10.1177/1461348417725953

El-Azab M.S.; El-Gamel M. A numerical algorithm for the solution of telegraph equations. Applied Mathematics and Computation, 2007, 190(1), 757-764. DOI: https://doi.org/10.1016/j.amc.2007.01.091

Pandit S.; Kumar M.; Tiwari S. Numerical simulation of second-order hyperbolic telegraph type equations with variable coefficients. Computer Physics Communications, 2015, 187, 83-90. DOI: https://doi.org/10.1016/j.cpc.2014.10.013

Evans D.J.; Bulut H.; The numerical solution of the telegraph equation by the alternating group explicit (AGE) method. International journal of computer mathematics. 2003, 80(10), 1289-1297. DOI: https://doi.org/10.1080/0020716031000112312

Ding H.F.; Zhang Y.X.; Cao J.X. Tian JH. A class of difference scheme for solving telegraph equation by new non-polynomial spline methods. Applied Mathematics and Computation. 2012, 218(9), 4671-4683. DOI: https://doi.org/10.1016/j.amc.2011.10.078

Evans D.J.; Raslan K.R.. The Adomian decomposition method for solving delay differential equation. International Journal of Computer Mathematics. 2005, 82(1), 49-54. DOI: https://doi.org/10.1080/00207160412331286815

Mohamed M.A.; Torky M.S. Numerical solution of nonlinear system of partial differential equations by the Kamal decomposition method and the Pade approximation. American Journal of Computational Mathematics. 2013, 3(3), 175-183. DOI: https://doi.org/10.4236/ajcm.2013.33026

Yindoula J.B; Youssouf P.; Bissanga G.; Bassono F. Application of the Adomian decomposition method and Laplace transform method to solving the convection diffusion-dissipation equation. International Journal of Applied Mathematics Research. 2014 3(1), 30-37. DOI: https://doi.org/10.14419/ijamr.v3i1.1596

Hamoud A.A.; Ghadle K.P. The combined modified Laplace with Adomian decomposition method for solving the nonlinear Volterra-Fredholm integro differential equations. Journal of the Korean Society for Industrial and Applied Mathematics. 2017, 21(1), 17-28.

Aggarwal S.; Gupta A.R.; Kumar D. Mohand transform of error function. International Journal of Research in Advent Technology. 2019, 7(5), 224-231.

Aggarwal S.; Sharma S.D. Sumudu transform of error function. Journal of Applied Science and Computations. 2019, 6(6),1222-1231.

Patil D.P.; Kandakar K.S.; Zankar T.V. Application of general integral transform of error function for evaluating improper integrals. International Journal of Advances in Engineering and Management. 2022, 4(6), 242-246.

Aggarwal S.; Gupta A.R.; Sharma S.D.; Chauhan R.; Sharma N. Mahgoub transform (Laplace-Carson transform) of error function. International Journal of Latest Technology in Engineering, Management & Applied Science. 2019, 8(4), 92-108.

Patil D.; Application of sawi transform of error function for evaluating improper integral. Journal of Applied Science and Computations. 2019, 6(5), 2235-2242.

Aggarwal S.; Sharma S.D. Application of Kamal transform for solving Abel’s integral equation. Global Journal of Engineering Science and Researches. 2019, 6(3), 82-90.

Aggarwal S.; Singh G.P. Kamal transform of error function. Journal of Applied Science and Computations. 2019, 6(5), 2223-2235.

Fariha M.; Muhammad Z.I.; Moazzam A.; Usman A.; Muhammad U.N. Subtituition Method using the Laplace transformation for solving partial differential equations involving more than two independent variables. Bulletin of Mathematics and Statistics Research. 2021, 9(3), 104-116.

Muhammad I.K.; Khurrem S.; Muhammad M.U.R.; Maria; Moazzam A. Influence of bioconvection and activation energy on maxwell flow of nano-fluid in the existence of motile microorganisms over a cylinder. International Journal of Multidisciplinary Research and Growth Evaluation. 2021, 2(5), 135-144.

Ammara A.; Madiha G.; Amina A.; Moazzam A. Laplace transforms techniques on equation of advection-diffussion in one-dimensional with semi-infinite medium to find the analytical solution. Bulletin of Mathematics and Statistics Research. 2021, 9(3), 86-96.

Muhammad K.; Khurrem S.; Moazzam A.; Ammara A.; Mariyam B. New transformation ‘AMK-transformation’ to solve ordinary linear differential equation of moment Pareto distribution. International Journal of Multidisciplinary Research and Growth Evaluation. 2021, 2(5), 125-134.

Moazzam A.; Kashif M.; Amjed U.; Khawar M.I. Devolpment of a new transformation to solve a new type of ordinary linear differential equation. Bulletin of Mathematics And Statistics Research (Bomsr). 2021, 9(3), 56-60.

Muhammad W.; Khurrem S.; Moazzam A.; Alizay B. Applications of Kamal transformation in temperature problems. Scholar Journal Engineering and Technology. 2022, 10(2), 5-8.

Eman A.; Mansour; Emad A. K.; Sadiq A.; Mehdi. On the SEE transform and system of ordinary differential equations. Periodicals of engineering and natural sciences, 2021, 9(3), 277-281.

Moazzam A.; Muhammad Z.I. Al-Zughair transformations on linear differential equations of Moment Pareto distribution. Proc. 19th International Conference on Statistical Science, 2022, 36, 199-206.

Moazzam A.; Muhammad Z.l. A new integral transform "Ali And Zafar" transformation and It's application in nuclear physics. Proc. 19th International Conference on Statistical Sciences, 2022, 36, 177-182.

Inokuti M.; Sekine H.; Mura T. General use of the Lagrange multiplier in nonlinear mathematical physics. Variational method in the mechanics of solids. 1978, 33(5):156-162. DOI: https://doi.org/10.1016/B978-0-08-024728-1.50027-6

He J.H. Variational iteration method–a kind of non-linear analytical technique: some examples. International journal of non-linear mechanics. 1999, 34(4), 699-708. DOI: https://doi.org/10.1016/S0020-7462(98)00048-1

He J.H. Variational iteration method for autonomous ordinary differential systems. Applied mathematics and computation. 2000, 114(3), 115-123. DOI: https://doi.org/10.1016/S0096-3003(99)00104-6

He J.H.; Wu X.H. Variational iteration method: new development and applications. Computers & Mathematics with Applications. 2007, 54(7-8), 881-894. DOI: https://doi.org/10.1016/j.camwa.2006.12.083

Odibat Ζ.Μ.; Momani S. Application of variational iteration method to nonlinear differential equations of fractional order. International Journal of Nonlinear Sciences and Numerical Simulation. 2006, 7(1), 27-34. DOI: https://doi.org/10.1515/IJNSNS.2006.7.1.27

He J.H. Some asymptotic methods for strongly nonlinear equations. International journal of Modern physics B. 2006, 20(10), 1141-1199. DOI: https://doi.org/10.1142/S0217979206033796

Nadeem M.; Li F.; Ahmad H. He’s variational iteration method for solving non-homogeneous Cauchy Euler differential equations. Nonlinear Sci Lett A Math Phys Mech. 2018, 9(3), 231-237.

Cheng H.; Yu Y.Y. Semi-analytical solutions of the nonlinear oscillator with a matrix Lagrange multiplier. Journal of Low Frequency Noise, Vibration and Active Control. 2019, 38(3-4), 1214-1219. DOI: https://doi.org/10.1177/1461348418769649

Biazar J.; Ebrahimi H.; Ayati Z. An approximation to the solution of telegraph equation by variational iteration method. Numerical Methods for Partial Differential Equations. 2009, 25(4), 797-801. DOI: https://doi.org/10.1002/num.20373

He J.H. Homotopy perturbation technique. Computer methods in applied mechanics and engineering. 1999, 178(3-4), 257-262. DOI: https://doi.org/10.1016/S0045-7825(99)00018-3

He J.H. Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and computation. 2003, 135(1), 73-79. DOI: https://doi.org/10.1016/S0096-3003(01)00312-5

He J.H. Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons & Fractals. 2005, 26(3), 695-700. DOI: https://doi.org/10.1016/j.chaos.2005.03.006

He J.H. Recent development of the homotopy perturbation method. Topological methods in nonlinear analysis. 2008, 31(2), 205-209.

Wu Y.; He J.H. Homotopy perturbation method for nonlinear oscillators with coordinate dependent mass. Results Phys. 2018, 10, 270-271.

Gupta S.; Kumar D.; Singh J. Analytical solutions of convection–diffusion problems by combining Laplace transform method and homotopy perturbation method. Alexandria Engineering Journal. 2015, 54(3), 645-651. DOI: https://doi.org/10.1016/j.aej.2015.05.004

Khuri S.A.; Sayfy A. A Laplace variational iteration strategy for the solution of differential equations. Applied Mathematics Letters. 2012, 25(12), 2298-2305. DOI: https://doi.org/10.1016/j.aml.2012.06.020

Medina G.D.; Ojeda N.R.; Pereira J.H.; Romero L.G. Fractional Laplace transform and fractional calculus. International Mathematical Forum. 2017, 12(20), 991-1000. DOI: https://doi.org/10.12988/imf.2017.71194