### Laplace Transform and Homotopy Perturbation Methods for Solving the Pseudohyperbolic Integrodifferential Problems with Purely Integral Conditions

Authors: A. NECIB AND A. MERAD

DOI: 10.46793/KgJMat2002.251N

Abstract:

In this paper we deﬁned and investigated the various properties of a class of pseudohyperbolic equation deﬁned on purely integral (nonlocal) conditions. We proved the uniqueness and the existence of the solution using energy inequality (A priori estimates). We found a semi analytical solution using the Laplace transform and Stehfest algorithm method. Next, we used another method called the Homotopy perturbation. Finally, we give some examples for illustration.

Keywords:

Pseudohyperbolic integrodiﬀerential equation, a priori estimate, Laplace transform method, Stehfest algorithm, homotopy perturbation method.

References:

[1]   M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.

[2]   S. Abbasbandy, Numerical solutions of the integral equations: homotopy perturbation method and Adomian’s decomposition method, Appl. Math. Comput. 173(2-3) (2006), 493–500.

[3]   G. A. Afrouzi, D. D. Ganji, H. Hosseinzadeh and R. A. Talarposhti, Fourth order Volterra integro-diﬀerential equations using modied homotopy-perturbation method, Turkish Journal of Mathematics and Computer Science 3(2) (2011), 179–191.

[4]   A. Bouziani and A. Merad, The Laplace transform method for one-dimensional hyperbolic equationwith purely integral conditions, Rom. J. Math. Comput. Sci. 3(2) (2013), 191–204.

[5]   A. Bouziani and R. Mechri, The Rothe method to a parabolic integro-diﬀerential equation with a nonclassical boundary conditions, Int. J. Stoch. Anal. (2010), Article ID 519684, 16 pages, DOI: 10.1155/519684/(2010)

[6]   A. Bouziani and M. S. Temsi, On a pseudohyperbolic equation with nonlocal boundary condition, Kobe J. Math. 21 (2004), 15–31.

[7]   D. D. Ganji, G. A. Afrouzi, H. Hosseinzadeh and R. A. Talarposhti, Application of Hmotopy perturbation method to the second kind of nonlinear integral equations, Phys. Lett. A 371(1-2) (2007), 20–25.

[8]   D. D. Ganji and A. Sadighi, Application of He’s homotopy-perturbation method to nonlinear coupled systems of reaction-diﬀusion equations, Int. J. Nonlinear Sci. Numer. Simul. 7(4) (2006), 411–418.

[9]    D. D. Ganji and A. Rajabi, Assessment of homotopy-perturbation and perturbation methods in heat radiation equations, International Communications in Heat and Mass Transfer 33 (2006), 391–400.

[10]   D. D. Ganji and M. Rafei, Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method, Phys. Lett. A 356 (2006), 131–137.

[11]   D. P. Graver, Observing stochastic processes and aproximate transform inversion, Oper. Res. 14 (1966), 444–459.

[12]   J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg. 178 (1999), 257–262.

[13]   J. H. He, New interpretation of homotopy perturbation method, Internat. J. Modern Phys. B 20 (2006), 2561–2568.

[14]   J. H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Non-Linear Mechanics 35(1) (2000), 37–43.

[15]   J. H. He, Limit cycle and bifurcation of nonlinear problems, Chaos Solitons Fractals 26 (2005), 827–833.

[16]   J. H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. Numer. Simul. 6(2) (2005), 207–208.

[17]   J. H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals 26 (2005), 695–700.

[18]   M. Madani, M. Fathizadeh, Y. Khan and A. Yildrim, On coupling the homotpy perturbation method ans Laplace transformation, Math. Comput. Modelling 53(2011) (1970), 1937–1945.

[19]   H. Hassanzadeh and M. Pooladi-Darvish, Comparision of diﬀerent numerical Laplace inversion methods for engineering applications, Appl. Math. Comput. 189 (2007), 1966–1981.

[20]   A. Merad and A. Bouziani, Laplace transform technique for pseudoparabolic equation with nonlocal conditiond, Transylvanian Journal of Mathematics and Mechanics 5(1) (2013), 59–64.

[21]   A. Merad and A. Bouziani, A method of solution of integro-diﬀerential parabolic equation with purely integral conditions, in: Advances in Applied Mathematics and Approximation Theory, Springer Proceeding in Mathematics and Statistics 41, New York, 2013.

[22]   A. Merad and A. Bouziani, Solvability the telegraph equation with purely integral conditions, TWMS J. Appl. Eng. Math. 3(2) (2013), 117–125.

[23]   M. El-Shahed, Application of He’s homotopy perturbation method to Volterra integrodiﬀerential equation, Int. J. Nonlinear Sci. Numer. Simul. 6(2) (2005), 163–168.

[24]   A. D. Shruti, Numerical solution for nonlocal Sobolev-type diﬀerential equations, Electron. J. Diﬀerential Equations 19 (2010), 75–83.

[25]    Z. Suying, Z. Minzhen, D. Zichen and L. Wencheng, Solution of nonlinear dynamic diﬀerential equations based on numerical Laplace transform inversion, Appl. Math. Comput. 189 (2007), 79–86.

[26]   H. Stehfest, Numerical inversion of the Laplace transform, Communications of the ACM 13 (1970), 47–49.