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

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:

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

   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.

   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.

   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.

   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)

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

   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.

   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.

    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.

   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.

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

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

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

   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.

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

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

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

   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.

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

   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.

   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.

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

   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.

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

    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.

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