Laplace transform and homotopy perturbation methods for solving the pseudohyperbolic integrodifferential problems with purely integral conditions. (English) Zbl 1494.35121

Summary: 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.


35L82 Pseudohyperbolic equations
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35R09 Integro-partial differential equations
44A10 Laplace transform
45J05 Integro-ordinary differential equations
65R20 Numerical methods for integral equations
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[1] M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions, Dover, New York, 1972. · Zbl 0543.33001
[2] S. Abbasbandy,Numerical solutions of the integral equations: homotopy perturbation method and Adomian’s decomposition method, Appl. Math. Comput.173(2-3) · Zbl 1090.65143
[3] G. A. Afrouzi, D. D. Ganji, H. Hosseinzadeh and R. A. Talarposhti,Fourth order Volterra integro-differential equations using modied homotopy-perturbation method,
[4] A. Bouziani and A. Merad,The Laplace transform method for one-dimensional hyperbolic equationwith purely integral conditions, Rom. J. Math. Comput. Sci. · Zbl 1313.35213
[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) · Zbl 1194.35201
[6] A. Bouziani and M. S. Temsi,On a pseudohyperbolic equation with nonlocal boundary condition, Kobe J. Math.21(2004), 15-31. · Zbl 1112.35129
[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, · Zbl 1209.65145
[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. · Zbl 1120.65108
[9] D. D. Ganji and A. Rajabi,Assessment of homotopy-perturbation and perturbation methods in heat radiation equations, International Communications in Heat and · Zbl 1236.65059
[10] D. D. Ganji and M. Rafei,Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method, Phys. Lett. A356(2006), 131-137. · Zbl 1160.35517
[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. · Zbl 0956.70017
[13] J. H. He,New interpretation of homotopy perturbation method, Internat. J. Modern Phys. B20(2006), 2561-2568.
[14] J. H. He,A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Non-Linear Mechanics35(1) (2000), · Zbl 1068.74618
[15] J. H. He,Limit cycle and bifurcation of nonlinear problems, Chaos Solitons Fractals 26(2005), 827-833. · Zbl 1093.34520
[16] J. H. He,Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. Numer. Simul.6(2) (2005), 207-208. · Zbl 1401.65085
[17] J. H. He,Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals26(2005), 695-700. · Zbl 1072.35502
[18] M. Madani, M. Fathizadeh, Y. Khan and A. Yildrim,On coupling the homotpy perturbation method ans Laplace transformation, Math. Comput. Modelling53(2011) · Zbl 1219.65121
[19] H. Hassanzadeh and M. Pooladi-Darvish,Comparision of different numerical Laplace inversion methods for engineering applications, Appl. Math. Comput. 189(2007), 1966-1981. · Zbl 1243.65151
[20] A. Merad and A. Bouziani,Laplace transform technique for pseudoparabolic equation with nonlocal conditiond, Transylvanian Journal of Mathematics and Mechan · Zbl 1291.35121
[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 Statistics41, · Zbl 1335.65102
[22] A. Merad and A. Bouziani,Solvability the telegraph equation with purely integral conditions, TWMS J. Appl. Eng. Math.3(2) (2013), 117-125. · Zbl 1309.65116
[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. · Zbl 1401.65150
[24] A. D. Shruti,Numerical solution for nonlocal Sobolev-type differential equations, Electron. J. Differential Equations19(2010), 75-83. · Zbl 1200.35012
[25] Z. Suying, Z. Minzhen, D. Zichen and L. Wencheng,Solution of nonlinear dynamic differential equations based on numerical Laplace transform inversion, Appl. Math. · Zbl 1123.65071
[26] H. Stehfest,Numerical inversion of the Laplace transform, Communications of the ACM13(1970), 47-49.
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