A COMPUTATIONAL METHOD FOR INTEGRO-DIFFERENTIAL HYPERBOLIC EQUATION WITH INTEGRAL CONDITIONS

Year 2013, Volume: 62 Issue: 1, 131 - 142, 01.02.2013

Ahcene Merad Abdelfatah Bouzıanı

https://doi.org/10.1501/Commua1_0000000692

Abstract

The subject of this work is to prove existence, uniqueness, and continuous dependence upon the data of solution to integrodifferential hyperbolicequation with integral conditions. The proofs are based on a priori estimatesand Laplace transform method. Finally, the solution by using a numericaltechnique for inverting the Laplace transforms is obtained

Keywords

Integro-diğerential parabolic equations, Laplace Transform Method, integral conditions

References

• M. Abramowitz, I. A. Stegun, Hand book of Mathematical Functions, Dover, New York, 1972. [2] W.T. Ang, A Method of Solution for the One-Dimentional Heat Equation Subject to Nonlocal Conditions, Southeast Asian Bull. Math. 26 (2002), 185–191.
• S.A. Beïlin, Existence of solutions for one-dimentional wave nonlocal conditions, Electron. J. Diğer. Equ. 2001 (2001), no. 76, 1–8.
• A. Bouziani, Problèmes mixtes avec conditions intégrales pour quelques équations aux dérivées partielles, Ph.D. thesis, Constantine University, (1996).
• A. Bouziani, Mixed problem with boundary integral conditions for a certain parabolic equa- tion, J. Appl. Math. Stochastic Anal. 9(1996), no. 3, 323–330.
• A. Bouziani, Solution forte d’un problème mixte avec une condition non locale pour une classe d’équations hyperboliques [Strong solution of a mixed problem with a nonlocal condition for a class of hyperbolic equations], Bull. Cl. Sci., VI. Sér., Acad. R. Belg. 8(1997), 53–70.
• A. Bouziani, Strong solution to an hyperbolic evolution problem with nonlocal boundary con- ditions, Maghreb Math. Rev. 9 (2000), no. 1-2, 71–84.
• A. Bouziani, On the quasi static *exur of thermoelastic rod, Commun. Appl. Anal. Theory Appl. 6(2002), no.4, 549–568.
• A. Bouziani, Initial-boundary value problem with nonlocal condition for a viscosity equation, Int. J. Math. & Math. Sci. 30 (2002), no. 6, 327–338.
• A. Bouziani, On the solvabiliy of parabolic and hyperbolic problems with a boundary integral condition, Internat. J. Math. Math. Sci. 31 (2002), 435–447.
• A. Bouziani, On a class of nonclassical hyperbolic equations with nonlocal conditions, J. Appl. Math. Stochastic Anal. 15 (2002), no. 2, 136–153.
• A. Bouziani, Mixed problem with only integral boundary conditions for an hyperbolic equation, Internat. J. Math. & Math. Sci. 26 (2004), 1279–1291.
• A. Bouziani, N. Benouar, Problème mixte avec conditions intégrales pour une classe d’équations hyperboliques, Bull. Belg. Math. Soc. 3 (1996), 137–145.
• A. Bouziani, N. Benouar, Sur un problème mixte avec uniquement des conditions aux limites intégrales pour une classe d’équations paraboliques, Maghreb Math. Rev. 9 (2000), no. 1-2, 55–70.
• A. Bouziani, R. Mechri, The Rothe Method to a Parabolic Integro-diğ erential Equation with a Nonclassical Boundary Conditions, Int. J. Stochastic Anal. Article ID 519684/ 16 page, doi: 10.1155/519684/ (2010).
• D.P. Graver, Observing stochastic processes and aproximate transform inversion, Oper. Res. 14(1966), 444–459.
• D.G. Gordeziani, G.A. Avalishvili, Solution of nonlocal problems for one-dimensional oscil- lations of a medium, Mat. Model. 12 (2000), no. 1, 94–103.
• H. Hassanzadeh, M. Pooladi-Darvish, Comparision of diğ erent numerical Laplace inversion methods for engineering applications, Appl. Math. Comp. 189 (2007), 1966–1981.
• J. Kac¼ur, Method of Rothe in Evolution Equations, Teubner-Texte zur Mathematik 80, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1985.
• S. Mesloub, A. Bouziani, On a class of singular hyperbolic equation with a weighted integral condition, Int. J. Math. & Math. Sci. 22 (1999), no. 3, 511–519.
• S. Mesloub, A. Bouziani, Mixed problem with integral conditions for a certain class of hyper- bolic equations, J. Appl. Math. 1 (2001), no. 3, 107–116.
• N. Merazga, A. Bouziani, Rothe time-discretization method for a nonlocal problem arising in thermoelasticity, J. Appl. Math. Stochastic Anal. 2005 (2005), no. 1, 13–28.
• A. Merad, Adomian decomposition method for solution of parabolic equation to nonlocal conditions, Int. J. Contemp. Math. Sci. 6 (2011), no. 29-32, 1491–1496.
• A. Merad, A.L. Marhoune, Strong solution for a high order boundary value problem with integral condition, Turk. J. Math. 37 (2013), no.3, 1–9.
• L.S. Pul’kina, A non-local problem with integral conditions for hyperbolic equations, Electron. J. Diğer. Equ. 1999 (1999), no. 45, 1–6.
• L.S. Pul’kina, On the solvability in L2 of a nonlocal problem with integral conditions for a hyperbolic equation, Diğer. Equ. 36 (2000), no. 2, 316–318.
• L.S. Pul’kina, A mixed problem with integral condition for the hyperbolic equation, Mat. Zametki 74 (2003), no. 3, 435–445.
• H. Stehfest, Numerical inversion of the Laplace transform, Comm. ACM 13 (1970), 47–49. [29] A.D. Shruti, Numerical solution for nonlocal Sobolev-type diğ erential equations, Electron. J. Diğer. Equ. Conf. 19 (2010), 75–83.
• Current address : A. Merad and A. Bouziani; Department of Mathematics, Larbi Ben M’hidi University, 04000, ALGERIA
• E-mail address : [e-mail], [e-mail]
• URL: http://communications.science.ankara.edu.tr/index.php?series=A1