Abstract
The objective of this paper is to prove existence, uniqueness, and continuous dependence upon the data of solution to integro-differential parabolic equation with purely integral conditions. The proofs are based on a priory estimates and Laplace transform method. Finally, we obtain the solution by using a numerical technique for inverting the Laplace transforms.
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References
M. Abramowitz and I. A. Stegun, Hand book of Mathematical Functions, Dover, New York, 1972.
W. T. Ang, A Method of Solution for the One-Dimentional Heat Equation Subject to Nonlocal Conditions, Southeast Asian Bulletin of Mathematics 26 2002, 185–191.
S. A. Beïlin, Existence of solutions for one-dimentional wave nonlocal conditions, Electron. J. Differential Equations 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 equation, J. Appl. Math. Stochastic Anal.09, no. 3, 323–330 1996.
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], Acad. Roy. Belg. Bull. Cl. Sci.8, 53–70 1997.
A. Bouziani, Strong solution to an hyperbolic evolution problem with nonlocal boundary conditions, Maghreb Math. Rev., 9, no. 1-2, 71–84 (2000).
A. Bouziani, On the quasi static flexure of a thermoelastic rod, Communications in Applied Analysis for Theory and Applications, 6, no. 4, 549-568, (2002).
A. Bouziani, Initial-boundary value problem with nonlocal condition for a viscosity equation, Int. J. Math. & Math. Sci.30, no. 6, 327–338.2002.
A. Bouziani, On the solvabiliy of parabolic and hyperbolic problems with a boundary integral condition, Internat. J. Math. & Math. Sci., 31, 435–447.2002.
A. Bouziani, On a class of nonclassical hyperbolic equations with nonlocal conditions, J. Appl. Math. Stochastic Anal.15, no. 2, 136–153.2002
A. Bouziani, Mixed problem with only integral boundary conditions for an hyperbolic equation, Internat. J. Math. & Math. Sci., 26, 1279–1291 (2004).
A. Bouziani and N. Benouar, Problème mixte avec conditions intégrales pour une classe d’équations hyperboliques, Bull. Belg. Math. Soc. 3, 137–145 (1996).
A. Bouziani and N.-E. Benouar, Sur un problème mixte avec uniquement des conditions aux limites intégrales pour une classe d’équations paraboliques, Maghreb Mathematical Review, 9, no. 1-2, 55–70, (2000).
A. Bouziani and R. Mechri, The Rothe Method to a Parabolic Integrodifferential Equation with a Nonclassical Boundary Conditions, Int. Jour. of Stochastic Analysis, Article ID 519684/16 page, doi: 10.1155/519684/2010.
D. G. Gordeziani and G. A. Avalishvili, Solution of nonlocal problems for one-dimensional oscillations of a medium, Mat. Model. 12, no. 1, 94–103 (2000)
D. P. Graver, Observing stochastic processes and aproximate transform inversion, Oper. Res. 14, 444–459.(1966).
H. Hassanzadeh and M. Pooladi-Darvish, Comparision of different numerical Laplace inversion methods for engineering applications, Appl. Math. Comp. 189 1966–1981(2007).
J.Kacŭr, Method of Rothe in Evolution Equations, Teubner-Texte zur Mathematik, vol. 80, BSB B. G. Teubner Verlagsgesellschaft, Leipzig,1985.
A. Merad, Adomian Decomposition Method for Solution of Parabolic Equation to Nonlocal Conditions, Int. Journal of Contemp. Math. Sciences, Vol 6, no. 29-32, 1491–1496 (2011).
A. Merad and A. L. Marhoune, Strong Solution for a High Order Boundary Value Problem with Integral Condition, Turk. Jour. Math., 1-9,doi:10.3906/mat-1105-34 2012.
N. Merazga and A. Bouziani, Rothe time-discretization method for a nonlocal problem arising in thermoelasticity, Journal of Applied Mathematics and Stochastic Analysis, 2005, no. 1, 13–28, (2005).
S.Mesloub and A. Bouziani, On a class of singular hyperbolic equation with a weighted integral condition, Int. J. Math. Math. Sci. 22, no. 3, 511–519 (1999).
S.Mesloub and A. Bouziani, Mixed problem with integral conditions for a certain class of hyperbolic equations, Journal of Applied Mathematics, Vol. 1, no. 3, 107–116.(2001)
L. S. Pul’kina, A non-local problem with integral conditions for hyperbolic equations, Electron. J. Differential Equations 1999, no. 45, 1–6(1999).
L. S. Pul’kina, On the solvability in L2 of a nonlocal problem with integral conditions for a hyperbolic equation, Differ. 36, no. 2, 316–318.(2000)
L. S. Pul’kina, A mixed problem with integral condition for the hyperbolic equation, Matematicheskie Zametki, vol. 74, no. 3,, pp. 435–445.2003.
A. D. Shruti, Numerical Solution for Nonlocal Sobolev-type Differential Equations, Electronic Journal of Differential Equations, Conf. 19, pp. 75–83.(2010)
H. Stehfest, Numerical Inversion of the Laplace Transform, Comm. ACM13, 47–49.(1970)
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Merad, A., Bouziani, A. (2013). A Method of Solution for Integro-Differential Parabolic Equation with Purely Integral Conditions. In: Anastassiou, G., Duman, O. (eds) Advances in Applied Mathematics and Approximation Theory. Springer Proceedings in Mathematics & Statistics, vol 41. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6393-1_20
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DOI: https://doi.org/10.1007/978-1-4614-6393-1_20
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