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A Galerkin-finite element method for diffusion equation with purely integral boundary conditions
by
Abdelfatah Bouziani
Centre Universitaire Larbi Ben M'hidi, Oum El Bouagui, B.P. 565, 04000, Algérie.
In the present work, we study the following problem: Let v₀, E, M and f be known functions, and T>0 and>0 be given constants. In Q=(0,1)×(0,T), find the function v=v(x,t) satisfying ((∂v)/(∂t))-((∂²v)/(∂x²))=f(x,t), (x,t)∈Q, v(x,0)=v₀(x), x∈(0,1), ∫₀¹v(x,t)dx=E(t), t∈(0,T), ∫₀¹xv(x,t)dx=M(t),t∈(0,T), This mathematical model, recently derived in <cite>B3</cite>, describes the quasi static flexure of a thermoelastic rod.
Although an increasing attention has been given recently to evolution problems which involve nonlocal boundary conditions (see, for instance, <cite>BY, B2, B3, B4, C, CH1, CH2, CH3, Y</cite> and references therein), only few works have been consecrated to mixed parabolic problems with purely integral boundary conditions over the spatial domain <cite>B1, B3, BB</cite>.
Differently to works cited above, we, first, employ a Galerkin method in a nonclassical function space, and construct an approximate sequence to solve the stated problem. A semi-discrete approximation will be then introduced by discretizing with respect to the space variable, using a finite element method. Finally, a total discretization procedure will also be presented.
BY : N. E. Benouar and N. I. Yurchuk, A mixed problem with an integral condition for parabolic equations with Bessel operator, Differentsial'nye Uravneniya 27 (1991), no. 12, 2094-2098 (Russian).
B1 : A. Bouziani, Mixed problem with boundary integral conditions for a certain parabolic equation, J. Appl. Math. Stochastic Anal. 9:3 (1996), 323-330.
B2 : A. Bouziani, On a class of composite equations with nonlocal boundary conditions, Electronic Modeling, 23 (2002), no. 6, 34-41.
B3 : A. Bouziani, On the quasi static flexure of a thermoelastic rod, Comm. Appl. Anal. 6:4 (2002), 549-568.
B4 : A. Bouziani, On the solvability of parabolic and hyperbolic problems with a boundary integral condition, Int. J. Math. Math. Sci. 31 (2002), no. 4, 201-213.
BB : A. Bouziani, N. E. Benouar, Sur un problème mixte avec uniquement des conditions aux limites intégrales pour une classe d'équations paraboliques, Maghreb Math. Rev., 9:1&2 (2000), 55-70.
C : J. R. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math. 21 (1963), 155-160.
CH1 : J. R. Cannon and J. van der Hoek, The existence of and continuous dependence result for the solution of the heat equation subject to the specification of energy, Boll. Un. Mat. Ital. Suppl. (1981), no. 1, 253-282.
CH2 : J. R. Cannon and J. van der Hoek, An implicit finite difference scheme for the diffusion equation subject to the specification of mass in a portion of the domain, Numerical Solutions of Partial Differential Equations (Parkville, 1981) (J. Noye, ed.), North-Holland, Amsterdam, 1982, pp. 527-539.
CH3 : J. R. Cannon, J. Van Der Hoek, Diffusion subject to the specification of mass, J. Math. Anal. Appl., 115 (1986), 517-529.
Y : N. I. Yurchuk, Mixed problem with an integral condition for certain parabolic equations, Differentsial'nye Uravneniya 22 (1986), no. 12, 2117-2126 (Russian).
Date received: May 6, 2004
Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caoe-29.