Geodesic
Three-dimensional integro-multipoint boundary value problem for loaded volterra-hyperbolic integro-differential equations of Bianchi type
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2012), pp. 8-20.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper the combined three-dimensional non-local boundary value problem with integro-multipoint conditions for loaded volterra-hyperbolic integro-differential equation of Bianchi type is explored. The matter of principle is the fact, that the considered equation has discontinuous coefficients which satisfy only some conditions of $P$-integrability type and boundedness, i.e. the considered hyperbolic differential operator has no traditional conjugate operator. In particular, for example, Riemann function under Goursat conditions for such equation cannot be constructed by classical method of characteristics.
Keywords: three-dimensional non-local boundary problem, loaded integro-differential equations, hyperbolic equation, equations with discontinuous coefficients. 
@article{VSGTU_2012_1_a0,
     author = {I. G. Mamedov},
     title = {Three-dimensional integro-multipoint boundary value problem for loaded volterra-hyperbolic integro-differential equations of {Bianchi} type},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {8--20},
     publisher = {mathdoc},
     number = {1},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2012_1_a0/}
}
TY  - JOUR
AU  - I. G. Mamedov
TI  - Three-dimensional integro-multipoint boundary value problem for loaded volterra-hyperbolic integro-differential equations of Bianchi type
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2012
SP  - 8
EP  - 20
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2012_1_a0/
LA  - ru
ID  - VSGTU_2012_1_a0
ER  - 
%0 Journal Article
%A I. G. Mamedov
%T Three-dimensional integro-multipoint boundary value problem for loaded volterra-hyperbolic integro-differential equations of Bianchi type
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2012
%P 8-20
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2012_1_a0/
%G ru
%F VSGTU_2012_1_a0
I. G. Mamedov. Three-dimensional integro-multipoint boundary value problem for loaded volterra-hyperbolic integro-differential equations of Bianchi type. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2012), pp. 8-20. http://geodesic.mathdoc.fr/item/VSGTU_2012_1_a0/

[1] Bateman H., “Logarithmic Solutions of Bianchi's Equation”, Proc. Natl. Acad. Sci. USA, 19:9 (1933), 852–854 | DOI | MR | Zbl

[2] Fage M. K., “The Cauchy problem for Bianchi's equation”, Mat. Sb. (N.S.), 45(87):3 (1958), 281–322 | MR | Zbl

[3] Zhegalov V. I., “A three-dimensional analogue of the Goursat problem”, Nonclassical equations and equations of mixed type, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1990, 94–98 | MR | Zbl

[4] Zhegalov V. I., “On the three-dimensional riemann function”, Siberian Math. J., 38:5 (1997), 929–934 | DOI | MR | Zbl

[5] Dzhokhadze O. M., “On the three-dimensional generalized Goursat problem for a third-order equation, and related general two-dimensional Volterra integral equations of the first kind”, Differ. Equ., 42:3 (2006), 412–421 | DOI | MR | Zbl

[6] Utkina E. A., “A problem with shifts for the three-dimensional Bianchi equation”, Differ. Equ., 46:4 (2010), 538–542 | DOI | MR | Zbl

[7] Bondarenko B. A., Basis systems of polynomial and quasipolynomial solutions of partial differential equations, Fan, Tashkent, 1987, 147 pp. | MR | Zbl

[8] Soldatov A. P., Shkhanukov M. Kh., “Boundary value problems with A. A. Samarskiĭ's general nonlocal condition for higher-order pseudoparabolic equations”, Soviet Math. Dokl., 36:3 (1988), 507–511 | MR | Zbl

[9] Nakhushev A. M., Problems with shifts for partial differential equations, Nauka, Moscow, 2006, 287 pp. | Zbl

[10] Repin O. A., “A nonlocal problem for a mixed-type equation with a singular coefficient”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2005, no. 34, 5–9 | DOI

[11] Repin O. A., Kumykova S. K., “Nonlocal problem for a equation of mixed type of third order with generalized operators of fractional integro-differentiation of arbitrary order”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2011, no. 4(25), 25–36 | DOI | MR

[12] Pul'kina L. S., “On the solvability in $L_2$ of a nonlocal problem with integral conditions for a hyperbolic equation”, Differ. Equ., 36:2 (2000), 316–318 | DOI | MR | Zbl

[13] Ogorodnikov E. N., Yashagin N. S., “Setting and Solving of the Cauchy type problems for the Second Order Differential Equations with Riemann–Liouville Fractional Derivatives”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2010, no. 1(20), 24–36 | DOI

[14] Arlanova E. Y., “On the problem for mixed type equation with M. Saigo operators”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2011, no. 3(24), 157–161 | DOI

[15] Danilkina O. Yu., “On one nonlocal problem for the heat equation with an integral condition”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2007, no. 1(14), 5–9 | DOI

[16] Tarasenko A. V., “The Boundary Value Problem for the Loaded Equation of Mixed Parabolic-Hyperbolic Type in Rectangular Area”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2010, no. 5(21), 263–267 | DOI

[17] Berezanskiy Yu. M., Roytberg Ya. A., “A theorem on homeomorphisms and Green's function for general elliptic boundary problems”, Ukrain. Mat. Z., 19:5 (1967), 3–32 | MR

[18] Zhitarashu N. V., “A theorem on the complete set of isomorphisms in the $L\sb 2$-theory of model initial parabolic boundary value problems”, Mat. Issled. No. 88, Kishinyov, 1986, 40–59 | MR | Zbl

[19] Akhiev S. S., “Fundamental solutions of some local and nonlocal boundary value problems and their representations”, Dokl. Akad. Nauk SSSR, 271:2 (1983), 265–269 | MR | Zbl

[20] Mamedov I. G., “One Goursat problem in a Sobolev space”, Russian Math. (Iz. VUZ), 55:2 (2011), 46–55 | MR | Zbl

[21] Mamedov I. G., Local and nonlocal boundary value problems for hyperbolic equations with purely mixed derivatives of third-order with non-smooth coefficients, 2000, 65 pp. (Deposited at AzNIINTI, No 2669-Az)

[22] Nakhushev A. M., Equations of Mathematical Biology, Moscow, 1995, 301 pp. | Zbl

[23] Mamedov I. G., “The optimal control problem in the processes described by the non-local problem with loadings for hyperbolic integro-differential equation”, Izv. NAN Azerb. (Ser. Fiz.-Tekhn. Matem. Nauk), 24:2 (2004), 74–79

[24] Mamedov I. G., “Generalization of multipoint boundary-value problems of Bitsadze–Samarski and Samarski–Ionkin type for fourth order loaded hyperbolic integro-differential equations and their operator generalization”, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 23 (2005), 77–84 | MR | Zbl

[25] Mamedov I. G., “Three-dimensional nonlocal boundary-value problem with integral conditions for loaded Volterra-hyperbolic integro-differential equations”, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 24 (2006), 153–162 | MR | Zbl

[26] Mamedov I. G., “A mixed problem with nonlocal boundary conditions of Bitsadze–Samarskiy and Samarskiy–Ionkin types that arises in modeling the filtration of a fluid in fractured media”, Izv. NAN Azerb. (Ser. Fiz.-Tekhn. Matem. Nauk), 26:3 (2006), 32–37 | MR

[27] Mamedov I. G., “The study of the problem with integro-multipoint boundary conditions for the generalized equation of moisture transfer”, Izv. NAN Azerb. (Ser. Fiz.-Tekhn. Matem. Nauk), 27:2–3 (2007), 121–126