Geodesic
Boundary value problems for degenerate and degenerate fractional
Ufa mathematical journal, Tome 11 (2019) no. 2, pp. 34-55 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the paper we study non-local boundary value problems for differential and partial differential equations of fractional order with a non-local linear source being mathematical models of the transfer of water and salts in soils with fractal organization. Apart of the Cartesian case, in the paper we consider one-dimensional cases with cylindrical and spherical symmetry. By the method of energy inequalities, we obtain apriori estimates of solutions to nonlocal boundary value problems in differential form. We construct difference schemes and for these schemes, we prove analogues of apriori estimates in the difference form and provide estimates for errors assuming a sufficient smoothness of solutions to the equations. By the obtained apriori estimates, we get the uniqueness and stability of the solution with respect to the the initial data and the right par, as well as the convergence of the solution of the difference problem to the solution of the corresponding differential problem with the rate of $O(h^2+\tau^2)$.
Keywords: boundary value problem, apriori estimate, the equation of moisture transfer, the differential equation of fractional order, Gerasimov-Caputo fractional derivative. 
@article{UFA_2019_11_2_a2,
     author = {M. Kh. Beshtokov},
     title = {Boundary value problems for degenerate and degenerate fractional},
     journal = {Ufa mathematical journal},
     pages = {34--55},
     year = {2019},
     volume = {11},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2019_11_2_a2/}
}
TY  - JOUR
AU  - M. Kh. Beshtokov
TI  - Boundary value problems for degenerate and degenerate fractional
JO  - Ufa mathematical journal
PY  - 2019
SP  - 34
EP  - 55
VL  - 11
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/UFA_2019_11_2_a2/
LA  - en
ID  - UFA_2019_11_2_a2
ER  - 
%0 Journal Article
%A M. Kh. Beshtokov
%T Boundary value problems for degenerate and degenerate fractional
%J Ufa mathematical journal
%D 2019
%P 34-55
%V 11
%N 2
%U http://geodesic.mathdoc.fr/item/UFA_2019_11_2_a2/
%G en
%F UFA_2019_11_2_a2
M. Kh. Beshtokov. Boundary value problems for degenerate and degenerate fractional. Ufa mathematical journal, Tome 11 (2019) no. 2, pp. 34-55. http://geodesic.mathdoc.fr/item/UFA_2019_11_2_a2/

[1] A.M. Nakhushev, Fractional calculus and its application, Fizmatgiz, M., 2003 (in Russian)

[2] V.V. Uchaikin, Method of fractional derivatives, Artishok Publ., Ulyanovsk, 2008 (in Russian)

[3] B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman and Company, New York | MR | Zbl

[4] R.L. Begli, P.J. Torvik, “Differential calculus based on fractional order derivatives: a new approach for calculating construction with viscoelastic damping”, Aerokosmich. Tekh., 2:2 (1984), 84–93 (in Russian)

[5] J. Feder, Fractals, Plenum Press, New York, 1988 | MR | Zbl

[6] O.Yu. Dinariev, “Flow in a fractured medium with fractal fracture geometry”, Fluid Dyn., 25:5 (1990), 704–708 | DOI | MR | Zbl

[7] R.R. Nigmatullin, “The realization of generalized transfer equation in a medium with fractal geometry”, Phys. Status Solidi. B, 133:1 (1986), 425–430 | DOI

[8] A.N. Kochubei, “Fractional-order diffusion”, Differ. Equat., 26:4 (1990), 485–492 | MR | Zbl

[9] K.V. Chukbar, “Stochastic transport and fractional derivatives”, J. Exp. Theor. Phys., 81:5 (1995), 1025–1029

[10] G.I. Barenblatt, Yu.P. Zheltov, I.N. Kochina, “Basic Concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata)”, J. Appl. Math. Mekh., 24:5 (1960), 1286–1303 | DOI | Zbl

[11] E.S. Dzektser, “Equations of motion of underground water with a free surface in stratified media”, DAN SSSR, 220:3 (1975), 540–543 (in Russian) | Zbl

[12] M. Hallaire, “Le potential efficace de l'eau dans le sol en régime de dessèchement”, L'eau et production vegetale, I.N.R.A., 1964, 27–62

[13] S.V. Nerpin, A.F. Chudnovsky, Energy- and mass-transfer in a system plant-soil-air, Gidrometeoizdat, L., 1975 (in Russian)

[14] P.J. Chen, M.E. Curtin, “On a theory of heat conduction involving two temperatures”, Jornal Angew. Math. Phys., 19 (1968), 614–627 | DOI | Zbl

[15] R.E. Showalter, T.W. Ting, “Pseudoparabolic partial differential equations”, SIAM J. Math. Anal., 1:1 (1970), 1–26 | DOI | MR | Zbl

[16] S.L. Sobolev, “On a new problem of mathematical physics”, Izv. AN SSSR. Ser. Matem., 18:1 (1954), 3–50 (in Russian) | MR | Zbl

[17] A.G. Sveshnikov, A.B. Al'shin, M.O. Korpusov, Yu.D. Pletner, Linear and nonlinear equations of Sobolev type, Fizmatlit, M., 2007 (in Russian)

[18] M.Kh. Shkhanukov, “Some boundary-value problems for a third-order equation, and extremal properties of its solutions”, Differ. Equat., 19:1 (1983), 124–130 | MR | MR | Zbl | Zbl

[19] V.A. Vodakhova, “On a certain boundary value problem for a third order equation with a non-local condition of A. M. Nakhushev”, Differ. Uravn., 19:1 (1983), 163–166 (in Russian) | MR

[20] M.Kh. Beshtokov, “On the numerical solution of a nonlocal boundary value problem for a degenerating pseudoparabolic equation”, Diff. Equat., 52:10 (2016), 1341–1354 | DOI | DOI | MR | Zbl

[21] M.Kh. Beshtokov, “Difference method for solving a nonlocal boundary value problem for a degenerating third-order pseudo-parabolic equation with variable coefficients”, Comp. Math. Math. Phys., 56:10 (2016), 1763–1777 | DOI | MR | Zbl

[22] M.Kh. Beshtokov, “On a non-local boundary value problem for the third order pseudo-parabolic equation”, Comp. Math. Model., 27:1 (2016), 60–79 | DOI | MR | Zbl

[23] M.Kh. Beshtokov, “The third boundary value problem for loaded differential Sobolev type equation and grid methods of their numerical implementation”, 11th International Conference on “Mesh methods for boundary-value problems and applications”, IOP Conference Series: Materials Science and Engineering, 158, 2016, 012019 | DOI

[24] M.Kh. Beshtokov, “Differential and difference boundary value problem for loaded third-order pseudo-parabolic differential equations and difference methods for their numerical solution”, Comp. Math. Math. Phys., 57:12 (2017), 1973–1993 | DOI | MR | Zbl

[25] M.Kh. Beshtokov, “Local and nonlocal boundary value problems for degenerating and nondegenerating pseudoparabolic equations with a Riemann-Liouville fractional derivative”, Diff. Equat., 54:6 (2018), 758–774 | DOI | DOI | MR | Zbl

[26] H. Caputo, “Lineal model of dissipation whose Q is almost frequency independent”, II Geophys J. Astronom. Soc., 13 (1967), 529–539 | DOI

[27] A.N. Gerasimov, “Generalization of linear deformation laws and their application to problems of internal friction”, Prikl. Matem. Mekh., 12 (1948), 251–260 (in Russian) | Zbl

[28] A.A. Alikhanov, “A priori estimates for solutions of boundary value problems for fractional-order equations”, Diff. Equat., 46:5 (2010), 660–666 | DOI | MR | Zbl

[29] A.A. Samarski, The theory of difference schemes, Marcel Dekker, New York, 2001 | MR | MR | Zbl

[30] A.A. Alikhanov, “A new difference scheme for the time fractional diffusion equation”, Journal of Computational Physics, 280 (2015), 424–438 | DOI | MR | Zbl

[31] D. Li, H. -L. Liao, W. Sun, J. Wang, J. Zhang, “Analysis of L1-Galerkin FEMs for time-fractional nonlinear parabolic problems”, Commun. Comput. Phys., 24:86 (2018), 86–103 | MR