ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR

doi:10.1016/S0252-9602(15)30031-X

数学物理学报(英文版) ›› 2015, Vol. 35 ›› Issue (5): 970-980.doi: 10.1016/S0252-9602(15)30031-X

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR

B. J. KADIRKULOV¹, M. KIRANE^2,3

1. Tashkent State Institute for Oriental Studies, Tashkent, Uzbekistan;
2. Laboratoire de Mathématiques, Image et Applications, Université de La Rochelle, Avenue M. Crépeau, 17042 La Rochelle Cedex, France;
3. NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

收稿日期:2014-10-10 修回日期:2015-03-08 出版日期:2015-09-01 发布日期:2015-09-01
作者简介:B. J. KADIRKULOV, E-mail: kadirkulovbj@gmail.com; M. KIRANE, E-mail: moktar.kirane@univ-lr.fr

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR

B. J. KADIRKULOV¹, M. KIRANE^2,3

1. Tashkent State Institute for Oriental Studies, Tashkent, Uzbekistan;
2. Laboratoire de Mathématiques, Image et Applications, Université de La Rochelle, Avenue M. Crépeau, 17042 La Rochelle Cedex, France;
3. NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

Received:2014-10-10 Revised:2015-03-08 Online:2015-09-01 Published:2015-09-01

摘要/Abstract

摘要：

In this work, we investigate the solvability of the boundary value problem for the Poisson equation, involving a generalized Riemann-Liouville and the Caputo derivative of fractional order in the class of smooth functions. The considered problems are generalization of the known Dirichlet and Neumann problems with operators of a fractional order.

关键词: operator of fractional integration and differentiation, solvability, boundary value problem, Riemann-Liouville operator, Caputo fractional derivative, Poisson equation, Dirichlet and Neumann problems

Abstract:

Key words: operator of fractional integration and differentiation, solvability, boundary value problem, Riemann-Liouville operator, Caputo fractional derivative, Poisson equation, Dirichlet and Neumann problems

中图分类号:

35J05

B. J. KADIRKULOV, M. KIRANE. ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR[J]. 数学物理学报(英文版), 2015, 35(5): 970-980.

B. J. KADIRKULOV, M. KIRANE. ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR[J]. Acta mathematica scientia,Series B, 2015, 35(5): 970-980.

参考文献

[1] Bavrin I I. Operators for harmonic functions and their applications. Differential Equations, 1985, 21(1):6-10
[2] Berdyshev A S, Nieto J J, Kadirkulov B J. Solvability of an elliptic partial differential equation with boundary condition involving fractional derivatives. Complex Variables and Elliptic Equations, 2014, 59(5):680-692
[3] Berdyshev A S, Turmetov B Kh, Kadirkulov B J. Some properties and applications of the integrodifferential operators of Hadamard-Marchaud type in the class of harmonic function. Siberian Mathematical J, 2012, 53(4):600-610
[4] Bitsadze A V. Equations of Mathematical Physics. Moscow:Mir Publishers, 1980
[5] Erdely A. On fractional integration and its applications to the theory of Hankel transforms. Quart J Math Oxford, 1940, 11(1):293-303
[6] Ferreira N M F, Duarte F B, Lima M F M, Marcos M G, Tenreiro Machado J A. Application of fractional calculus in the dynamical analysis and control of mechanical manipulations. Frac Calc Appl Anal, 2008, 11(1):91-113
[7] Gilbarg D, Trudinger N. Elliptical Partial Differential Equations of the Second Order. Berlin:Springer-Verlag, 1998
[8] Hilfer R. Fractional calculus and regular variation in thermodynamics//Hilfer R, ed. Applications of Fractional Calculus in Physics. Singapore:World Scientific, 2000
[9] Jesus I S, Tenreiro Machado J A, Cunha J B. Fractional impedance in botanical elements. J Vibration and Control, 2008, 14(9/10):1389-1402
[10] Jesus I S, Tenreiro Machado J A. Fractional control of heat diffusion systems. Nonlinear Dyn, 2008, 54:263-282
[11] Karachik V V, Turmetov B Kh, Torebek B T. On some integro-differential operators in the class of harmonic functions and their applications. Siberian Adv Math, 2012, 22(2):115-134
[12] Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. Amsterdam:Elsevier, 2006
[13] Kirane M, Tatar N E. Nonexistence for the Laplace equation with a dynamical boundary condition of fractional type. Siberian Math J, 2007, 48(5):849-856
[14] Kober H. On a fractional integral and derivative. Quart J Math Oxford, 1940, 11(1):193-211
[15] Koeller R C. Application of fractional calculus to the theory of viscoelasticity. J Appl Mech, 1984, 51:299-307
[16] Serbina L I. A model of substance transfer in fractal media. J Math Model, 2003, 15:17-28
[17] Sneddon I N. The use in mathematical analysis of the Erdelyi-Kober operators and some of their applications//Lect Notes Math 457. New York:SpringerVerlag, 1975:3779
[18] Tenreiro Machado J A. Theory of fractional integrals and derivatives:application to motion control//Proceedings of International Conference on recent advances in Mechatronics (ICRAM 95). Istanbul, Turkey, 14-16 August 1995
[19] Torebek B T, Turmetov B Kh. On Solvability of a Boundary Value Problem for the Poisson Equation with the Boundary operator of a fractional order. Boundary Value Problems, 2013, 93:1-18
[20] Umarov S R, Luchko Yu F, Gorenflo R. On boundary value problems for elliptic equations with boundary operators of fractional order. Frac Calc Appl Anal, 2000, 3(4):454-468
[21] Umarov S R. On certain boundary-value-problems for elliptic-equations with boundary operator of fractional order. Doklady Akademii Nauk, 1993, 333(6):708-710
[22] Turmetov B Kh. A boundary value problem for the harmonic equation. Differential Equations, 1996, 32(8):1093-1096
[23] Berdyshev A S, Cabada A, Turmetov B Kh. On solvability of a boundary value problem for a nonhomo-geneous biharmonic equation with a boundary operator of a fractional order. Acta Mathematica Scientia, 2014, 34B(6):1695-1706

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	Alberto CABADA, Nikolay D. DIMITROV. EXISTENCE OF SOLUTIONS OF NONLOCAL PERTURBATIONS OF DIRICHLET DISCRETE NONLINEAR PROBLEMS[J]. 数学物理学报(英文版), 2017, 37(4): 911-926.
[2]	侯飞. ON THE GLOBAL EXISTENCE OF SMOOTH SOLUTIONS TO THE MULTI-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS WITH TIME-DEPENDING DAMPING IN HALF SPACE[J]. 数学物理学报(英文版), 2017, 37(4): 949-964.
[3]	肖羽, 范恩贵, 徐建. THE FOKAS-LENELLS EQUATION ON THE FINITE INTERVAL[J]. 数学物理学报(英文版), 2017, 37(3): 852-876.
[4]	王丽霞, 马世旺, 许娜. MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS SCHRODINGER-POISSON EQUATIONS WITHÖSIGN-CHANGING POTENTIAL[J]. 数学物理学报(英文版), 2017, 37(2): 555-572.
[5]	于海波. GLOBAL REGULARITY TO THE 2D INCOMPRESSIBLE MHD WITH MIXED PARTIAL DISSIPATION AND MAGNETIC DIFFUSION IN A BOUNDED DOMAIN[J]. 数学物理学报(英文版), 2017, 37(2): 395-404.
[6]	Nermina MUJAKOVIC, Nelida CRNJARIC-ZIC. GLOBAL SOLUTION TO 1D MODEL OF A COMPRESSIBLE VISCOUS MICROPOLAR HEAT-CONDUCTING FLUID WITH A FREE BOUNDARY[J]. 数学物理学报(英文版), 2016, 36(6): 1541-1576.
[7]	Bashir AHMAD, Sotiris K. NTOUYAS, Jessada TARIBOON. A NONLOCAL HYBRID BOUNDARY VALUE PROBLEM OF CAPUTO FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS[J]. 数学物理学报(英文版), 2016, 36(6): 1631-1640.
[8]	Yaghoub JALILIAN. FRACTIONAL INTEGRAL INEQUALITIES AND THEIR APPLICATIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS[J]. 数学物理学报(英文版), 2016, 36(5): 1317-1330.
[9]	Batirkhan TURMETOV. ON SOME BOUNDARY VALUE PROBLEMS FOR NONHOMOGENOUS POLYHARMONIC EQUATION WITH BOUNDARY OPERATORS OF FRACTIONAL ORDER[J]. 数学物理学报(英文版), 2016, 36(3): 831-846.
[10]	连汝续, 刘健. FREE BOUNDARY VALUE PROBLEM FOR THE CYLINDRICALLY SYMMETRIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2016, 36(1): 111-123.
[11]	刘华. ANALYTIC BOUNDARY VALUE PROBLEMS ON CLASSICAL DOMAINS[J]. 数学物理学报(英文版), 2015, 35(5): 1037-1045.
[12]	Rinaldo M. COLOMBO, Elena ROSSI. RIGOROUS ESTIMATES ON BALANCE LAWS IN BOUNDED DOMAINS[J]. 数学物理学报(英文版), 2015, 35(4): 906-944.
[13]	Eid H. DOHA, Waleed M. ABD-ELHAMEED, Mahmoud A. BASSUONY. ON THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS AND DERIVATIVES OF CHEBYSHEV POLYNOMIALS OF THE THIRD AND FOURTH KINDS[J]. 数学物理学报(英文版), 2015, 35(2): 326-338.
[14]	A.S. BERDYSHEV, A. CABADA, B.Kh. TURMETOV. ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR A NONHOMOGENEOUS BIHARMONIC EQUATION WITH A BOUNDARY#br# OPERATOR OF A FRACTIONAL ORDER[J]. 数学物理学报(英文版), 2014, 34(6): 1695-1706.
[15]	韦忠礼. SOME NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE OF POSITIVE SOLUTIONS FOR THIRD ORDER SINGULAR SUBLINEAR MULTI-POINT BOUNDARY VALUE PROBLEMS[J]. 数学物理学报(英文版), 2014, 34(6): 1795-1810.