Fractional Parabolic Differential and Difference Equations with <strong>the</strong> Dirichlet-Neumann Condition A. Ashyralyev 1 , N. Emirov 1 and Z. Cakir 2 1 Department of Ma<strong>the</strong>matics, Fatih University, Istanbul, Turkey 2 Department of Ma<strong>the</strong>matical <strong>Abstract</strong> Eng<strong>in</strong>eer<strong>in</strong>g, <strong>Gumushane</strong> University,<strong>Gumushane</strong>, Turkey The multidimensional fractional parabolic equation with <strong>the</strong> Dirichlet-Neumann condition is studied. Stability estimates for <strong>the</strong> solution of <strong>the</strong> <strong>in</strong>itial-boundary value problem for this fractional parabolic equation are established. The stable difference schemes for this problem are presented. Stability estimates for <strong>the</strong> solution of <strong>the</strong> first order of accuracy difference scheme are obta<strong>in</strong>ed. A procedure of modified Gauss elim<strong>in</strong>ation method is applied for <strong>the</strong> solution of first and second order of accuracy difference schemes of one-dimensional fractional parabolic differential equations. References [1] I. Podlubny, Fractional differential Equations,vol. 198 of Mahematics <strong>in</strong> Science and Eng<strong>in</strong>eer<strong>in</strong>g, Academic Press, San Diego, California, USA, 1999. [2] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, Switzerland, 1993. [3] A. A. Kilbas, H. M. Sristava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Ma<strong>the</strong>matics Studies, 2006. [4] J. L. Lavoie, T. J. Osler, and R. Tremblay, SIAM Review 18(2), 240–268 (1976). [5] V. E. Tarasov, Internatioanl Journal of Ma<strong>the</strong>matics 18(3), 281–299 (2007). [6] M. De la Sen, R. P. Agarwal, A. Ibeas, et al. Advances <strong>in</strong> Difference Equations, 2011, Article ID 748608, 32 pages (2011). [7] R. P. Agarwal, B. de Andrade, C. Cuevas, Nonl<strong>in</strong>ear Analysis Series B: Real World Applications 11, pp 3532–3554 (2010). [8] R. Gorenflo, and F. Ma<strong>in</strong>ardi, Fractional Calculus: Integral and Differential Equations of Frac- tional Order, <strong>in</strong> Fractals and Fractional Calculus <strong>in</strong> Cont<strong>in</strong>uum Mechanics, Edited by A.Carp<strong>in</strong>teri and F.Ma<strong>in</strong>ardi, 378 of CISM Courses and Lectures, Spr<strong>in</strong>ger, Vienna, Austria 1997, pp. 223–276 [9] A. S. Berdyshev, A. Cabada, E. T. Karimov, Nonl<strong>in</strong>ear Anal.,75(6) 3268–3273 (2011). [10] A. Ashyralyev, D. Amanov, <strong>Abstract</strong> and Applied Analysis , <strong>2012</strong>, Article ID 594802, 14 pages (<strong>2012</strong>). [11] A. Ashyralyev, B. Hicdurmaz, Kybernetes 40(5-6), 736–750 (2011). [12] F. Ma<strong>in</strong>ardi, Fractional Calculus: Some Basic Problems <strong>in</strong> Cont<strong>in</strong>uum and Statistical Mechanics, <strong>in</strong> Fractals and Fractional Calculus <strong>in</strong> Cont<strong>in</strong>uum Mechanics , Edited by A. Carp<strong>in</strong>teri and F.Ma<strong>in</strong>ardi, Spr<strong>in</strong>ger-Verlag, New-York,USA, 1997, pp. 291–348. [13] M. Kirane, Y. Laskri, Applied Ma<strong>the</strong>matics and Computation, 167(2), 1304–1310 (2005). [14] A. Ashyralyev, F. Dal, and Z. Pınar, Appl. Math. Comput. 217(9), 4654–4664 (2011). [15] V. Lakshimikantham, A. Vatsala, Appl.Anal., 11(3-4), 395–402 (2007). [16] A. Ashyralyev, Applied Ma<strong>the</strong>matics Letters 24, 1176–1180 (2011). Page 10 [17] A. Ashyralyev, Journal of Ma<strong>the</strong>matical Analysis and Applications357(1), 232–236 (2009).
[18] S. G. Kre<strong>in</strong>, L<strong>in</strong>ear Differential Equations <strong>in</strong> a Banach Space, Nauka, Moscow, 1966(Russian). [19] G. Da Prato, P. Grisvard, J. Math. Pures et Appl., 54, 305–387 (1975). [20] P. E. Sobolevskii, Dokl. Akad. Nauk, 225(6), 1638–1641 (1975). [21] Ph. Clement, On (L p -L a ) coerciveness for a class of <strong>in</strong>tegrodifferential equation on <strong>the</strong> l<strong>in</strong>e p Prepr<strong>in</strong>t 5-4-90, VGU, Voronezh, 1990. [22] Z. Cakir <strong>Abstract</strong> and Applied Analysis , <strong>2012</strong>, Article ID 463746, 17 pages (<strong>2012</strong>). Page 11
- Page 3 and 4: TABLE OF CONTENTS P a g e | i Shado
- Page 5 and 6: P a g e | iii Sturm Liouville Probl
- Page 7 and 8: P a g e | v Real Time 3D Palmprint
- Page 9 and 10: ��������� ��
- Page 11 and 12: Abstract On the Classifications of
- Page 13 and 14: A Note on the Numerical Solution of
- Page 15 and 16: Abstract A Third-Order of Accuracy
- Page 17: Boundary Value Problem for a Third
- Page 21 and 22: Positivity of Two-dimensional Ellip
- Page 23 and 24: On the Numerical Solution of Ultra
- Page 25 and 26: Abstract Optimal Control Problem fo
- Page 27 and 28: On Stability Of Hyperbolic- Ellipti
- Page 29 and 30: Abstract FUZZY CONTINUOUS DYNAMICAL
- Page 31 and 32: [10] D.L. DeAngelis, R.A. Goldstein
- Page 33 and 34: Bright and dark soliton solutions f
- Page 35 and 36: [22] H., Triki, A.M.,Wazwaz, Bright
- Page 37 and 38: [5] Khadjiev Dj., Çavu¸s A., Four
- Page 39 and 40: Paths of Minimal Length on Suborbit
- Page 41 and 42: On A SUBCLASS OF UNIVALENT FUNCTION
- Page 43 and 44: Approximation by Certain Linear Pos
- Page 45 and 46: PARABOLIC PROBLEMS WITH PARAMETER O
- Page 47 and 48: On the Numerical Solution of a Diff
- Page 49 and 50: [3] Y. Ma, and Y. Ge, Applied Mathe
- Page 51 and 52: A Fuzzy Approach to Multi Objective
- Page 53 and 54: Derivation and numerical study of r
- Page 55 and 56: [11] G.A. Baker, P. Graves-Morris,
- Page 57 and 58: Three Models based Fusion Approach
- Page 59 and 60: The normal inverse Gaussian distrib
- Page 61 and 62: Compact and Fredholm Operators on M
- Page 63 and 64: Abstract Extended Eigenvalues of Di
- Page 65 and 66: A New General Inequality for double
- Page 67 and 68: Exact Solutions of the Schrödinger
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[9] Titeux I., Yakubov Y., Complete
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Positivity of Elliptic Difference O
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Transient and Cycle Structure of El
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Abstract Characterizations of Slant
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Page 69 [20] Struik, D. J., Differe
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Applied Mathematics Analysis of the
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Multibody Railway Vehicle Dynamics
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Existence of Global Solutions for a
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On the stability of the steady-stat
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On the Numerical Solution of Diffus
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Using expanding method of (G ′ /G
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Weighted Bernstein Inequality for T
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Abstract An Application on Suborbit
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On The First Fundamental Theorem fo
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Abstract Almost Convergence and Gen
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Forecast results in contrast with o
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Page 93 [15] M. El-Shahed, A. Salem
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yields a system of equations which
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Abstract Application of the Trial E
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Oscillation Theorems for Second-Ord
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Abstract On Hadamard Type Integral
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A geometrical approach of an optima
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A Perturbation Solution Procedure f
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Solution of Differential Equations
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Aproximation Properties of a Genera
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Blow up of a solution for a system
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A Note on Some Elementary Geometric
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Abstract Solving Crossmatching Puzz
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Polygonal Approximation of Digital
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Panoramic Image Mosaicing Using Mul
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[9] Naimark, M. A. Linear Differant
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Abstract Chaos in cubic-quintic non
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Semismooth Newton method for gradie
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Multiple solutions for quasilinear
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The Modified Bi-quintic B-spline ba
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The Modified Kudryashov Method for
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Study of an inverse problem that mo
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Abstract Finding Global minima with
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PARAMETER DEPENDENT NOVIER-STOKES L
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7. D. J. Fyfe, The use of cubic spl
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Page 141 An error correction method
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Classification of exact solutions f
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Abstract Numerical Solution of a Hy
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generalized hyperbolic tangent func
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EQUIVALENCE OF AFFINE CURVES Yasemi
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Modified trial equation method for
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Normal Extensions of a Singular Dif