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[13] Alibekov Kh.A. and Sobolevskii P.E., Stability and convergence of difference schemes of a high order for parabolic differential equations, Ukrain. Math. Zh. 31(6) 627–634, 1979 (Russian). [14] Ashyralyev A. and Sobolevskii P.E., The linear operator interpolation theory and the stability of the difference schemes, Dokl. Acad. Nauk SSSR 275(6) 1289–1291, 1984 (Russian). [15] Ashyralyev A., Method of Positive Operators of Investigations of the High Order of Accuracy Difference Schemes for Parabolic and Elliptic Equations, Doctor of Sciences Thesis, Kiev: Inst. of Math. of Acad. Sci. Kiev, 1992 (Russian). [16] Simirnitskii Yu.A. and Sobolevskii P.E., Positivity of multidimensional difference operators in the C−norm, Usp. Mat. Nauk. 36(4) 202–203, 1981 (Russian). [17] Danelich S.I., Fractional Powers of Positive Difference Operators, PhD Thesis, Voronezh: Voronezh State University 1989 (Russian). [18] Ashyralyev A. and N. Yaz, On Structure of Fractional Spaces Generated by Positive Operators with the Nonlocal Boundary Value Conditions, in Proceedings of the Conference Differential and Differ- ence Equations and Applications,, Hindawi Publishing Corporation, New York, edited by R.F. Agarwal and K. Perera, 91–101, 2006. Page 14
On the Numerical Solution of Ultra Parabolic Equations with Neumann Condition A. Ashyralyev and S. Yılmaz Abstract Department of Mathematics, Fatih University, Istanbul, Turkey In this paper, our interest is studying the stability of first order difference scheme for the approximate solution of the initial boundary value problem for ultra parabolic equations ⎧ ∂u(t,s) ∂u(t,s) ⎪⎨ ∂t + ∂s + Au(t, s) = f(t, s), 0 < t, s < T, u(0, s) = ψ(s), 0 ≤ s ≤ T, (1) ⎪⎩ u(t, 0) = ϕ(t), 0 ≤ t ≤ T in an arbitrary Banach space E with a strongly positive operator A.We refer to [1, 2] and the references therein for a series of papers by the authors, dealing with ultra parabolic equations , arising in diffusion theory, probability and finance. Some new results about numerical methods for ultra-parabolic equations are also announced, see [3-5]. For approximately solving problem (1), the first-order of accuracy difference scheme ⎧ uk,m−uk−1,m τ + uk−1,m−uk−1,m−1 τ +Auk,m = fk,m , ⎪⎨ fk,m = f(tk, sm), tk = kτ, sm = mτ, 1 ≤ k, m ≤ N, Nτ = 1, (2) u0,m = ψm, ψm = ψ(sm), 0 ≤ m ≤ N, ⎪⎩ uk,0 = ϕk, ϕk = ϕ(tk), 0 ≤ k ≤ N is presented. The stability estimates and almost coercive stability estimates for the solution of difference schemes (2) is established. In applications, the stability in maximum norm of difference shemes for multidimensional ultra parabolic equations with Neumann condition is established. Applying the difference schemes, the numerical methods are proposed for solving one dimensional ultra parabolic equations. References [1] Ashyralyev A. and Yılmaz S., An approximation of ultra-parabolic equations, Abstract and Applied Analysis, Article ID 840621, 13 pages, 2012.(in press) [2] Lanconelli E., Pascucci A. and Polidoro S, Linear and nonlinear ultraparabolic equations of Kol- mogorov type arising in diffusion theory and in finance, in Proceedings of the International Mathematical Series Conference, Nonlinear Problems in Mathematical Physics and Related Topics VOL. II in Honor of Professor O.A. Ladyzhenskya, 243-265, 2002. [3] Akrivis G., Crouzeix M. and Thom˙ee V., Numerical methods for ultraparabolic equations,Calcolo 31 , 179-190, 1994. [4] Ashyralyev A. and Yılmaz S., Second Order of Accuracy Difference Schemes for Ultra Parabolic Equations, in Proceedings of the International Conference on Numerical Analysis and Applied Mathe- matics, AIP Conference Proceedings, Volume 1389, 601-604, 2011. [5] Ayati B. P., A variable time step method for an age-dependent population model with nonlinear diffusion, Sıam J. Numer. Anal., Volume 37(5), 1571-1589, 2000. Page 15
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 and 18: Boundary Value Problem for a Third
Page 19 and 20: [18] S. G. Krein, Linear Differenti
Page 21: Positivity of Two-dimensional Ellip
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
Page 69 and 70: [9] Titeux I., Yakubov Y., Complete
Page 71 and 72: Positivity of Elliptic Difference O
Page 73 and 74:
Transient and Cycle Structure of El
Page 75 and 76:
Abstract Characterizations of Slant
Page 77 and 78:
Page 69 [20] Struik, D. J., Differe
Page 79 and 80:
Applied Mathematics Analysis of the
Page 81 and 82:
Multibody Railway Vehicle Dynamics
Page 83 and 84:
Existence of Global Solutions for a
Page 85 and 86:
On the stability of the steady-stat
Page 87 and 88:
On the Numerical Solution of Diffus
Page 89 and 90:
Using expanding method of (G ′ /G
Page 91 and 92:
Weighted Bernstein Inequality for T
Page 93 and 94:
Abstract An Application on Suborbit
Page 95 and 96:
On The First Fundamental Theorem fo
Page 97 and 98:
Abstract Almost Convergence and Gen
Page 99 and 100:
Forecast results in contrast with o
Page 101 and 102:
Page 93 [15] M. El-Shahed, A. Salem
Page 103 and 104:
yields a system of equations which
Page 105 and 106:
Abstract Application of the Trial E
Page 107 and 108:
Oscillation Theorems for Second-Ord
Page 109 and 110:
Abstract On Hadamard Type Integral
Page 111 and 112:
A geometrical approach of an optima
Page 113 and 114:
A Perturbation Solution Procedure f
Page 115 and 116:
Solution of Differential Equations
Page 117 and 118:
Aproximation Properties of a Genera
Page 119 and 120:
Blow up of a solution for a system
Page 121 and 122:
A Note on Some Elementary Geometric
Page 123 and 124:
Abstract Solving Crossmatching Puzz
Page 125 and 126:
Polygonal Approximation of Digital
Page 127 and 128:
Panoramic Image Mosaicing Using Mul
Page 129 and 130:
[9] Naimark, M. A. Linear Differant
Page 131 and 132:
Abstract Chaos in cubic-quintic non
Page 133 and 134:
Semismooth Newton method for gradie
Page 135 and 136:
Multiple solutions for quasilinear
Page 137 and 138:
The Modified Bi-quintic B-spline ba
Page 139 and 140:
The Modified Kudryashov Method for
Page 141 and 142:
Study of an inverse problem that mo
Page 143 and 144:
Abstract Finding Global minima with
Page 145 and 146:
PARAMETER DEPENDENT NOVIER-STOKES L
Page 147 and 148:
7. D. J. Fyfe, The use of cubic spl
Page 149 and 150:
Page 141 An error correction method
Page 151 and 152:
Classification of exact solutions f
Page 153 and 154:
Abstract Numerical Solution of a Hy
Page 155 and 156:
generalized hyperbolic tangent func
Page 157 and 158:
EQUIVALENCE OF AFFINE CURVES Yasemi
Page 159 and 160:
Modified trial equation method for
Page 161 and 162:
Normal Extensions of a Singular Dif
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