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Abstract:
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The field of research in this dissertation is consideration of convergence of finite differnce method for parabolic problems with generalized solutions. The research in this dissertation is connected with the following subjects: Numerical Analysis and Partial Differential Equations.
This dissertation, beside Preface and References with 56 items, consists of five chapters: 1. Introductory Topics; 2. Parabolic Problems with Variable Operator: Convergence in W(2,1)-norm; 3. Parabolic Problems with Variable Operator: Convergence in W(1,1/2)-norm; 4. Convergence in L-2 norm; 5. Application of Interpolatyion theory
In Chapter 1 a brief review of the Sobolev spaces, anisotropic Sobolev spaces, multipliers in Sobolev spaces, interpolation theory of Banach spaces and existence of generalized solution of parabolic problems are presented. Initial-boundary-value problems with variable (time-dependent) operator are considered in Chapters 2 and 3. In Chapter 2 is proved convergence of finite difference scheme in discrete W(2,1) Sobolev norm. Convergence in W(1,1/2) norm is proved in Chapter 3. In Chapter 4, parabolic problem with variable coefficients is considered and convergence in L-2 norm is proved. Finally, in Chapter 5 , interpolation theory is applied to the convergence analysis. |