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Numerical solutions to integral equations equivalent to differential equations with fractional time

100%
Bandrowski B., Karczewska A., Rozmej P.
International Journal of Applied Mathematics and Computer Science
|
2010
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tom 20
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nr 2
261-269
EN
This paper presents an approximate method of solving the fractional (in the time variable) equation which describes the processes lying between heat and wave behavior. The approximation consists in the application of a finite subspace of an infinite basis in the time variable (Galerkin method) and discretization in space variables. In the final step, a large-scale system of linear equations with a non-symmetric matrix is solved with the use of the iterative GMRES method.
2
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Applications of the Carathéodory theorem to PDEs

100%
Holly K., Orewczyk J.
Annales Polonici Mathematici
|
2000
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tom 73
|
nr 1
1-27
EN
We discuss and exploit the Carathéodory theorem on existence and uniqueness of an absolutely continuous solution x: ℐ (⊂ ℝ) → X of a general ODE $ẋ {(*)\over=} ℱ(t,x)$ for the right-hand side ℱ : dom ℱ ( ⊂ ℝ × X) → X taking values in an arbitrary Banach space X, and a related result concerning an extension of x. We propose a definition of solvability of (*) admitting all connected ℐ and unifying the cases "dom ℱ is open" and "dom ℱ = ℐ × Ω for some Ω ⊂ X". We show how to use the theorems mentioned above to get approximate solutions of a nonlinear parabolic PDE and exact solutions of a linear evolution PDE with distribution data.
3
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Numerical Solution of Time Fractional Schrödinger Equation by Using Quadratic B-Spline Finite Elements

100%
Esen A., Tasbozan O.
Annales Mathematicae Silesianae
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2017
|
tom 31
|
nr 1
83-98
EN
In this article, quadratic B-spline Galerkin method has been employed to solve the time fractional order Schrödinger equation. Numerical solutions and error norms L2 and L∞ are presented in tables.
4
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On the convergence of the wavelet-Galerkin method for nonlinear filtering

100%
Nowak Ł., Pasławska-Południak M., Twardowska K.
International Journal of Applied Mathematics and Computer Science
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2010
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tom 20
|
nr 1
93-108
EN
The aim of the paper is to examine the wavelet-Galerkin method for the solution of filtering equations. We use a wavelet biorthogonal basis with compact support for approximations of the solution. Then we compute the Zakai equation for our filtering problem and consider the implicit Euler scheme in time and the Galerkin scheme in space for the solution of the Zakai equation. We give theorems on convergence and its rate. The method is numerically much more efficient than the classical Galerkin method.
5
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A fixed point method in dynamic processes for a class of elastic-viscoplastic materials

88%
Amassad A.
Annales Polonici Mathematici
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1998
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tom 68
|
nr 3
237-247
EN
Two problems are considered describing dynamic processes for a class of rate-type elastic-viscoplastic materials with or without internal state variable. The existence and uniqueness of the solution is proved using classical results of linear elasticity theory together with a fixed point method.
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