An efficient Legendre-Galerkin spectral method and its error analysis for a one-dimensional parabolic equation with Dirichlet-type non-local boundary conditions are presented in this paper. The spatial discretization is based on Galerkin formulation and the Legendre orthogonal polynomials, while the time derivative is discretized by using the symmetric Euler finite difference schema. The stability and convergence of the semi-discrete spectral approximation are rigorously set up by following a novel approach to overcome difficulties caused by the non-locality of the boundary condition. Several numerical tests are included to confirm the efficacy of the proposed method and to support the theoretical results.
We show the existence of Banach spacesX,Ysuch that the set of strictly singular operatorsđź(X,Y)(resp., the set of strictly cosingular operatorsđ đź(X,Y))would be strictly included inâ±+(X,Y)(resp.,â±â(X,Y))for the nonempty class of closed densely defined upper semi-Fredholm operatorsΊ+(X,Y)(resp., for the nonempty class of closed densely defined lower semi-Fredholm operatorsΊâ(X,Y)).
The present paper is devoted to the numerical approximation for the diffusion equation subject to non-local boundary conditions. For the space discretization, we apply the Legendre-Chebyshev pseudospectral method, so that, the problem under consideration is reduced to a system of ODEs which can be solved by the second order Crank-Nicolson schema. Optimal error estimates for the semi-discrete scheme are derived in L2-norm. Numerical tests are included to demonstrate the effectiveness of the proposed method.
AbstractFractional calculus has been shown to improve the dynamics of differential system models and provide a better understanding of their dynamics. This paper considers the timeâfractional version of the DegnâHarrison reactionâdiffusion model. Sufficient conditions are established for the local and global asymptotic stability of the model by means of invariant rectangles, the fundamental stability theory of fractional systems, the linearization method, and the direct Lyapunov method. Numerical simulation results are used to illustrate the theoretical results.
Duca and Lupsa [6] show that the results obtained in Youness [2] concerning the characterization of an E-convex function f in terms of its E-epigraph are incorrect. In this paper we introduce the correct form of this Theorem wich will be used in our study.
Let A be a bounded linear operator in a complex Banach space X. We show that Id X â A is a Fredholm operator provided that A has a sufficiently small polynomially measure of noncompactness. In our general framework, we note that the case of Riesz operator becomes a particular one as it is for the other results in the domain. This enable us to obtain a new characterization for the Weyl essential spectrum of a closed densely defined operators.
Mathematics Subject Classification: 47A53, 47A55
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