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 operators𝒮(X,Y)(resp., the set of strictly cosingular operators𝒞 𝒮(X,Y))would be strictly included inℱ+(X,Y)(resp.,ℱ−(X,Y))for the nonempty class of closed densely defined upper semi-Fredholm operatorsΦ+(X,Y)(resp., for the nonempty class of closed densely defined lower semi-Fredholm operatorsΦ−(X,Y)).
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 time–fractional version of the Degn–Harrison reaction–diffusion 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.
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