In the present paper the Riesz fractional coupled Schr6dinger-Boussinesq （S-B） equations have been solved by the time-splitting Fourier spectral （TSFS） method. This proposed technique is utilized for discretizing ...In the present paper the Riesz fractional coupled Schr6dinger-Boussinesq （S-B） equations have been solved by the time-splitting Fourier spectral （TSFS） method. This proposed technique is utilized for discretizing the Schrodinger like equation and further, a pseudospectral discretization has been employed for the Boussinesq-like equation. Apart from that an implicit finite difference approach has also been proposed to compare the results with the solutions obtained from the time-splitting technique. Furthermore, the time-splitting method is proved to be unconditionally stable. The error norms along with the graphical solutions have also been presented here.展开更多
In this paper,a Riesz space fractional advection-dispersion equation with fractional Robin boundary condition is considered.By applying the fractional central di erence formula and the weighted and shifted Grunwald-L...In this paper,a Riesz space fractional advection-dispersion equation with fractional Robin boundary condition is considered.By applying the fractional central di erence formula and the weighted and shifted Grunwald-Letnikov formula,we derive a weighted implicit nite difference scheme with accuracy O(△t^2+h^2).The solvability,stability,and convergence of the proposed numerical scheme are proved.A numerical example is presented to confirm the accuracy and efficiency of the scheme.展开更多
In this paper,we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations.The finite difference method is employed to approximate the m...In this paper,we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations.The finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives,which generates symmetric positive definite ill-conditioned multi-level Toeplitz matrices.The preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear system.Theoretically,we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval(12,32)and thus the preconditioned conjugate gradient method converges linearly within an iteration number independent of the discretization step-size.Moreover,the proposed method can be extended to handle ill-conditioned multi-level Toeplitz matrices whose blocks are generated by functions with zeros of fractional order.Our theoretical results fill in a vacancy in the literature.Numerical examples are presented to show the convergence performance of the proposed preconditioner that is better than other preconditioners.展开更多
基金Supported by NBHM,Mumbai,under Department of Atomic Energy,Government of India vide Grant No.2/48(7)/2015/NBHM(R.P.)/R&D Ⅱ/11403
文摘In the present paper the Riesz fractional coupled Schr6dinger-Boussinesq （S-B） equations have been solved by the time-splitting Fourier spectral （TSFS） method. This proposed technique is utilized for discretizing the Schrodinger like equation and further, a pseudospectral discretization has been employed for the Boussinesq-like equation. Apart from that an implicit finite difference approach has also been proposed to compare the results with the solutions obtained from the time-splitting technique. Furthermore, the time-splitting method is proved to be unconditionally stable. The error norms along with the graphical solutions have also been presented here.
基金Supported by the Nation Natural Science Foundation of China(No.11271141)the Chongqing Science and Technology Commission(cstc2018jcyjAX0787)。
文摘In this paper,a Riesz space fractional advection-dispersion equation with fractional Robin boundary condition is considered.By applying the fractional central di erence formula and the weighted and shifted Grunwald-Letnikov formula,we derive a weighted implicit nite difference scheme with accuracy O(△t^2+h^2).The solvability,stability,and convergence of the proposed numerical scheme are proved.A numerical example is presented to confirm the accuracy and efficiency of the scheme.
基金supported in part by research grants of the Science and Technology Development Fund,Macao SAR(No.0122/2020/A3)University of Macao(No.MYRG2020-00224-FST)+1 种基金the HKRGC GRF(No.12306616,12200317,12300218,12300519,17201020))China Postdoctoral Science Foundation(Grant 2020M682897).
文摘In this paper,we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations.The finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives,which generates symmetric positive definite ill-conditioned multi-level Toeplitz matrices.The preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear system.Theoretically,we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval(12,32)and thus the preconditioned conjugate gradient method converges linearly within an iteration number independent of the discretization step-size.Moreover,the proposed method can be extended to handle ill-conditioned multi-level Toeplitz matrices whose blocks are generated by functions with zeros of fractional order.Our theoretical results fill in a vacancy in the literature.Numerical examples are presented to show the convergence performance of the proposed preconditioner that is better than other preconditioners.
