Numerical Solution Of Time Research Articles

A transformed $ L1 $ Legendre-Galerkin spectral method for time fractional Fokker-Planck equations

<abstract><p>The numerical solutions of time $ \alpha $-order $ (\alpha \in (0, 1)) $ Caputo fractional Fokker-Planck equations is considered. The constructed method is consist of the transformed $ L1 $ ($ TL1 $) scheme in the temporal direction and the Legendre-Galerkin spectral method in the spatial direction. It has been shown that the $ TL1 $ Legendre-Galerkin spectral method in $ L^2 $-norm is exponential order convergent in space and ($ 2-\alpha $)-th order convergent in time. Several numerical examples are given to verify the obtained theoretical results.</p></abstract>

Numerical solutions of time and space fractional coupled Burgers equations using time–space Chebyshev pseudospectral method

The aim of the paper is to develop and analyze a spectrally accurate time–space pseudospectral method to the approximate solution of nonlinear time and space fractional coupled Burgers equations. Liouville–Caputo fractional derivative formula is used to evaluate the fractional derivatives matrix at CGL points. Using the Chebyshev fractional derivative matrices, the given problem is reduced to a system of nonlinear algebraic equations. A mapping is used to transform the nonhomogeneous initial–boundary values to homogeneous initial–boundary values. Error analysis of the proposed method for the equation is presented. A model example of fractional coupled Burgers equations is tested for a set of fractional‐order derivatives. For the proposed method, highly accurate numerical results are obtained, which confirm the accuracy and efficiency of the proposed method.

Local meshless differential quadrature collocation method for time-fractional PDEs

This paper is concerned with the numerical solution of time- fractional partial differential equations (PDEs) via local meshless differential quadrature collocation method (LMM) using radial basis functions (RBFs). For the sake of comparison, global version of the meshless method is also considered. The meshless methods do not need mesh and approximate solution on scattered and uniform nodes in the domain. The local and global meshless procedures are used for spatial discretization. Caputo derivative is used in the temporal direction for both the values of $ \alpha \in (0,1) $ and $ \alpha\in(1,2) $. To circumvent spurious oscillation casued by convection, an upwind technique is coupled with the LMM. Numerical analysis is given to asses accuracy of the proposed meshless method for one- and two-dimensional problems on rectangular and non-rectangular domains.

Numerical Solution of Time and Space Fractional Burger's Equation with Finite Difference Method

In this study, fractional Burger’s Equation, which has Dirichlet Boundary Conditions, is solved with the Finite Difference Method. Fractional Burger Equation is found by S. Momani, which is made with changing time and space terms with fractional terms. This equation is solved with the finite difference method and analysis of this scheme is discussed with examples. Stability and Uniqueness are discussed with using matrix method. We compare analytical and numerical solutions with error analysis of them.

Analytical and numerical solutions of time and space fractional advection–diffusion–reaction equation

In this paper, analytical and numerical solutions of time and space fractional advection-diffusion-reaction equations are found. In general, by using Lie transformations, it is possible to reduce the fractional partial differential equations into fractional ordinary differential equations, if the symmetries admitted by target equations allow to determine the Lie transformations. In the case of the time and space fractional advection–diffusion–reaction model, the Lie symmetries do not lead to reduce the equation into fractional ordinary one. So we propose an alternative strategy to find the analytical and numerical solutions starting from the analytical and numerical results recently obtained by the authors for the time fractional advection-diffusion-reaction equation and for the space fractional advection–diffusion–reaction equation, separately, by using the Lie symmetries. The numerical results prove the efficiency and the applicability of the proposed procedure that results to be, for its high precision, a good tool to find solutions of a wide class of problems involving the fractional differential equations.

Numerical solution of time- and space-fractional coupled Burgers’ equations via homotopy algorithm

In this paper, we constitute a homotopy algorithm basically extension of homotopy analysis method with Laplace transform, namely q-homotopy analysis transform method to solve time- and space-fractional coupled Burgers’ equations. The suggested technique produces many more opportunities by appropriate selection of auxiliary parameters ℏ and n(n⩾1) to solve strongly nonlinear differential equations. The proposed technique provides ℏ and n-curves, which describe that the convergence range is not a local point effects and finds elucidated series solution that makes it superior than HAM and other analytical techniques.

Creation of high energy electronic excitations in inorganicinsulators by intense femtosecond laser pulses

We measured photoemission spectra for a number of insulators (CsI, Diamond, SiO2, CeF3) excited by femtosecond Ti-Sapphire laser pulses at peak intensities, from 0.5 to 6 TW/cm2, which are at least one order of magnitude below the optical breakdown threshold. An intense and pronounced plateau of high energy electrons appears in the photoelectron spectra in this intensity range, which extends up to 30-40 eV at the highest intensities, which we used. The excitation of electrons at high energies is treated in terms of direct interbranch transitions in the conduction band of insulator. These processes are described using calculations based on numerical solution of time dependant Schrodinger equation (TDSE) for Bloch electrons in electric fields. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)