The fractional differential equations involving different types of fractional derivatives are currently used in many fields of science and engineering. Therefore, the first purpose of this study is to investigate the qualitative properties including the stability, asymptotic stability, as well as Mittag–Leffler stability of solutions of fractional differential equations with the new generalized Hattaf fractional derivative, which encompasses the popular forms of fractional derivatives with non-singular kernels. These qualitative properties are obtained by constructing a suitable Lyapunov function. Furthermore, the second aim is to develop a new numerical method in order to approximate the solutions of such types of equations. The developed method recovers the classical Euler numerical scheme for ordinary differential equations. Finally, the obtained analytical and numerical results are applied to a biological nonlinear system arising from epidemiology.

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On the Stability and Numerical Scheme of Fractional Differential Equations with Application to Biology

Heat transfer and irreversibility evaluation of non-Newtonian nanofluid density-driven convection within a hexagonal-shaped domain influenced by an inclined magnetic field

A GFDM with supplementary nodes for thin elastic plate bending analysis under dynamic loading

The radial point interpolation method (RPIM) is combined with the appropriate implicit time integration technique for transient wave propagation analysis in this work. The dispersion error issue of the numerical results is investigated detailedly by performing the dispersion analysis. It is found that the RPIM with sufficiently large support domains of quadrature points is able to yield almost no spatial dispersion errors which are much lower than those from the traditional finite element (FE) approach with the identical node distributions, hence we can monotonically improve the computation accuracy of the numerical results by using the decreasing time steps, namely the present method shows the important and attractive monotonic convergence property for transient wave analysis. This property makes the present method clearly superior to the standard FE approach in transient wave analysis and much more accurate solutions can be achieved. Several typical benchmark problems of transient wave propagations are studied to assess the robust and superior performance of the present method over the conventional FE for transient wave analysis. 

Performance of the radial point interpolation method (RPIM) with implicit time integration scheme for transient wave propagation dynamics

The generalized finite difference method (GFDM) is a typical meshless collocation method based on the Taylor series expansion and the moving least squares technique. In this paper, we first provide theoretical results of the meshless function approximation in the GFDM. Properties, stability and error estimation of the approximation are studied theoretically, and a stabilized approximation is proposed by revising the computational formulas of the original approximation. Then, we provide theoretical results consisting of error bound and condition number of the GFDM. Numerical results are finally provided to confirm these theoretical results. 

Theoretical analysis of the generalized finite difference method

Integrating Krylov Deferred Correction and Generalized Finite Difference Methods for Dynamic Simulations of Wave Propagation Phenomena in Long-Time Intervals

An Analytical Solution for Nonlinear Vibration Analysis of Functionally Graded Rectangular Plate in Contact with Fluid

Convergence Analysis and Error Estimate for Distributed Optimal Control Problems Governed by Stokes Equations with Velocity-Constraint

Multiple-Relaxation-Time Lattice Boltzmann Simulation of Magnetic Field Effect on Natural Convection of Non-Newtonian Nanofluid in Rectangular Enclosure

Unconditional Stability and Error Estimates of the Modified Characteristics FEM for the Time-Dependent Viscoelastic Oldroyd Flows

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