A computational approach based on finite difference scheme and a redefined extended B-spline functions is presented to study the approximate solution of time fractional advection diffusion equation. The Caputo time-fractional derivative and redefined extended B-spline functions have been used for the time and spatial discretization, respectively. The numerical scheme is shown to be O ( h 2 + Δ t 2 - α ) accurate and unconditionally stable. The proposed method is tested through some numerical experiments involving homogeneous/non-homogeneous boundary conditions which concluded that it is more accurate than existing methods. The simulation results show superior agreement with the exact solution as compared to existing methods.

A computational approach for solving time fractional differential equation via spline functions

In this paper, we investigate a fully implicit finite difference scheme for solving the time fractional advection–diffusion equation. The time fractional derivative is estimated using Caputo’s formulation, and the spatial derivatives are discretized using extended cubic B-spline functions. The convergence and stability of the fully implicit scheme are analyzed. Numerical experiments conducted indicate that the scheme is feasible and accurate.

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A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection–diffusion equation

In this work, the nonlocal strain gradient theory is applied to study the free vibration response of a Timoshenko beam made of triclinic material. The governing equations of the problem and the associated boundary conditions are obtained by means of the Hamiltonian principle, whereby the generalized differential quadrature (GDQ) method is implemented as numerical tool to solve the eigenvalue problem in a discrete form. Different combinations of boundary conditions are also considered, which include simply-supports, clamped supports and free edges. Starting with some pioneering works from the literature about isotropic nanobeams, a convergence analysis is first performed, and the accuracy of the proposed size-dependent anisotropic beam model is checked. A large parametric investigation studies the effect of the nonlocal, geometry, and strain gradient parameters, together with the boundary conditions, on the vibration response of the anisotropic nanobeams, as useful for practical engineering applications.

Free Vibration Analysis of Triclinic Nanobeams Based on the Differential Quadrature Method

In this research paper, our main objective is to represent a direct numerical approach for solving time-fractional Burger’s equation using modified hybrid B-spline basis function. The Caputo derivative is used to discretize the time-fractional derivative and for Space derivative Differential Quadrature Method (DQM) based on B-Spline is used. The DQM method has its own inherited advantage being a simple and programable method. The embedding of B-spline basis makes it more practical to approximate the solution curve. DQM with B-spline basis is a simple and efficient technique based on the matrix approach. The problem is discretized in the system of non-linear equations and then further solved by a programming tools. The stability is examined by the matrix-based approach. The presented method has been applied to three test problems. The obtained results showed that the proposed method is good for solving non-linear time-fractional Burger’s equation. The approximated solutions are graphically represented and the results showed that solutions are closed to the exact solution.

Cubic spline based differential quadrature method: A numerical approach for fractional Burger equation

The present attempt is to design a novel approach for the numerical solution of fractional telegraph equation. The novelty of the paper exist in solving the time fractional telegraph equation with differential quadrature method based on cubic B-spline (MHB-DQM) for the fractional parameter 1<γ<2. Telegraph equation is used in electric transmission line to find distance and time. The fractional derivative involved in the equation is discretized using Caputo derivative. On the other hand, the terms involving space derivatives are approximated by fusion of differential quadrature method with modified version of cubic B-spline. Here B-spline acts as a basis to compute the weighted coefficients of differential quadrature method. This phenomenon reduce the PDE to the system of equations, which then are solved using suitable numerical technique. A matrix based technique is used to ensure the stability of the proposed scheme. Feasibility and applicability of this algorithm is performed using test problems. Approximate solutions obtained by MHB-DQM is portrayed through graph and tables in an interactive way. These results show that solution converges for smaller values of time and at various values of fractional parameter. 

An efficient numerical scheme for fractional model of telegraph equation

This article studies a direct numerical approach for fractional advection–diffusion equations (ADEs). Using a set of cubic trigonometric B-splines as test functions, a differential quadrature (DQ) method is firstly proposed for the 1D and 2D time-fractional ADEs of order (0, 1]. The weighted coefficients are determined, and with them, the original equation is transformed into a group of general ordinary differential equations (ODEs), which are discretized by an effective difference scheme or Runge–Kutta method. The stability is investigated under a mild theoretical condition. Secondly, based on a set of cubic B-splines, we develop a new Crank–Nicolson type DQ method for the 2D space-fractional ADEs without advection. The DQ approximations to fractional derivatives are introduced, and the values of the fractional derivatives of B-splines are computed by deriving explicit formulas. The presented DQ methods are evaluated on five benchmark problems and the simulations of the unsteady propagation of solitons and Gaussian pulse. In comparison with the algorithms in the open literature, numerical results finally illustrate the validity and accuracy.

An efficient differential quadrature method for fractional advection–diffusion equation

This article studies a direct numerical approach for fractional advection-diffusion equations (ADEs). Using a set of cubic trigonometric B-splines as test functions, a differential quadrature (DQ) method is firstly proposed for the 1D and 2D time-fractional ADEs of order $(0,1]$. The weighted coefficients are determined, and with them, the original equation is transformed into a group of general ordinary differential equations (ODEs), which are discretized by an effective difference scheme or Runge-Kutta method. The stability is investigated under a mild theoretical condition. Secondly, based on a set of cubic B-splines, we develop a new Crank-Nicolson type DQ method for the 2D space-fractional ADEs without advection. The DQ approximations to fractional derivatives are introduced and the values of the fractional derivatives of B-splines are computed by deriving explicit formulas. The presented DQ methods are evaluated on five benchmark problems and the concrete simulations of the unsteady propagation of solitons and Gaussian pulse. In comparison with the existing algorithms in the open literature, numerical results finally illustrate the validity and accuracy.

X. G. Zhu

Papers

An efficient differential quadrature method for fractional advection–diffusion equation

An efficient differential quadrature method for fractional advection-diffusion equation