Cubic B-spline collocation method and its application for anomalous fractional diffusion equations in transport dynamic systems

Abstract

In this paper, we approximate the solution of the initial and boundary value problems of anomalous second-and fourthorder sub-diffusion equations of fractional order. The fractional derivative is used in the Caputo sense. To solve these equations, we will use a numerical method based on B-spline basis functions and the collocation method. It will be shown that the proposed scheme is unconditionally stable and convergent. Three numerical examples are adopted to demonstrate the performance of the proposed scheme.

The third kind Chebyshev wavelets collocation method for solving the time-fractional convection diffusion equations with variable coefficients

“…Luo et al developed a quadratic spline collocation method for the constant coefficient case without convection [16]. Sayevand et al considered the same type equation by a cubic B-spline collocation method [23]. Pirkhedri and Javadi gave a collocation approach for the variable coefficient case via expanding its solution as the elements of Haar and Sinc functions [20].…”

An efficient differential quadrature method for fractional advection–diffusion equation

“…When α = 1, because the analytic solutions still remain unknown and the Newton's procedure relies heavily on its initial values, we instead employ the trust-regiondogleg algorithm built into Matlab to improve the convergence of iteration. At first, taking τ = 2.0 × 10 −3 , M = 100, β = 2, and Ω = [−10, 10], the mean square errors at t = 0.1 with the initial condition (38) for various α are reported in Table 5, where the solutions computed by using the coefficients (13) on a very fine time-space lattice, i.e., τ = 2.5 × 10 −4 , M = 400, are adopted as reference solutions (α = 1). As seen from 20,20], we display the evolution of the amplitude of the mobile soliton created by (23) for α = 0.98 and 1.0 in Fig.…”

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

“…B-spline collocation methods were proposed for the solutions of time fractional diffusion problems by Esen et al [19,20]. Sayevand et al [21] solved anomalous time fractional diffusion problems in transport dynamic systems using a B-spline collocation scheme. In [21], the fractional derivative in Caputo sense was utilized to represent the time derivative.…”

“…Sayevand et al [21] solved anomalous time fractional diffusion problems in transport dynamic systems using a B-spline collocation scheme. In [21], the fractional derivative in Caputo sense was utilized to represent the time derivative. A cubic trigonometric B-spline collocation scheme for the time fractional diffusion problem was presented by Yaseen et al [22].…”