A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

Abstract

A high-accuracy numerical approach for a nonhomogeneous time-fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time-fractional derivative is described in the sense of Riemann-Liouville and discretized by the backward Euler scheme. A fourth-order optimal cubic B-spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx 4 + Δt 2 −) convergence, where ∈ (0,1). Moreover, the impact of fractional-order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B-spline collocation method. The CPU time for present numerical method and the method based on cubic B-spline collocation method are provided.

An efficient numerical scheme and its analysis for the multiterm time‐fractional convection‐diffusion‐reaction equation

“…Nayied et al [29] considered Fisher‐type nonlinear reaction‐diffusion equation and constructed a numerical scheme by utilizing collocation method based on Fibonacci wavelet. In previous works [30–32], the authors used L1 scheme for discretization of temporal fractional derivatives appearing in the governing differential equations. It should be noted that the weak singularity at $$$ t=0 $$$ was not considered in above stated papers.…”

A hybrid B-spline collocation technique for the Caputo time fractional nonlinear Burgers’ equation

“…The time fractional partial differential equations have grown more attention outstanding to several real-life applications in electrical network systems, signal processing, optics, mathematical biology, financial evaluation and prediction, material science, electromagnetic control theory, multidimensional fluid flow, acoustics, pre-predator modeling in biological systems, and many more [2,3,[5][6][7][8]. The delayed time fractional predatorprey model with feedback control has been studied by Hopf bifurcation [10,11].…”

Design and analysis of efficient computational techniques for solving a temporal‐fractional partial differential equation with the weakly singular solution

“…The fractional differential equations (FDEs) have gained much attention in recent years and are widely used to describe many important phenomena and dynamic processes in applied sciences, engineering and other fields, for example, see previous works [1][2][3][4][5][6][7][8][9][10] and references therein. In this study, we consider the following TFCD equation:…”

“…The fractional differential equations (FDEs) have gained much attention in recent years and are widely used to describe many important phenomena and dynamic processes in applied sciences, engineering and other fields, for example, see previous works [1–10] and references therein. In this study, we consider the following TFCD equation: $$$$ {D}_t^{\gamma }Z\left(x,t\right)-\hat{A}(x)\frac{\partial^2Z\left(x,t\right)}{\partial {x}^2}+\hat{B}(x)\frac{\partial Z\left(x,t\right)}{\partial x}+\hat{C}(x)Z(x)=F\left(x,t\right),\gamma \in \left(0,1\right),\left(x,t\right)\in \left(c,d\right)\times \left(0,T\right], $$$$ with initial condition (IC) $$$$ Z\left(x,0\right)=\theta (x) $$$$ and boundary conditions (BCs) $$$$ Z\left(c,t\right)=l(t),Z\left(d,t\right)=r(t).$$…”