An unconditionally stable algorithm for multiterm time fractional advection–diffusion equation with variable coefficients and convergence analysis

Abstract

This paper focuses on the numerical solution of the variable coefficient multiterm time fractional advection-diffusion equation via exponential B-splines. We discretize the temporal part by using the Crank-Nicolson method and spatial part by the exponential B-splines. The unconditional stability is obtained by the Von-Neumann method. The convergence rates are also studied. Numerical simulations confirm the theoretically expected accuracy in both time and space directions. A comparative analysis with the other methods shows the superiority of the proposed algorithm.

Natural transform decomposition method for the numerical treatment of the time fractional Burgers–Huxley equation

“…Development of accurate fractional differential equation solution is a critical endeavor since it aids in understanding the qualitative characteristics of relevant phenomena and mechanisms in science and engineering. A significant amount of work has gone into developing numerical and analytical solutions for various fractional differential equations, including as linear and nonlinear fractional differential equations [16], non‐linear fractional Schrodinger equation [17], fractional telegraph equation [18], nonlinear fractional Klein–Gordon equation [19], time‐fractional gas dynamics equations [20], third‐order dispersive fractional partial differential equation [21], time‐fractional nonlinear coupled Boussinesq–Burger's equations [22], time fractional Black–Scholes European option pricing model [23], fractional vibration equation [24], time fractional diffusion model [25], fractional Drinfeld–Sokolov–Wilson equation [26], time fractional mobile–immobile advection–dispersion model [27], time‐fractional Fisher's equation [28], fractional order multi‐dimensional telegraph equation [29], multi‐term time‐fractional diffusion model [30], non‐linear space–time fractional Burgers–Huxley and reaction–diffusion equation [31], fractional immunogenetic tumors model [32], time‐fractional reaction–diffusion equation [33], fractional Kaup–Kupershmidt equation [34], time fractional multi‐dimensional heat equations [35], time‐fractional Brusselator reaction–diffusion system [36], Lotka–Volterra system of fractional differential equations [37], conformable Klein–Gordon equation [38], Kaup–Kupershmidt equation [39], time fractional Klein–Fock–Gordon equation [40], multi‐term time‐fractional advection–diffusion equation [41], fractional Lotka–Volterra population model [42] and others (see, e.g., [43–50]). The aim of this paper is to extend the scope of the natural transform decomposition method (NTDM) to the following time‐fractional Burgers–Huxley equation …”

An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options

“…McCartin also showed that the exponential splines accept a basis of B-splines. They are used in approximating the solutions of various classes of problems in differential equations [38][39][40][41].…”

A computational procedure and analysis for multi‐term time‐fractional Burgers‐type equation

“…The exponential B‐spline technique is based on piece‐wise non‐polynomial functions and was developed by McCartin, 23 and it is the generalization of the cubic spline. Several differential equations have been solved by utilizing this method 24–30 . Using this technique, one can avail of the solution even between nodes, which is an advantage over the finite difference scheme.…”