Numerical Solution of Fractional Order Anomalous Subdiffusion Problems Using Radial Kernels and Transform

Abstract

By coupling of radial kernels and localized Laplace transform, a numerical scheme for the approximation of time fractional anomalous subdiffusion problems is presented. The fractional order operators are well suited to handle by Laplace transform and radial kernels are also built for high dimensions. The numerical computations of inverse Laplace transform are carried out by contour integration technique. The computation can be done in parallel and no time sensitivity is involved in approximating the time fractional operator as contrary to finite differences. The proposed numerical scheme is stable and accurate.

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

“…For better accuracy in real-life models, the applications of fractional models are growing and indicate significant requirements for better fractional mathematical models. Radial basis functions and Laplace transformation are used for the approximation of fractional anomalous sub-diffusion equation [12]. This process's advantage is handling many matrix data efficiently and accurately.…”

A Numerical Method for Solving Fractional Differential Equations

“…erefore, the numerical methods are essential for approximation solution of many FDEs. Many approximations have sprung up recently, such as the numerical scheme by coupling of radial kernels and localized Laplace transform [6], the radial basis function (RBF)-based numerical scheme which uses the Coimbra variable time fractional derivative of order 0 < α(t, x) < 1 [7], the time-space numerical technique based on time-space radial kernels [8], the local meshless method based on Laplace transform [9], nite di erence methods [10], nite element methods [11][12][13], spectral methods [14], shooting method [15], approximation formula [16], pseudo-spectral method [17], variational iteration method [18], Adomian decomposition method (ADM) [19,20], trapezoidal methods [21], reproducing kernel method [22][23][24][25][26], and so on. e development, however, for e cient numerical methods to solve linear and nonlinear FDEs is still an important issue.…”

Solving Some Partial Differential Equations by Using Double Laplace Transform in the Sense of Nonconformable Fractional Calculus

“…One can see that the last two defnitions satisfy the classical properties mentioned above. Many researchers solved nonlinear fractional partial diferential equations in the sense of conformable derivative and Caputo derivative [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. Tere is an important study [32], where the authors treat the price adjustment equation in many senses of fractional derivatives, such as truncated M-derivative including the Mittag-Lefer function, beta-derivative, and conformable derivative defned in the form of limit for α -diferentiable functions.…”