Discrete Hahn polynomials for numerical solution of two-dimensional variable-order fractional Rayleigh–Stokes problem

Abstract

The main aim of this paper is to find the numerical solutions of 2D Rayleigh–Stokes problem with the variable-order fractional derivatives in the Riemann–Liouville sense. The presented method is based on collocation procedure in combination with the new operational matrix of the variable-order fractional derivatives, in the Caputo sense, for the discrete Hahn polynomials. The main advantage of the proposed method is obtaining a global approximation for spatial and temporal discretizations, and it reduced the problem to an algebraic system, which is easier to solve. Also, the profit of approximating a continuous function by Hahn polynomials is that for computing the coefficients of the expansion, we only have to compute a summation and the calculation of coefficients is exact. The error bound for the approximate solution is estimated. Finally, we evaluate results of the presented method with other numerical methods.

Appendix

Definition 4

The Pochhammer symbol, $(a)_k$, is defined by the following relations:

$$\begin{aligned} (a)_0&:=1,\nonumber \\ (a)_k&:=a(a+1)(a+2)\ldots (a+k-1), \ \ k\in \mathbb {N}, \ a\in \mathbb {C},\ Re(a)>0. \end{aligned}$$

(40)

Definition 5

(Abramowitz and Stegun 1972, p. 824) Stirling numbers of the first kind is defined as

$$\begin{aligned} S_k^{(i)}=\sum _{r=0}^{k-i}(-1)^{r}{k-1+r \atopwithdelims ()k-i+r}{2k-i \atopwithdelims ()k-i-r}s_ {k-i+r}^{(r)},\quad i,k,r\in \mathbb {N}, \end{aligned}$$

(41)

where $s_k^{(i)}$ are the Stirling numbers of the second kind, namely

$$\begin{aligned} s_k^{(i)}=\frac{1}{i!} \sum _{r=0}^{i}(-1)^{i-r}{r \atopwithdelims ()i}r^{k}. \end{aligned}$$

(42)

Definition 6

The Kronecker product of the $m\times n$ matrix $\mathbf {A}$ and the $p\times q$ matrix $\mathbf {B}$ is $np\times mq$ matrix $\mathbf {A}\otimes \mathbf {B}$, which is defined by

$$\begin{aligned} \mathbf {A}_{m\times n}= & {} \left[ \begin{array}{cccc} a_{11} &{} a_{12} &{} {\dots }&{} a_{1n}\\ a_{21} &{} a_{22} &{} {\dots }&{} a_{2n}\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ a_{m1} &{} a_{m2} &{} {\dots }&{} a_{mn}\\ \end{array} \right] , \quad \mathbf {B}_{p\times q}= \left[ \begin{array}{cccc} b_{11} &{} b_{12} &{} {\dots }&{} b_{1q}\\ b_{21} &{} b_{22} &{} {\dots }&{} b_{2q}\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ b_{p1} &{} b_{p2} &{} {\dots }&{} b_{pq}\\ \end{array} \right] ,\\ \mathbf {A}\otimes \mathbf {B}= & {} \left[ \begin{array}{cccc} a_{11}\mathbf {B} &{} a_{12}\mathbf {B} &{} {\dots }&{} a_{1n}\mathbf {B}\\ a_{21}\mathbf {B} &{} a_{22}\mathbf {B} &{} {\dots }&{} a_{2n}\mathbf {B}\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ a_{m1}\mathbf {B} &{} a_{m2}\mathbf {B} &{} {\dots }&{} a_{mn}\mathbf {B}\\ \end{array} \right] . \end{aligned}$$