Modeling and Computing of Fractional Convection Equation

Abstract

In this paper, we derive the fractional convection (or advection) equations (FCEs) (or FAEs) to model anomalous convection processes. Through using a continuous time random walk (CTRW) with power-law jump length distributions, we formulate the FCEs depicted by Riesz derivatives with order in (0, 1). The numerical methods for fractional convection operators characterized by Riesz derivatives with order lying in (0, 1) are constructed too. Then the numerical approximations to FCEs are studied in detail. By adopting the implicit Crank–Nicolson method and the explicit Lax–Wendroff method in time, and the second-order numerical method to the Riesz derivative in space, we, respectively, obtain the unconditionally stable numerical scheme and the conditionally stable numerical one for the FCE with second-order convergence both in time and in space. The accuracy and efficiency of the derived methods are verified by numerical tests. The transport performance characterized by the derived fractional convection equation is also displayed through numerical simulations.

Appendices

Appendix A

If $v^{(i)}(x)$ is continuous on $\mathbb {R}$ and $v^{(i)}(x)\in L^1(\mathbb {R})$ for $i=0,1,\ldots ,p+2$, where $p=6$ for the sixth-order method and $p=8$ for the eighth-order method, then the sixth- and eighth-order methods [9] formulated through the fractional centered difference scheme (13) for the Riesz derivative $\frac{\partial ^\gamma v(x)}{\partial |x|^\gamma }$, $\gamma \in (0,1)$ are given as

$$\begin{aligned} \frac{\partial ^\gamma v(x)}{\partial |x|^\gamma }=\, & {} \frac{\Lambda _s ^\gamma v(x)}{h^\gamma } + O(h^6),\\ \frac{\partial ^\gamma v(x)}{\partial |x|^\gamma }=\, & {} \frac{\Lambda _e ^\gamma v(x)}{h^\gamma } + O(h^8). \end{aligned}$$

Here $\Lambda _s ^\gamma$ and $\Lambda _e ^\gamma$ are given by

$$\begin{aligned} \Lambda _s ^\gamma v(x)=\, & {} a_1\Delta _h^\gamma v(x+2h)+a_2\Delta _h^\gamma v(x+h) +a_3\Delta _h^\gamma v(x) \\&+a_2\Delta _h^\gamma v(x-h)+a_1\Delta _h^\gamma v(x-2h),\\ \Lambda _e ^\gamma v(x)=\, & {} b_1\Delta _h^\gamma v(x+3h)+ b_2\Delta _h^\gamma v(x+2h) +b_3\Delta _h^\gamma v(x+h)\\&+b_4\Delta _h^\gamma v(x) + b_3\Delta _h^\gamma v(x-h) +b_2v(x-2h)+b_1\Delta _h^\gamma v(x-3h), \end{aligned}$$

in which $\Delta _h^\gamma$ is defined by (14) and

$$\begin{aligned} a_1=\, & {} -\left( \frac{\gamma }{1\,152}+\frac{11}{2\,880}\right) \gamma , \; a_2= \left( \frac{\gamma }{288}+\frac{41}{720}\right) \gamma , \; a_3=-\left( \frac{\gamma ^2}{192}+\frac{17\gamma }{160}+1\right) ;\\ b_1=\, & {} \left( \frac{\gamma ^2}{82\,944}+\frac{11\gamma }{69\,120}+\frac{191}{362\,880}\right) \gamma , \; b_2=-\left( \frac{\gamma ^2}{13\,824}+\frac{7\gamma }{3\,840}+\frac{211}{30\,240}\right) \gamma ,\\ b_3=\, & {} \left( \frac{5\gamma ^2}{27\,648}+\frac{3\gamma }{512}+\frac{7\,843}{120\,960}\right) \gamma , \; b_4=-\left( \frac{5\gamma ^3}{20\,736}+\frac{29\gamma ^2}{3\,456}+\frac{5\,297\gamma }{45\,360}+1\right) . \end{aligned}$$

For v(x) defined on (a, b), through the zero extension of v(x), $\frac{\partial ^\gamma v(x)}{\partial |x|^\gamma }$ at $x_i$ can be approximated as follows:

$$\begin{aligned} \frac{\partial ^\gamma v(x_i)}{\partial |x|^\gamma }=\, & {} \frac{1}{h^\gamma } \left\{ a_1\sum ^{[\frac{b-a}{h}]-2}_{k=-2}g_{i-k}^{(\gamma )} v(x_{k+2})+\, a_2\sum ^{[\frac{b-a}{h}]-1}_{k=-1}g_{i-k}^{(\gamma )} v(x_{k+1}) +a_3\sum ^{[{\frac{b-a}{h}}]}_{k=0}g_{i-k}^{(\gamma )} v(x_{k})\right. \\&\left. +a_2 \sum ^{[\frac{b-a}{h}]+1}_{k=1}g_{i-k}^{(\gamma )}v(x_{k-1}) +a_1\sum ^{[\frac{b-a}{h}]+2}_{k=2}g_{i-k}^{(\gamma )}v(x_{k-2}) \right\} +O(h^6),\\ \frac{\partial ^\gamma v(x)}{\partial |x|^\gamma }=\, & {} \frac{1}{h^\gamma } \left\{ b_1\sum ^{[\frac{b-a}{h}]-3}_{k=-3}g_{i-k}^{(\gamma )} v(x_{k+3})+ b_2\sum ^{[\frac{b-a}{h}]-2}_{k=-2}g_{i-k}^{(\gamma )} v(x_{k+2}) +b_3\sum ^{[\frac{b-a}{h}]-1}_{k=-1}g_{i-k}^{(\gamma )} v(x_{k+1})\right. \\&\left. +\,b_4 \sum ^{[{\frac{b-a}{h}}]}_{k=0}g_{i-k}^{(\gamma )}v(x_{k}) +b_3\sum ^{[\frac{b-a}{h}]+1}_{k=1}g_{i-k}^{(\gamma )} v(x_{k-1}) +b_2\sum ^{[\frac{b-a}{h}]+2}_{k=2}g_{i-k}^{(\gamma )} v(x_{k-2})\right. \\&\left. +\,b_1\sum ^{[\frac{b-a}{h}]+3}_{k=3}g_{i-k}^{(\gamma )} v(x_{k-3})\right\} + O(h^8). \end{aligned}$$

Appendix B

In this appendix, we prove Lemma 3.6.

Proof

The conclusion presented in (i) can be readily obtained by direct calculations. Next we show (ii).

Denote

$$\begin{aligned} c_{1,k}=-3(\gamma -k+1),\; c_{2,k}=\frac{3}{2}(2\gamma -k+2),\; c_{3,k}= -\frac{1}{3}(3\gamma -k+3), \end{aligned}$$

and

$$\begin{aligned} B_k=\frac{q_{3,k-1}^{(\gamma )}}{q_{3,k}^{(\gamma )}},\;m(k)=\frac{k+2}{k-3\gamma -1}. \end{aligned}$$

By (27), one has

$$\begin{aligned} q_{3,k}^{(\gamma )}=\, & {} \frac{6}{11k}\left( c_{1,k}\,q_{3,k-1}^{(\gamma )}+c_{2,k} \,q_{3,k-2}^{(\gamma )} +c_{3,k}\,q_{3,k-3}^{(\gamma )}\right) ,\\ \frac{1}{B_k}=\, & {} \frac{6}{11k}\left( c_{1,k}+c_{2,k}\, B_{k-1}+c_{3,k} \,B_{k-1}B_{k-2}\right) . \end{aligned}$$

We aim to prove that for $k\ge 5$, $1\le B_k\le m(k)$ by mathematical induction.

For $k=5,6,\ldots ,17$, we display the figures in Fig. 3. One can find from this figure that $B_k$ satisfies $1\le B_k\le m(k)$, and $q_{3,k}^{(\gamma )}<0$ and $q_{3,k}^{(\gamma )}\le q_{3,k+1}^{(\gamma )}$ when $k=5,6,\ldots ,17.$

Fig. 3

Comparison of $B_k$ and m(k) for $k=5,6,\ldots ,17.$

Suppose that $1\le B_{k}\le m(k)$ for $k=l-2,l-1,\; l\ge 18$. When $k=l$, $B_l$ is computed as

$$\begin{aligned} \frac{1}{B_l}=\, & {} \frac{6}{11l}\left( c_{1,l}+c_{2,l} B_{l-1}+c_{3,l} B_{l-1}B_{l-2}\right) \nonumber \\=\, & {} \frac{6}{11l}\left[ c_{1,l}+ B_{l-1}(c_{2,l}+c_{3,l} B_{l-2})\right] \nonumber \\\le & {} \frac{6}{11l}\left[ c_{1,l}+B_{l-1}(c_{2,l}+c_{3,l} m(l-2))\right] \nonumber \\\le & {} \frac{6}{11l}\left[ c_{1,l}+ c_{2,l}+c_{3,l} m(l-2)\right] \nonumber \\\le & {} 1, \end{aligned}$$

(55)

since $m(l-2)=\frac{l}{l-3\gamma -3}<2$ for $l\ge 18$, and $c_{2,l}+2c_{3,l}= \gamma -\frac{5}{6}l+1 <0$.

On the other hand,

$$\begin{aligned} \frac{1}{B_l}=\, & {} \frac{6}{11l}\left[ c_{1,l}+ B_{l-1}(c_{2,l}+c_{3,l} B_{l-2})\right] \\\ge & {} \frac{6}{11l}\left[ c_{1,l}+B_{l-1}(c_{2,l}+c_{3,l})\right] \\\ge & {} \frac{6}{11l}\left[ c_{1,l}+ m(l-1)(c_{2,l}+c_{3,l})\right] ,\\ \end{aligned}$$

due to $c_{2,l}+c_{3,l}=2\gamma -\frac{7}{6}l+2<0$ for $l\ge 18$. Multiplying m(l) on both sides of the above inequality gives

$$\begin{aligned} \frac{m(l)}{B_l}\ge & {} \frac{6}{11l}\left[ c_{1,l}+ m(l-1)(c_{2,l}+c_{3,l})\right] m(l)\nonumber \\=\, & {} \frac{6}{11l} I_1, \end{aligned}$$

(56)

where $I_1=\left[ c_{1,l}+ m(l-1)(c_{2,l}+c_{3,l})\right] m(l)$.

Substituting the expressions of $c_{1,l},c_{2,l},c_{3,l}$ and $m(l),m(l-1)$ in $I_1$ yields

$$\begin{aligned} I_1=\, & {} \frac{(l+2)(3l-3\gamma -3)}{l-3\gamma -1} + \frac{(l+1)(l+2)(2\gamma -\frac{7}{6}l+2)}{(l-3\gamma -1)(l-3\gamma -2)}\\=\, & {} \frac{ \left( \frac{11}{6}l^3-10\gamma l^2-\frac{9}{2}l^2+9\gamma ^2 l-3\gamma l -\frac{25}{3}l +18\gamma ^2+34\gamma +16\right) }{(l-3\gamma -1)(l-3\gamma -2)}. \end{aligned}$$

Therefore,

$$\begin{aligned} I_1-\frac{11l}{6} = \frac{\left[ (1+\gamma )l^2 -\left( \frac{15}{2}\gamma ^2+\frac{39}{2}\gamma +12\right) l +18\gamma ^2+34\gamma +16\right] }{(l-3\gamma -1)(l-3\gamma -2)}. \end{aligned}$$

(57)

Define f(l) as

$$\begin{aligned} f(l)=(1+\gamma )l^2 -\left( \frac{15}{2}\gamma ^2+\frac{39}{2}\gamma +12\right) l +18\gamma ^2+34\gamma +16. \end{aligned}$$

By simple calculation, we have

$$\begin{aligned} f'(l)=\, & {} 2(1+\gamma )l-\left( \frac{15}{2}\gamma ^2+\frac{39}{2}\gamma +12\right) \\\ge & {} 2(1+\gamma )\times 18-\left( \frac{15}{2}\gamma ^2+\frac{39}{2}\gamma +12\right) \\=\, & {} 2(-15\gamma ^2+33\gamma +48)>0 \end{aligned}$$

for ${\text{all}}\, \gamma \in (0,1)$.

Hence,

$$\begin{aligned} f(l)\ge f(18)=-117\gamma ^2+7\gamma +124 >0, \end{aligned}$$

for ${\text{all}}\, \gamma \in (0,1)$.

In view of (56) and (57), and $m(l)>0$, it leads to

$$\begin{aligned} \frac{m(l)}{B_l}> 1, \end{aligned}$$

i.e.,

$$\begin{aligned} \frac{1}{B_l}> \frac{1}{m(l)}. \end{aligned}$$

Combining with (55), we obtain

$$\begin{aligned} 1\le B_k\le m(k), \;\mathrm {for}\; k\ge 5,\;\forall \gamma \in (0,1), \end{aligned}$$

which yields $q_{3,k-1}^{(\gamma )}\le q_{3,k}^{(\gamma )},\;\mathrm {for}\; k\ge 5$.

It also implies that the sign of $q_{3,k}^{(\gamma )}$ is same as that of $q_{3,k-1}^{(\gamma )}$. Since $q_{3,4}^{(\gamma )}<0$, $q_{3,k}^{(\gamma )}<0$ for $k\ge 4$.

All this completes the proof.

Appendix C

Here we introduce $(2-\gamma )$ and $(3-\gamma )$ orders approximations to Riesz derivatives $\frac{\partial ^\gamma v(x)}{\partial |x|^\gamma }$ defined on (a, b) with $\gamma \in (0,1)$.

For $0<\gamma <1$, the left and right Riemann–Liouville derivatives can be rewritten as

$$\begin{aligned} _{RL}\mathrm {D}^\gamma _{a,x}v(x)=\, & {} \frac{(x-a)^{-\gamma }v(a)}{\Gamma (1-\gamma )} +\frac{1}{\Gamma (1-\gamma )}\int ^x_a(x-s)^{-\gamma }v'(s)\mathrm {d}s \\=\, & {} \frac{(x-a)^{-\gamma }v(a)}{\Gamma (1-\gamma )} +\,_C\mathrm {D}^\gamma _{a,x}v(x),\\ _{RL}\mathrm {D}^\gamma _{x,b}v(x)=\, & {} \frac{(b-x)^{-\gamma }v(b)}{\Gamma (1-\gamma )} -\frac{1}{\Gamma (1-\gamma )}\int ^b_x(s-x)^{-\gamma }v'(s)\mathrm {d}s\\=\, & {} \frac{(b-x)^{-\gamma }v(b)}{\Gamma (1-\gamma )} +\,_{C}\mathrm {D}^\gamma _{x,b}v(x), \end{aligned}$$

where $_C\mathrm {D}^\gamma _{a,x}v(x)$ and $_{C}\mathrm {D}^\gamma _{x,b}v(x)$ represent the left and right Caputo derivatives, respectively.

(i) Choosing $v'(s)\approx \frac{v(x_{k+1})-v(x_k)}{h}$, $s\in [x_k,x_{k+1}]$, then the L1 method for the left Riemann–Liouville derivative is presented as [15, 22]

$$\begin{aligned} _{RL}\mathrm {D}^\gamma _{a,x_i}v(x_i)=\, & {} \frac{(x_i-a)^{-\gamma }v(a)}{\Gamma (1-\gamma )} +\frac{1}{\Gamma (1-\gamma )}\sum _{k=0}^{i-1}\int ^{x_{k+1}}_{x_k}(x_i-s)^{-\gamma }v'(s)\mathrm {d}s\\=\, & {} \frac{(x_i-a)^{-\gamma }v(a)}{\Gamma (1-\gamma )}+\sum _{k=0}^{i-1}b^{(\gamma )}_{i-k-1}[v(x_{k+1})-v(x_k)] +O(h^{2-\gamma }), \end{aligned}$$

where $x_0=a$, $b_k^{(\gamma )}=\frac{h^{-\gamma }}{\Gamma (2-\gamma )}\left[ (k+1)^{1-\gamma }-k^{1-\gamma }\right]$.

We can naturally extend this method to the right Riemann–Liouville derivative as follows:

$$\begin{aligned} _{RL}\mathrm {D}^\gamma _{x_i,b}v(x_i)=\, & {} \frac{(b-x_i)^{-\gamma }v(b)}{\Gamma (1-\gamma )} -\frac{1}{\Gamma (1-\gamma )}\sum _{k=i}^{M-1}\int ^{x_{k+1}}_{x_k}(s-x_i)^{-\gamma }v'(s)\mathrm {d}s\\=\, & {} \frac{(b-x_i)^{-\gamma }v(b)}{\Gamma (1-\gamma )}-\sum _{k=i}^{M-1}b^{(\gamma )}_{k-i}[v(x_{k+1})-v(x_k)] +O(h^{2-\gamma }), \end{aligned}$$

where $x_M=b$, $b_k^{(\gamma )}=\frac{h^{-\gamma }}{\Gamma (2-\gamma )}\left[ (k+1)^{1-\gamma }-k^{1-\gamma }\right]$.

Thus, the Riesz derivative can be approximated by

$$\begin{aligned} \begin{aligned} \frac{\partial ^\gamma v(x_i)}{\partial |x|^\gamma }=&- \frac{1}{2\cos (\frac{\pi }{2}\gamma )} \left( \frac{(x_i-a)^{-\gamma }v(a)}{\Gamma (1-\gamma )} +\frac{(b-x_i)^{-\gamma }v(b)}{\Gamma (1-\gamma )}\right. \\&\left. +\sum _{k=0}^{i-1}b^{(\gamma )}_{i-k-1}[v(x_{k+1})-v(x_k)] -\sum _{k=i}^{M-1}b^{(\gamma )}_{k-i}[v(x_{k+1})-v(x_k)]\right) \\&+O(h^{2-\gamma }). \end{aligned} \end{aligned}$$

(ii) In [1], the author derived the L2-$1_\sigma$ method for the left Caputo derivatives with $(3-\gamma )$th order for $\gamma \in (0,1)$, we can also apply this method to the left and right Riemann–Liouville derivatives.

Considering this integral at node $x_{i+\sigma }$, $i\in \{0,1,\dots ,M-1\}$, choosing $v'(s)\approx \frac{v(x_{k+1})-v(x_k)}{h} +\frac{v(x_{k+1})-2v(x_{k})+v(x_{k-1})}{h^2}(s-x_{k+1/2})$ for $s\in [x_{k-1},x_{k}]$ and $v'(s)\approx \frac{v(x_{i+1})-v(x_i)}{h}$ for $s\in [x_{i},x_{i+\sigma }]$, we can obtain from [1] that

$$\begin{aligned} _C\mathrm {D}^\gamma _{a,x_{i+\sigma }} v(x_{i+\sigma })=\, & {} \frac{1}{\Gamma (1-\gamma )}\left( \sum _{k=1}^{i}\int ^{x_{k}}_{x_{k-1}}(x_{i+\sigma }-s)^{-\gamma }v'(s)\mathrm {d}s + \int ^{x_{i+\sigma }}_{x_i}(x_{i+\sigma }-s)^{-\gamma }v'(s)\mathrm {d}s\right) \nonumber \\=\, & {} \frac{h^{-\gamma }}{\Gamma (2-\gamma )} \sum _{k=0}^i d_{i-k}^{(\gamma ,\sigma )}[v(x_{k+1})-v(x_k)]+O(h^{3-\gamma })\nonumber \\&+\frac{v''(x_{i+1/2})(2\sigma -2+\gamma )}{2(1-\gamma )(2-\gamma )}\sigma ^{1-\gamma }h^{2-\gamma }, \end{aligned}$$

(58)

where

$$\begin{aligned} \left\{ \begin{aligned} c_{0}^{(\gamma ,\sigma )}&=\sigma ^{1-\gamma },\\ c_{k}^{(\gamma ,\sigma )}&=(k+\sigma )^{1-\gamma }-(k-1+\sigma )^{1-\gamma }, \; k\ge 1,\\ {\tilde{c}}_{k}^{(\gamma ,\sigma )}&= \frac{1}{2-\gamma }\left[ (k+\sigma )^{2-\gamma }-(k-1+\sigma )^{2-\gamma }\right] \\&\quad -\frac{1}{2}\left[ (k+\sigma )^{1-\gamma }+(k-1+\sigma )^{1-\gamma }\right] ,\; k\ge 1, \end{aligned}\right. \end{aligned}$$

and for $i=0$, $d_{0}^{(\gamma ,\sigma )}=c_{0}^{(\gamma ,\sigma )}$; for $i\ge 1$,

$$\begin{aligned} d_{k}^{(\gamma ,\sigma )}=\left\{ \begin{aligned}&c_{0}^{(\gamma ,\sigma )} + {\tilde{c}}_{1}^{(\gamma ,\sigma )},&k=0,\\&c_{k}^{(\gamma ,\sigma )} + {\tilde{c}}_{k+1}^{(\gamma ,\sigma )} -{\tilde{c}}_{k}^{(\gamma ,\sigma )},&1\le k\le i-1,\\&c_{i}^{(\gamma ,\sigma )} - {\tilde{c}}_{i}^{(\gamma ,\sigma )},&k=i.\\ \end{aligned}\right. \end{aligned}$$

(59)

If we choose $\sigma =1-\frac{\gamma }{2}$, then the scheme (58) is just the L2-$1_\sigma$ method [1] for the left Caputo derivatives.

Similarly, one has

$$\begin{aligned} \begin{aligned} _C\mathrm {D}^\gamma _{x_{i+\sigma '},b} v(x_{i+\sigma '})&=- \frac{1}{\Gamma (1-\gamma )}\left( \int _{x_{i+\sigma '}}^{x_{i+1}}(s-x_{i+\sigma '})^{-\gamma }v'(s)\mathrm {d}s\right. \\&\quad \left. +\sum _{k=i+1}^{M-1}\int ^{x_{k+1}}_{x_k}(s-x_{i+\sigma '})^{-\gamma }v'(s)\mathrm {d}s \right) \\&=-\frac{h^{-\gamma }}{\Gamma (2-\gamma )} \sum _{k=i}^{M-1} {\tilde{d}}_{k-i}^{(\gamma ,\sigma ')}[v(x_{k+1})-v(x_k)] +O(h^{3-\gamma })\\&\quad +\frac{v''(x_{i+1/2})(2\sigma '-\gamma )}{2(1-\gamma )(2-\gamma )}\sigma '^{1-\gamma }h^{2-\gamma }, \end{aligned} \end{aligned}$$

(60)

where

$$\begin{aligned} \left\{ \begin{aligned} c_{0}^{(\gamma ,\sigma ')}&=(1-\sigma ') ^{1-\gamma },\\ c_{k}^{(\gamma ,\sigma ')}&=(k-\sigma ')^{1-\gamma }-(k-1-\sigma ')^{1-\gamma }, \; k\ge 1,\\ {\tilde{c}}_{k}^{(\gamma ,\sigma ')}&= \frac{1}{2-\gamma }\left[ (k-\sigma ')^{2-\gamma }-(k-1-\sigma ')^{2-\gamma }\right] \\&\quad -\frac{1}{2}\left[ (k-\sigma ')^{1-\gamma }+(k-1-\sigma ')^{1-\gamma }\right] ,\; k\ge 1, \end{aligned}\right. \end{aligned}$$

and for $i=M-1$, ${\tilde{d}}_{0}^{(\gamma ,\sigma )}=c_{0}^{(\gamma ,\sigma ')}$; for $i\ge 1$,

$$\begin{aligned} {\tilde{d}}_{k}^{(\gamma ,\sigma ')}=\left\{ \begin{aligned}&c_{0}^{(\gamma ,\sigma ')} + {\tilde{c}}_{2}^{(\gamma ,\sigma ')},&k=0,\\&c_{k+1}^{(\gamma ,\sigma ')} - {\tilde{c}}_{k+1}^{(\gamma ,\sigma ')} +{\tilde{c}}_{k+2}^{(\gamma ,\sigma ')},&1\le k\le M-2-i,\\&c_{k+1}^{(\gamma ,\sigma ')} - {\tilde{c}}_{k+1}^{(\gamma ,\sigma ')},&k=M-1-i.\\ \end{aligned}\right. \end{aligned}$$

(61)

If we choose $\sigma '=\frac{\gamma }{2}$, then the scheme (60) is of $(3-\gamma )$th order for the right Caputo derivatives.

Combine (58) with (60) and let $2\sigma -2+\gamma +2\sigma '-\gamma =0$ which follows that $\sigma +\sigma '=1$. Furthermore, assume that $\sigma =\sigma '$, i.e., $\sigma =\sigma '=\frac{1}{2}$, then $x_{i+\sigma }=x_{i+1/2}=x_{i+\sigma '}$. Thus, one can get the following $(3-\gamma )$th order scheme for Riesz derivatives:

$$\begin{aligned} \frac{\partial ^\gamma v(x_{i+1/2})}{\partial |x|^\gamma }=\, & {} - \frac{1}{2\cos (\frac{\pi }{2}\gamma )} \left[ \frac{(x_{i+1/2}-a)^{-\gamma }v(a)}{\Gamma (1-\gamma )} +\frac{(b-x_{i+1/2})^{-\gamma }v(b)}{\Gamma (1-\gamma )}\right. \nonumber \\&+\frac{h^{-\gamma }}{\Gamma (2-\gamma )}\left( \sum _{k=0}^i d_{i-k}^{(\gamma ,1/2)}[v(x_{k+1})-v(x_k)]\right. \nonumber \\&\left. \left. -\sum _{k=i}^{M-1} {\tilde{d}}_{k-i}^{(\gamma ,1/2)}[v(x_{k+1})-v(x_k)]\right) \right] +O(h^{3-\gamma }), \end{aligned}$$

(62)

in which $d_{k}^{(\gamma ,1/2)}$ and ${\tilde{d}}_{k}^{(\gamma ,1/2)}$ are defined by (59) and (61), respectively, where $\sigma =\sigma '=\frac{1}{2}.$