Now, we consider the applications of Theorems 4.1, 4.3 and 4.4.

First, initial value problem (45) is considered. The discretization of problem (45) is carried out in two steps. In the first step, the grid space

${\mathbb{R}}_{h}^{n}$ (

$0<h\le {h}_{0}$) is defined as the set of all points of the Euclidean space

${\mathbb{R}}^{n}$ whose coordinates are given by

${x}_{k}={s}_{k}h,\phantom{\rule{1em}{0ex}}{s}_{k}=0,\pm 1,\pm 2,\dots ,k=1,\dots ,n.$

(98)

The difference operator

${A}_{h}^{t,x}={B}_{h}^{t,x}+\sigma {I}_{h}$ is assigned to the differential operator

${A}^{x}={B}^{x}+\sigma I$, defined by (52). The operator

${B}_{h}^{t,x}={h}^{-2m}\sum _{2m\le |s|\le S}{b}_{s}^{t,x}{\mathrm{\Delta}}_{1-}^{{s}_{1}}{\mathrm{\Delta}}_{1+}^{{s}_{2}}\cdots {\mathrm{\Delta}}_{n-}^{{s}_{2n-1}}{\mathrm{\Delta}}_{n+}^{{s}_{2n}}$

(99)

acts on functions defined on the entire space

${\mathbb{R}}_{h}^{n}$. Here

$s\in {\mathbb{R}}^{2n}$ is a vector with nonnegative integer coordinates,

${\mathrm{\Delta}}_{k\pm}{f}^{h}(x)=\pm ({f}^{h}(x\pm {e}_{k}h)-{f}^{h}(x)),$

(100)

where ${e}_{k}$ is the unit vector of the axis ${x}_{k}$.

An infinitely differentiable function

$\phi (x)$ of the continuous argument

$x\in {\mathbb{R}}^{n}$ that is continuous and bounded together with all its derivatives is said to be smooth. We say that the difference operator

${A}_{h}^{t,x}$ is a

*λ* th order (

$\lambda >0$) approximation of the differential operator

${A}^{t,x}$ if the inequality

$\underset{x\in {\mathbb{R}}_{h}^{n}}{sup}|{A}_{h}^{t,x}\phi (x)-{A}^{t,x}\phi (x)|\le M(\phi ){h}^{\lambda}$

(101)

holds for any smooth function $\phi (x)$. The coefficients ${b}_{s}^{t,x}$ are chosen in such a way that the operator ${A}_{h}^{t,x}$ approximates in a specified way the operator ${A}^{t,x}$. It will be assumed that the operator ${A}_{h}^{t,x}$ approximates the differential operator ${A}^{t,x}$ with any prescribed order [57, 58].

The function ${A}^{t,x}(\xi h,h)$ is obtained by replacing the operator ${\mathrm{\Delta}}_{k\pm}$ in the right-hand side of equality (99) with the expression $\pm (exp\{\pm i{\xi}_{k}h\}-1)$, respectively, and is called the symbol of the difference operator ${B}_{h}^{t,x}$.

It will be assumed that for

$|{\xi}_{k}h|\le \pi $ and fixed

*x*, the symbol

${A}^{t,x}(\xi h,h)$ of the operator

${B}_{h}^{t,x}={A}_{h}^{t,x}-\sigma {I}_{h}$ satisfies the inequalities

${(-1)}^{m}{A}^{t,x}(\xi h,h)\ge M{|\xi |}^{2m},\phantom{\rule{2em}{0ex}}|arg{A}^{t,x}(\xi h,h)|\le \varphi <{\varphi}_{0}\le \frac{\pi}{2}.$

(102)

Suppose that the coefficient

${b}_{s}^{x}$ of the operator

${B}_{h}^{t,x}={A}_{h}^{t,x}-\sigma {I}_{h}$ is bounded and satisfies the inequalities

$|{b}_{s}^{t,x+{e}_{k}h}-{b}_{s}^{t,x}|\le M{h}^{\u03f5},\phantom{\rule{1em}{0ex}}x\in {\mathbb{R}}_{h}^{n},\u03f5\in (0,1].$

(103)

With the help of

${A}_{h}^{t,x}$, we arrive at the nonlocal boundary value problem

$\{\begin{array}{c}\frac{d{v}^{h}(t,x)}{dt}+{D}_{t}^{\frac{1}{2}}{v}^{h}(t,x)+{A}_{h}^{t,x}{v}^{h}(t,x)={f}^{h}(t,x),\phantom{\rule{1em}{0ex}}0<t<1,x\in {\mathbb{R}}_{h}^{n},\hfill \\ {v}^{h}(0,x)=0,\phantom{\rule{1em}{0ex}}x\in {\mathbb{R}}_{h}^{n}\hfill \end{array}$

(104)

for an infinite system of ordinary differential equations.

In the second step, problem (104) is replaced by the difference scheme

$\{\begin{array}{c}\frac{{u}_{k}^{h}(x)-{u}_{k-1}^{h}(x)}{\tau}+\frac{1}{\sqrt{\pi}}{\sum}_{m=1}^{k}\frac{\mathrm{\Gamma}(k-m+\frac{1}{2})}{(k-m)!}\frac{{u}_{m}^{h}-{u}_{m-1}^{h}}{{\tau}^{\frac{1}{2}}}+{A}_{h}^{k,x}{u}_{k}^{h}={f}_{k}^{h}(x),\hfill \\ {f}_{k}^{h}(x)={f}^{h}({t}_{k},x),\phantom{\rule{2em}{0ex}}{t}_{k}=k\tau ,\phantom{\rule{1em}{0ex}}1\le k\le N-1,\phantom{\rule{2em}{0ex}}N\tau =1,\phantom{\rule{1em}{0ex}}x\in {\mathbb{R}}_{h}^{n},\hfill \\ {u}_{0}^{h}(x)=0,\phantom{\rule{1em}{0ex}}x\in {\mathbb{R}}_{h}^{n}.\hfill \end{array}$

(105)

Based on the number of corollaries of the abstract theorems given in the above, to formulate the result, one needs to introduce the spaces

${C}_{h}=C({\mathbb{R}}_{h}^{n})$ and

${C}_{h}^{\beta}={C}^{\beta}({\mathbb{R}}_{h}^{n})$ of all bounded grid functions

${u}^{h}(x)$ defined on

${\mathbb{R}}_{h}^{n}$, equipped with the norms

**Theorem 5.1** *Suppose that assumptions* (102)

*and* (103)

*for the operator* ${A}_{h}^{k,x}$ *hold*.

*Then*,

*the solutions of difference scheme* (105)

*satisfy the following stability estimates*:

$\begin{array}{r}\underset{1\le k\le N}{max}{\parallel {D}_{\tau}^{\frac{1}{2}}{u}_{k}^{h}\parallel}_{{C}_{h}^{\mu}}\le {M}_{1}(\mu )\underset{1\le k\le N}{max}{\parallel {f}_{k}^{h}\parallel}_{{C}_{h}^{\mu}},\phantom{\rule{1em}{0ex}}0\le \mu \le 1,\\ {\parallel {\left\{{\tau}^{-1}({u}_{k}^{h}-{u}_{k-1}^{h})\right\}}_{1}^{N}\parallel}_{{C}_{\tau}({C}_{h}^{\mu +2m\alpha})}+{\parallel {\{{A}_{k}{u}_{k}\}}_{1}^{N}\parallel}_{{C}_{\tau}({C}_{h}^{\mu +2m\alpha})}\\ \phantom{\rule{1em}{0ex}}\le M(\alpha ,\mu )\underset{1\le k\le N}{max}{\parallel {f}_{k}^{h}\parallel}_{{C}_{h}^{\mu +2m\alpha}},\phantom{\rule{1em}{0ex}}0<2m\alpha +\mu <1.\end{array}$

(108)

The proof of Theorem 5.1 is based on the abstract Theorems 4.1, 4.3, 4.4 and the strong positivity of the operator ${A}_{h}^{x}$ defined by (114) in ${C}_{h}^{\mu}$ and on the following two theorems on the coercivity inequality for the solution of the elliptic difference equation in ${C}_{h}^{\beta}$ and on the structure of the fractional space ${E}_{\alpha}^{\mathrm{\prime}}({C}_{h},{A}_{h}^{x})$.

**Theorem 5.2** *Suppose that assumptions* (102)

*and* (103)

*for the operator* ${A}_{h}^{k,x}$ *hold*.

*Then for the solutions of the elliptic difference equation* ${A}_{h}^{k,x}{u}^{h}(x)={\omega}^{h}(x),\phantom{\rule{1em}{0ex}}x\in {\mathbb{R}}_{h}^{n},$

(109)

*the estimates* [

54]

$\sum _{2m\le |s|\le S}{h}^{-2m}{\parallel {\mathrm{\Delta}}_{1-}^{{s}_{1}}{\mathrm{\Delta}}_{1+}^{{s}_{2}}\cdots {\mathrm{\Delta}}_{n-}^{{s}_{2n-1}}{\mathrm{\Delta}}_{n+}^{{s}_{2n}}{u}^{h}\parallel}_{{C}_{h}^{\beta}}\le M(\sigma ,\beta ){\parallel {\omega}^{h}\parallel}_{{C}_{h}^{\beta}}$

(110)

*are valid*.

**Theorem 5.3** *Suppose that assumptions* (102) *and* (103) *for the operator* ${A}_{h}^{k,x}$ *hold*. *Then for any* $0<\alpha <\frac{1}{2m}$, *the norms in the spaces* ${E}_{\alpha}^{\mathrm{\prime}}({C}_{h},{A}_{h}^{x})$ *and* ${C}_{h}^{2m\alpha}$ *are equivalent uniformly in* *h* [51].

Second, we consider mixed boundary value problem (56). The discretization of problem (56) is carried out in two steps. In the first step, let us define the grid space

${[0,1]}_{h}=\{x:{x}_{r}=rh,0\le r\le K,Kh=1\}.$

(111)

We introduce the Banach space

${C}_{h}^{\beta}={C}^{\beta}({[0,1]}_{h})$ (

$0<\beta <1$) of the grid functions

${\phi}^{h}(x)={\{{\phi}_{r}\}}_{1}^{K-1}$ defined on

${[0,1]}_{h}$, equipped with the norm

${\parallel {\phi}^{h}\parallel}_{{C}_{h}^{\beta}}={\parallel {\phi}^{h}\parallel}_{{C}_{h}}+\underset{1\le k<k+\tau \le K-1}{sup}\frac{|{\phi}_{k+r}-{\phi}_{k}|}{{\tau}^{\beta}},$

(112)

where

${C}_{h}=C({[0,1]}_{h})$ is the space of the grid functions

${\phi}^{h}(x)={\{{\phi}_{r}\}}_{1}^{K-1}$ defined on

${[0,1]}_{h}$, equipped with the norm

${\parallel {\phi}^{h}\parallel}_{{C}_{h}}=\underset{1\le k\le K-1}{max}|{\phi}_{k}|.$

(113)

To the differential operator

*A* generated by problem (56), we assign the difference operator

${A}_{h}^{x}$ by the formula

${A}_{h}^{t,x}{\phi}^{h}(x)={\{-{(a(t,x){\phi}_{\stackrel{-}{x}})}_{x,r}+\delta {\phi}_{r}\}}_{1}^{K-1},$

(114)

acting in the space of grid functions

${\phi}^{h}(x)={\{{\phi}_{r}\}}_{0}^{K}$ satisfying the conditions

${\phi}_{0}={\phi}_{K}$,

${\phi}_{1}-{\phi}_{0}={\phi}_{K}-{\phi}_{K-1}$. With the help of

${A}_{h}^{x}$, we arrive at the initial boundary value problem

$\{\begin{array}{c}\frac{d{v}^{h}(t,x)}{dt}+{D}_{t}^{\frac{1}{2}}{v}^{h}(t,x)+{A}_{h}^{t,x}{v}^{h}(t,x)={f}^{h}(t,x),\phantom{\rule{1em}{0ex}}0<t<1,x\in {[0,1]}_{h},\hfill \\ {v}^{h}(0,x)=0,\phantom{\rule{1em}{0ex}}x\in {[0,1]}_{h}\hfill \end{array}$

(115)

for an infinite system of ordinary fractional differential equations. In the second step, we replace problem (115) by difference scheme (3)

**Theorem 5.4** *Let* *τ* *and* *h* *be sufficiently small numbers*.

*Then*,

*the solutions of difference scheme* (116)

*satisfy the following stability estimates*:

$\begin{array}{r}\underset{1\le k\le N}{max}{\parallel {D}_{\tau}^{\frac{1}{2}}{u}_{k}^{h}\parallel}_{{C}_{h}^{\mu}}\le {M}_{1}(\mu )\underset{1\le k\le N}{max}{\parallel {f}_{k}^{h}\parallel}_{{C}_{h}^{\mu}},\phantom{\rule{1em}{0ex}}0\le \mu \le 1,\\ {\parallel {\left\{{\tau}^{-1}({u}_{k}^{h}-{u}_{k-1}^{h})\right\}}_{1}^{N}\parallel}_{{C}_{\tau}({C}_{h}^{\mu +2\alpha})}+{\parallel {\{{A}_{k}{u}_{k}\}}_{1}^{N}\parallel}_{{C}_{\tau}({C}_{h}^{\mu +2\alpha})}\\ \phantom{\rule{1em}{0ex}}\le M(\alpha ,\mu )\underset{1\le k\le N}{max}{\parallel {f}_{k}^{h}\parallel}_{{C}_{h}^{\mu +2\alpha}},\phantom{\rule{1em}{0ex}}0<2\alpha +\mu <1.\end{array}$

(117)

The proof of Theorem 5.4 is based on the abstract Theorems 4.1, 4.3, 4.4 and the strong positivity of the operator ${A}_{h}^{t,x}$ defined by (114) in ${C}_{h}^{\mu}$ and on the following theorem on the structure of the fractional space ${E}_{\alpha}^{\mathrm{\prime}}({C}_{h},{A}_{h}^{t,x})$.

**Theorem 5.5** *For any* $0<\alpha <\frac{1}{2}$, *the norms in the spaces* ${E}_{\alpha}^{\mathrm{\prime}}({C}_{h},{A}_{h}^{t,x})$ *and* ${C}_{h}^{2\alpha}$ *are equivalent uniformly in* *h* *and* $t\in [0,1]$ [60].