**2.2. ABM method for the VOFDDEs**

In this section, the ABM algorithm has been extended to study the VOFDDEs, where the derivative is defined in the Caputo VOF sense. Special attention is given to prove the error estimate of the proposed method. Numerical test examples are presented to demonstrate utility of the method. Chaotic behaviors are observed in variable-order one-dimensional delayed systems.

In the following, we apply the ABM predictor–corrector method to implement the numerical solution of VOFDDEs.

Let us consider the following VOF system:

$$D\_r^{a(t)}\mathbf{y}(t) = f\left(t, \mathbf{y}(t), \mathbf{y}(t-\tau)\right), \qquad \qquad t \in [o, T],\\0 < a(t) \le 1. \tag{15}$$

$$\mathbf{y}(t) = \mathbf{g}(t), \qquad \qquad t \in [-\tau, 0], \tag{16}$$

where *f* is in general a nonlinear function.

Also, consider a uniform grid {*tn* = *nh*: *n* = −*k*, −*k* + 1,…, −1,0,1,…, *N*} where *k* and *N* are integers such that *h* = *τ*/*k*. Let

$$\mathbf{y}\_h(t\_\nearrow) = \mathbf{g}\left(t\_\nearrow\right), \qquad \qquad j = -k, -k+1, \ldots, -1, 0,\tag{17}$$

and note that

**Figure 10.** The waveforms of the gating variables n and m in 3D.

122 Numerical Simulation - From Brain Imaging to Turbulent Flows

**(DDEs)**

**2.1. Introduction**

**2. Real-life models governed by nonlinear delay differential equations**

DDEs are differential equations in which the derivatives of some unknown functions at present time are dependent on the values of the functions at previous times. In real-world systems, delay is very often encountered in many practical systems, such as control systems [18], lasers, traffic models [19], metal cutting, epidemiology, neuroscience, population dynamics [20], and chemical kinetics [21]. Recent theoretical and computational advancements in DDEs reveal that DDEs are capable of generating rich and plausible dynamics with realistic parameter values. Naturally, occurrence of complex dynamics is often generated by well-formulated DDE models. This is simply due to the fact that a DDE operates on an infinite-dimensional space

For example, the Lotka–Volterra predator prey model [22] with crowding effect does not produce sustainable oscillatory solutions that describe population cycles. However, the Nicholson's blowflies model [23] can generate rich and complex dynamics. Delayed fractional differential equations (DFDEs) are correspondingly used to describe dynamical systems [23]. In recent years, DFDEs begin to arouse the attention of many researchers [7, 19, 20, 25–27]. Simulating these equations is an important technique in the research, and accordingly, finding effective numerical methods for the DFDEs is a necessary process. Several methods based on Caputo or Riemann–Liouville definitions [28] have been proposed and analyzed. For instance, based on the predictor–corrector scheme, Diethelm et al. introduced the ABM algorithm [29– 31] and mean while some error analysis was presented to improve the numerical accuracy. In

consisting of continuous functions that accommodate high-dimensional dynamics.

$$\mathbf{y}\_h(t\_j - \tau) = \mathbf{y}\_h(jh - kh) = \mathbf{y}\_h(t\_{j-k}), \qquad \qquad j = 0, 1, 2, \dots, N. \tag{18}$$

Applying the integral *Jtn*+1 *α*(*tn*+1) , which is defined by the Riemann–Liouville variable-order integral as

$$J\_{\boldsymbol{x}}^{a(\boldsymbol{x})}f\left(\mathbf{x}\right) = \frac{1}{\Gamma(a(\boldsymbol{x}))} \int\_{a}^{\boldsymbol{x}} (\mathbf{x} - t)^{a(\boldsymbol{x}) - 1} f\left(t\right) dt,\tag{19}$$

on both sides of (15) and using (16), we claim to:

$$\mathbf{y}\left(t\_{n+1}\right) = \mathbf{g}\left(0\right) + \frac{1}{\Gamma\left(a\left(t\_{n+1}\right)\right)} \left. \begin{matrix} t\_0 \\ \end{matrix} \left(t\_{n+1} - \boldsymbol{\zeta}\right)^{a\left(t\_{n+1}\right)-1} f\left(\boldsymbol{\zeta}, \mathbf{y}\left(\boldsymbol{\zeta}\right), \mathbf{y}\left(\boldsymbol{\zeta} - \boldsymbol{\tau}\right)\right) d\boldsymbol{\zeta} \end{matrix} \tag{20}$$

Further, the integral in Eq. (20) is evaluated using product trapezoidal quadrature formula [29– 31, 33]. Then, we have the following corrector formula:

$$\begin{aligned} \mathcal{Y}\_h(t\_{n+1}) &= \mathbf{g}\left(0\right) + \frac{h^{\alpha\left(t\_{n+1}\right)}}{\Gamma\left(\alpha\left(t\_{n+1}\right) + 2\right)} f\left(t\_{n+1}, \mathcal{Y}\_h\left(t\_{n+1}\right), \mathcal{Y}\_h\left(t\_{n+1} - \tau\right)\right) + \mathbf{0}, \\\frac{h^{\alpha\left(t\_{n+1}\right)}}{\Gamma\left(\alpha\left(t\_{n+1}\right) + 2\right)} &\sum\_{j=0}^{s} a\_{j, n+1} f\left(t\_j, \mathcal{Y}\_h\left(t\_j\right), \mathcal{Y}\_h\left(t\_j - \tau\right)\right), \end{aligned}$$

or

$$\begin{aligned} \mathbf{y}\_h(t\_{s+1}) &= \mathbf{g}\left(0\right) + \frac{h^{a\left(t\_{s+1}\right)}}{\Gamma\left(a\left(t\_{s+1}\right) + 2\right)} f\left(t\_{s+1}, \mathbf{y}\_h\left(t\_{s+1}\right), \mathbf{y}\_h\left(t\_{s+1-k}\right)\right) + \\ &\frac{h^{a\left(t\_{s+1}\right)}}{\Gamma\left(a\left(t\_{s+1}\right) + 2\right)} \sum\_{j=0}^{s} a\_{j,s+1} f\left(t\_j, \mathbf{y}\_h\left(t\_j\right), \mathbf{y}\_h\left(t\_{j-k}\right)\right), \end{aligned} \tag{21}$$

where

$$a\_{j,n+1} = \begin{cases} n^{a(t\_{n+1})+1} - \left(n - \alpha\left(t\_{n+1}\right)\right)(n+1)^{a(t\_{n+1})}, & j=0, \\ (n-j+2)^{a(t\_{n+1})+1} + (n-j)^{a(t\_{n+1})+1} - 2\left(n-j+1\right)^{a(t\_{n+1})+1}, & 1 \le j \le n, \\ 1, & j=n+1, \end{cases} \tag{22}$$

$$\mathbb{V}\_{h}\left(t\_{f-k}\right) \approx \mathbb{V}\_{n+1} = \begin{cases} \delta\_{\mathbb{V}\_{s-u+l}} + (1-\delta)\_{\mathbb{V}\_{s-u+l}}, & \text{if} \qquad m > 1, \\\delta\_{\mathbb{V}\_{s+l}^{p}} + (1-\delta)\_{\mathbb{V}\_{s}}, & \text{if} \qquad m = 1, \end{cases} \tag{23}$$

0 ≤ *δ* < 1 and the unknown term *yh*(*tn*+1) appears on both sides of (21).

Due to nonlinearity off, Eq. (21) cannot be solved explicitly for *yh*(*tn*+1), so we replace the term *yh*(*tn*+1) on the right-hand side by an approximation *yh p* (*tn*+1), which is called predictor. The product rectangle rule is used in (20) to evaluate the predictor term

$$\mathbb{E}\left(\mathbf{y}\_h^{\rho}\left(t\_{n+1}\right)\right) = \mathbf{g}\left(0\right) + \frac{1}{\Gamma\left(\alpha\left(t\_{n+1}\right)\right)} \sum\_{j=0}^{n} b\_{j,n+1} f\left(t\_j, \mathbf{y}\_h\left(t\_j\right), \mathbf{y}\_h\left(t\_j - \tau\right)\right),$$

or

( ) ( ) ( )

a

124 Numerical Simulation - From Brain Imaging to Turbulent Flows

on both sides of (15) and using (16), we claim to:

1 0 1 1

a

+ + +

31, 33]. Then, we have the following corrector formula:

( ) ( )

( )

+

*n*

*t*

( ) ( )

( )

+

*n*

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a

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1

+

a

a

or

where

, 1

+

*j n*

a

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1 0

+ =

*n j*

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ò

ò

( ) ( ) ( ( )) ( ) ( ) ( ( ) ( )) <sup>1</sup> <sup>1</sup> <sup>1</sup>

( )

+


*n*

*t h n n hn hn n*

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1

( ( ) ) ( ) ( )

(, , , Γ 2

( )

+

*n*

*t*

+

a

1

1 11 1 1

+ + + + -

*h n n hn hn k n*

+

<sup>0</sup> , , Γ 2

= + +

*jn j h j h j k j*

( ) ( ) ( ) ( )

+ + +

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*n m n m <sup>p</sup> <sup>n</sup> <sup>n</sup> y y*


 d

ì ü + - <sup>&</sup>gt; ï ï » = í ý + - <sup>=</sup> ï ï î þ

 d

*y y*

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*a nj nj nj j n*

1 1 1

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*t t n*

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0 ≤ *δ* < 1 and the unknown term *yh*(*tn*+1) appears on both sides of (21).

+

d

1

1 1

1

+

 a

(, , , Γ 2

*<sup>h</sup> a ft y t y t*

, 1 0

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+

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a

*yt g t fy y d*

*<sup>n</sup> <sup>n</sup> <sup>t</sup> <sup>t</sup>*

 z

Further, the integral in Eq. (20) is evaluated using product trapezoidal quadrature formula [29–

1 11 1 1

+ ++ + +

+

*jn j h j h j*

<sup>0</sup> , , Γ 2

<sup>+</sup> <sup>+</sup> -

( ) ( ) <sup>1</sup> <sup>1</sup> () , Γ( )

<sup>1</sup> <sup>0</sup> ,, . <sup>Γ</sup>

 z z

= +- - (20)

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= + - +

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t

1 , 0,

a

*if m*

(1 ) , 1, (1 ) , 1,

*j n*

a

 a


 zt

 z

t

(21)

(22)

(23)

$$\mathbf{y}\_h^p(t\_{n+1}) = \mathbf{g}\left(\mathbf{0}\right) + \frac{1}{\Gamma\left(a\left(t\_{n+1}\right)\right)} \sum\_{j=0}^n b\_{j,n+1} f(t\_j, \mathbf{y}\_h(t\_j), \mathbf{y}\_h(t\_{j-k}), \tag{24}$$

where *<sup>b</sup> <sup>j</sup>*,*n*+1 <sup>=</sup> *<sup>h</sup> <sup>α</sup>*(*<sup>t</sup> <sup>n</sup>*+1) *<sup>α</sup>*(*tn*+1) ((*n* − *j* + 1) *α*(*tn*+1) + (*n* − *j*) *α*(*tn*+1) .

#### **2.3. Error analysis of the algorithm**

In this subsection, we aim to introduce the following lemma, which will be used in the proof of main theorem.

Lemma 2.1. [31] (a) Let *z* ∈ *C*<sup>1</sup> [0, *T*], then

$$\left| \int\_{0}^{t\_{n+1}} \left( t\_{n+1} - t \right)^{a(t\_{n+1})-1} z \left( t \right) dt - \sum\_{j=0}^{n} b\_{j, n+1} z(t\_j) \right| \le \frac{1}{a\left( t\_{n+1} \right)} \left\| z' \right\|\_{\alpha} t\_{n+1}^{a(t\_{n+1})} h. \tag{25}$$

(b) Let *z* ∈ *C*<sup>2</sup> [0, *T*], then

$$\left| \int\_{0}^{t\_{s+1}} (t\_{s+1} - t)^{a(t\_{s+1})-1} z \left( t \right) dt - \sum\_{j=0}^{n+1} a\_{j, n+1} z(t\_j) \right| \le C\_{a(t\_{s+1})}^{r\_{s}} \left\| z \right\|\_{o} t\_{u+1}^{a(t\_{s+1})} h. \tag{26}$$

Proof. To prove statement (a), by construction of the product rectangle formula, we find that the quadrature error has the representation [31]

$$\int\_0^{t\_{s+1}} \left(t\_{s+1} - t\right)^{a(t\_{s+1})-1} \boldsymbol{z}\left(t\right) dt - \sum\_{j=0}^{n} \, \_0b\_{j,n+1} \boldsymbol{z}\left(t\_j\right) = \sum\_{j=0}^{n} \int\_{jh}^{(j+1)h} \left(t\_{s+1} - t\right)^{a(t\_{s+1})-1} \left(\boldsymbol{z}\left(t\right) - \boldsymbol{z}\left(t\_j\right)\right) dt.$$

Apply the mean value theorem of differential calculus to the second factor of the integrand on the right-hand side of the above equation

$$\begin{aligned} & \left| \int\_{0}^{t\_{n}} \left( t\_{n+1} - t \right)^{a(t\_{n+1})-1} z \left( t \right) dt - \sum\_{j=0}^{n} b\_{j,n+1} z \left( t\_{j} \right) \right| \\ \\ & \le \left\| z' \right\|\_{\alpha} \sum\_{j=0}^{n} \int\_{j\beta}^{(j+1)\alpha} \left( t\_{n+1} - t \right)^{a(t\_{n+1})-1} \left( t - jh \right) dt, \\\\ & = \left\| z' \right\|\_{\alpha} \frac{h^{1+a(t\_{n+1})}}{a(t\_{n+1})} \sum\_{j=0}^{n} \alpha \left( \frac{1}{1+\alpha \left( t\_{n+1} \right)} \right] \left( n+1 - j \right)^{1+a(t\_{n+1})} \\ \\ & \quad - \left( n - j \right)^{1+a(t\_{n+1})} \left] - \left( n - j \right)^{a(t\_{n+1})}, \\\\ & = \left\| z' \right\|\_{\alpha} \frac{h^{1+a(t\_{n+1})}}{\alpha \left( t\_{n+1} \right)} \left( \frac{n+1}{1+\alpha \left( t\_{n+1} \right)} \right) - \sum\_{j=0}^{n} j^{a(t\_{n+1})}, \\\\ & = \left\| z' \right\|\_{\alpha} \frac{h^{1+a(t\_{n+1})}}{a(t\_{n+1})} (\int\_{0}^{t\_{n+1}} j^{a(t\_{n+1})} dt - \sum\_{j=0}^{n} j^{a(t\_{n+1})}). \end{aligned}$$

Here, the term in parentheses is simply the remainder of the standard rectangle quadrature formula, applied to the function *t α*(*tn*+1) and taken over the interval [0, n + 1]. Since the integrand is monotonic, we may apply some standard results from quadrature theory to find that this term is bounded by the total variation of the integrand, thus

$$\left| \int\_0^{t\_{s+1}} (t\_{s+1} - t)^{\alpha(t\_{s+1}) - 1} z(t) dt - \sum\_{j=0}^n b\_{j, n+1} z \left( t\_j \right) \right| \le \frac{1}{\alpha \left( t\_{n+1} \right)} \left\| z' \right\|\_\alpha \left( n + 1 \right)^{\alpha(t\_{s+1}) h^{1 + \alpha(t\_{s+1})}} \qquad \square$$

Similarly, we can prove (b).

Now, let us consider that *f*(⋅) in (15) satisfies the following Lipschitz conditions with respect to its variables

$$\begin{aligned} \left| f\left(t, \mathbf{y}\_1, \mu\right) - f\left(t, \mathbf{y}\_2, \mu\right) \right| &\leq L\_1 \left| \mathbf{y}\_1 - \mathbf{y}\_2 \right|, \\\\ \left| f\left(t, \mathbf{y}, \mu\_1\right) - f\left(t, \mathbf{y}, \mu\_2\right) \right| &\leq L\_2 \left| \mu\_1 - \mu\_2 \right|, \end{aligned} \tag{27}$$

where *L*1 and *L*2 are positive constants.

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*<sup>n</sup> j,n j t t z t dt b z t*

( ) ( ) ( )

<sup>1</sup> <sup>1</sup>

<sup>+</sup> <sup>+</sup>

 a

( ) ( ) ( ) <sup>1</sup> 1 1 ( ) ] , *n n t t nj nj* <sup>+</sup>a

> ( ) ( ) ( ) ( ) <sup>1</sup> <sup>1</sup>

*<sup>t</sup> <sup>t</sup> <sup>n</sup> <sup>t</sup>*

a

1 1 1

 a

*<sup>h</sup> <sup>z</sup> j dt j*

¥ =

a

Here, the term in parentheses is simply the remainder of the standard rectangle quadrature

is monotonic, we may apply some standard results from quadrature theory to find that this

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) <sup>1</sup> <sup>1</sup> <sup>1</sup> <sup>1</sup> <sup>1</sup> <sup>1</sup>

*j n*

= +

Now, let us consider that *f*(⋅) in (15) satisfies the following Lipschitz conditions with respect

*f tyu f ty u L y y* ( ,, ,, <sup>1</sup> ) - £- ( 2 11 2 ) ,

a

*<sup>n</sup> tn <sup>n</sup> <sup>n</sup> <sup>t</sup> <sup>n</sup> <sup>t</sup> t h*

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*t t z t dt b z t z n <sup>t</sup>*

+ + ¥


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*n n <sup>h</sup> <sup>n</sup> z j t t*

*n n*

+ +

**+1**

1

+

*n*

a

1

and taken over the interval [0, n + 1]. Since the integrand

.1

a

*f t yu f t yu L u u* ( ,, ,, <sup>1</sup> ) - £- ( 2 21 2 ) , (27)

a

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+

*n*

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 a

0

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*j*

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( ) ( ) ( ) ( ) <sup>1</sup>

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<sup>1</sup> ( [1

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ò - - å **+1 <sup>1</sup>**

¥ <sup>=</sup> <sup>+</sup> £ -- ¢ å ò

0 1 1

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*j n n <sup>h</sup> <sup>z</sup> n j t t*

1

*n <sup>t</sup> <sup>n</sup> α t*

**0**

**+1**

*n*

126 Numerical Simulation - From Brain Imaging to Turbulent Flows

**+1**

( )

¥ =

( ) ( )

> ( ) ( )

1

+ <sup>=</sup> ¢ ò -å

*n*

*t* a

a

a

1

+

a

*α*(*tn*+1)

term is bounded by the total variation of the integrand, thus

1 , 1 0 0 1

a

*n jn j*

formula, applied to the function *t*

Similarly, we can prove (b).

to its variables

1 1

+ +

+

*n*

a

a

Theorem 1. Suppose the solution y(t) ∈ C2 [0, T], of the Eqs. (15) and (16), satisfies the following two conditions:

$$\begin{vmatrix} \int\_{0}^{t\_{s+1}} \left(t\_{s+1} - t\right)^{a(\iota\_{s+1})-1} D\_{\iota}^{a(\iota\_{s+1})} \mathcal{y}\left(t\right) dt - \\\\ \sum\_{j=0}^{s} b\_{j,s+1} D\_{\iota}^{a(\iota\_{s+1})} \mathcal{y}\left(t\_{j}\right) \end{vmatrix} \leq C t\_{s+1}^{\gamma\_{1}} h^{\delta\_{1}},\tag{28}$$

$$\begin{aligned} \left| \int\_{0}^{t\_{s+1}} \left( t\_{n+1} - t \right)^{\alpha(t\_{s+1})-1} D\_{l}^{\alpha(t\_{s+1})} \mathbf{y} \left( t \right) dt - \right| & \leq C t\_{n+1}^{\boldsymbol{\gamma}\_{2}} h^{\boldsymbol{\delta}\_{2}}, \\ \left| \sum\_{j=0}^{n} a\_{j, n+1} D\_{l}^{\alpha(t\_{s+1})} \mathbf{y} \left( t\_{j} \right) \right| & \end{aligned}$$

with some γ1,γ2 ≥ 0, and *δ*1, *δ*2 < 0, then for some suitable *T* > 0, we have

$$\max\_{0 \le j \le N} \left| \mathcal{y}(t\_j) - \mathcal{y}\_h(t\_j) \right| = O\left(h^q\right),$$

where *q* = min (*δ*1+ *α* (*t*), *δ*2), *N* = [*T*/*h*], and *C* is a positive constant.

Proof. We will use the mathematical induction to prove the result. Suppose that the conclusion is true for j = 0, 1,…, n, then we have to prove that the inequality also holds for j = n + 1. To do this, we first consider the error of the predictor *yn*+1 *<sup>p</sup>* . From (27), we have

$$\begin{aligned} \left| f\left(t\_j, \mathbf{y}(t\_j), \mathbf{y}(t\_j - \tau)\right) - f\left(t\_j, \mathbf{y}\_j, \boldsymbol{\nu}\_j\right) \right| &\leq \left| f\left(t\_j, \mathbf{y}(t\_j), \mathbf{y}(t\_j - \tau)\right) - f\left(t\_j, \mathbf{y}\_j, \mathbf{y}(t\_j - \tau)\right) \right| + \\ \left| f\left(t\_j, \mathbf{y}\_j, \mathbf{y}(t\_j - \tau)\right) - f\left(t\_j, \mathbf{y}\_j, \boldsymbol{\nu}\_j\right) \right| &\leq L\_1 h^q + L\_2 h^q = \left(L\_1 + L\_2\right) h^q. \end{aligned}$$

So

$$\left| \mathbf{y}(t\_{s+1}) - \mathbf{y}\_{s+1}^{\rho} \right| = \frac{1}{\Gamma(\alpha(t\_{s+1}))} \left| \int\_{0}^{t\_{s+1}} \left( t\_{s+1} - t \right)^{\alpha(t\_{s+1})-1} f\left( t, \mathbf{y}(t), \mathbf{y}(t-\tau) \right) dt - \sum\_{j=0}^{s} b\_{j,s+1} f\left( t\_{j}, \mathbf{y}\_{j}, \mathbf{0}\_{j} \right) \right| $$

$$\begin{split} &\leq \frac{1}{\Gamma(\alpha\left(t\_{n+1}\right))} \Big| \int\_{0}^{t\_{n+1}} \left(t\_{n+1}-t\right)^{\alpha\left(t\_{n+1}\right)-1} D\_{t}^{\alpha\left(t\_{n+1}\right)} \chi\left(t\right) dt - \sum\_{j=0}^{n} b\_{j,n+1} D\_{t}^{\alpha\left(t\_{n+1}\right)} \mathcal{Y}\left(t\_{j}\right) \Big| + \\ & \quad \frac{1}{\Gamma(\alpha\left(t\_{n+1}\right))} \sum\_{j=0}^{n} b\_{j,n+1} \Big| f\left(t\_{j}, \mathcal{Y}\left(t\_{j}\right), \mathcal{Y}\left(t\_{j}-\tau\right)\right) - f\left(t\_{j}, \mathcal{Y}\_{j}, \mathcal{Y}\left(t\_{j}-\tau\right)\right) \Big| + \\ & \quad \Big| \mathcal{F}\left(t\_{j}, \mathcal{Y}\_{j}, \mathcal{Y}\left(t\_{j}-\tau\right)\right) - f\left(t\_{j}, \mathcal{Y}\_{j}, \mathcal{U}\_{j}\right) \Big| \\ & \leq \frac{C t\_{n+1}^{\eta}}{\Gamma(\alpha\left(t\_{n+1}\right))} + \frac{1}{\Gamma(\alpha\left(t\_{n+1}\right))} \sum\_{j=0}^{n} b\_{j,n+1} \Big(L\_{1} + L\_{2}\big) Ch^{\alpha}, \end{split}$$

and from

$$\sum\_{j=0}^{n} \, \_{j=0}^{n} b\_{j, n+1} = \sum\_{j=0}^{n} \, \_{j=0}^{n} \left( t\_{n+1} - t \right)^{a(t\_{n+1}) - 1} dt = \frac{1}{a\left( t\_{n+1} \right)} t\_{n+1}^{a(t\_{n+1})} \le \frac{1}{a\left( t\_{n+1} \right)} T^{a(t\_{n+1})},$$

we have

$$\left|\boldsymbol{\chi}(t\_{n+1}-\boldsymbol{\chi}\_{n+1}^{\boldsymbol{\rho}})\right| \leq \frac{CT^{\boldsymbol{\gamma}}}{\Gamma\left(\alpha\left(t\_{n+1}\right)\right)}h^{\boldsymbol{\delta}} + \frac{CT^{\alpha\left(t\_{n+1}\right)}}{\Gamma\left(\alpha\left(t\_{n+1}\right)+1\right)}h^{\boldsymbol{\eta}}.$$

Since

$$\begin{aligned} \sum\_{j=0}^{n} a\_{j, n+1} &\leq \sum\_{j=0}^{n} \frac{h^{a(\iota\_{s+1})}}{\alpha \left(t\_{s+1}\right) (\alpha \left(t\_{s+1}\right) + 1)} \\\\ \left[ \left(k - j + 2\right)^{a(\iota\_{s+1}) + 1} - \left(k - j + 1\right)^{a(\iota\_{s+1}) + 1} - (k - j + 1)^{a(\iota\_{s+1}) + 1} + (k - j)^{a(\iota\_{s+1}) + 1} \right], \\\\ &= \int\_{t\_{s+1}}^{t\_{s+1}} \left(t\_{s+2} - t\right)^{a(\iota\_{s+1})} dt - \int\_{t}^{t\_{s+1}} \left(t\_{s+1} - t\right)^{a(\iota\_{s+1})} dt, \end{aligned}$$

$$\begin{aligned} &= \int\_0^\cdot \left( t\_{n+2} - t \right)^{\alpha(t\_{s+1})} dt - \int\_0^\cdot \left( t\_{n+1} - t \right)^{\alpha(t\_{s+1})} dt \\ &= \frac{1}{\alpha\left( t\_{n+1} \right)} \int\_0^{t\_{s+1}} \left[ \left( t\_{n+1} - t \right)^{\alpha(t\_{s+1})} \right] (t)^\cdot dt, \\ &= \frac{1}{\alpha\left( t\_{n+1} \right)} t\_{n+1}^{\alpha(t\_{s+1})} \le T^{\alpha(t\_{s+1})}, \end{aligned}$$

we have

Numerical Simulations of Some Real-Life Problems Governed by ODEs http://dx.doi.org/10.5772/63958 129

( ) ( ) ( ) ( ) ( ) ( ) <sup>1</sup> <sup>1</sup> 1 <sup>1</sup> <sup>0</sup> 1 1 <sup>1</sup> , 1 1, 1 1 1 1 <sup>0</sup> , ( ), ( ) <sup>1</sup> ( ) , Γ( ) , , , , *<sup>n</sup> <sup>n</sup> <sup>t</sup> <sup>t</sup> <sup>n</sup> <sup>p</sup> n n <sup>n</sup> <sup>p</sup> <sup>n</sup> jn j j j n n n n n <sup>j</sup> t t f t y t y t dt yt y t a ft y a ft y* a t a u u <sup>+</sup> <sup>+</sup> - + + + <sup>+</sup> <sup>=</sup> <sup>+</sup> ++ + + + - - - - = ò å ( ( )) ( ) ( ) ( ( ) ( )) ( ( )) ( ( )) ( ) ( ) ( ) ( ( )) 1 1 2 2 1 1 <sup>1</sup> <sup>0</sup> <sup>1</sup> , 1 <sup>0</sup> , 1 0 1, 1 1 1 1 111 1 , 1 0 1 1, 1 , , <sup>1</sup> [ <sup>Γ</sup> , , , , , , , ( ), ( ) , , ] <sup>1</sup> [ <sup>Γ</sup> Γ *<sup>n</sup> <sup>n</sup> <sup>t</sup> <sup>t</sup> n <sup>n</sup> <sup>n</sup> jn j j j <sup>j</sup> n jn j j j jjj j p nn n n n nnn n q n j n j n q n n t t f t y t y t dt t a f t y yt a f t y yt f t y a f t yt yt f t y Ct h Ch a t CT a Lh* a g d g t a t t u t u a <sup>+</sup> <sup>+</sup> - + <sup>+</sup> <sup>=</sup> <sup>+</sup> + = ++ + + + +++ + + = + + + - - £ + - - -- + - - £ ++ + ò å å å ( ( )) ( ) ( ( ) ) 1 1 1 1 ] Γ 1 *nt q n n CT h h t t* a d a a + + + æ ö ç ÷ <sup>+</sup> + è ø ( ( )) ( )( ( ) ) ( ( ) ) ( ) ( ) 2 1 1 1 1 1 1 1 <sup>1</sup> [ <sup>Γ</sup> 1 Γ 2 *n n n h n n n t t CT t Lh t t CLT t CLT h h Ch* g g a a a a a a + + + + + + £ + + + + +
