*Example 1:*

3 5 2 3 15

1 1 *m i i i*

1 1 (2 1) ( ) ( ), 2 ( 1) ( 1) *Lm m xL x mL x m m m m m* + - = +- ³

For Eq. (1), the trial solution can be written as defined in the study of Lagaris and Fotiadis [28],

(, ) *tr y t A tN* y

> yy

*<sup>t</sup>* æ ö =+ + - + + + ç ÷

3 5

 y

6 5 2 24 2 2 12 12

3 32 2 4 23 2 1 2 12 1 2

( ( ) ( ( )))

yy

<sup>1</sup> ( ) , , , , 0,1

4 3 5 33

The mean square error (MSE) of the Eq. (1) will be calculated from the following:

*i tr j i j tr j i*

*MSE y t Ft y t t <sup>n</sup>*

 a

G G æ ö æ ö <sup>+</sup> - + ç ÷ - ç ÷ G - è ø G - è ø

æ ö G G <sup>æ</sup> <sup>ö</sup> <sup>+</sup> <sup>+</sup> - + ç ÷ <sup>ç</sup> <sup>÷</sup> è ø G- G- <sup>è</sup> <sup>ø</sup>

Trial solution can be written in expanded form for ChSANN at *m*=2 as follows:

1 2 (, ) 3 <sup>15</sup> *tr yt At t t*

<sup>2</sup> 2 26 (, ) <sup>2</sup> 3 15 3 7

*tr y t t t*

y

2 7 <sup>3</sup> 2 15 7 <sup>3</sup> <sup>3</sup>


4 4 5 4


*t t*

*t t*

y

G G æ ö æ ö Ñ = ç ÷ -+ + ç ÷ G - è ø è ø G -

 yy

Fractional derivative in Caputo sense of Eq. (10) is as follows:

( ) ( ) ( )

y

a

y y yy

1

=

*j*

*n*

a

a

a

 a

( ) ( )

a  y  h

(6)

<sup>=</sup> å <sup>=</sup> *<sup>L</sup>* - (7)

+ + (8)

(*x*)= *x*, and value of *m* is adjustable to reach the utmost accuracy.

( ) ( ) 3 5

 yy

è ø (10)

 a

> yy

12 12 12

( ) ( ) ( )


 a

 a

1 1 1 5 4 1 1 2

> y yy

> > a

y y

2

=Ñ - Îé ù å ë û (12)

= + (9)

3

y a

y y

(11)

h

 =- + h

*N*

h

whereas hereas *L <sup>i</sup>*−1 are the Legendre polynomials with the recursive formula:

1 1

where here *L* <sup>0</sup>

(*x*)=1 and *L* <sup>1</sup>

102 Numerical Simulation - From Brain Imaging to Turbulent Flows

but *N* will be used according to the method.

y

a

y

a

y

Consider the following Riccati differential equation with initial condition as:

$$\frac{d^\alpha \mathbf{y}\left(t\right)}{dt^\alpha} + \mathbf{y}^2\left(t\right) - \mathbf{l} = \mathbf{0}, \quad \mathbf{y}\left(0\right) = \mathbf{0}, \ \mathbf{0} < \alpha \le \mathbf{b}$$

The exact solution for *α* =1 is given by the following:

$$\mathcal{Y}\left(t\right) = \frac{e^{2x} - 1}{e^{2x} + 1}$$

The above fractional Riccati differential equation is solved by implementing the ChSANN and LSANN algorithms for various values of *α* and the results are compared with several methods to exhibit the strength of proposed neural network algorithms. The ChSANN and LSANN methods are employed on the above equation for *α* =1 with 20 equidistant training points and 6 NAC and attained the mean square error up to 5.501631×10<sup>−</sup><sup>9</sup> and 1.21928×10<sup>−</sup><sup>9</sup> for ChSANN and LSANN, respectively. **Figure 2** shows the combined results of ChSANN for different

**Figure 2.** ChSANN results at different values of *α* =1.

values of *α*. **Table 1** depicts the comparison of results obtained from both the methods with exact solution and the absolute error values for both the methods. Absolute error (AE) values for ChSANN and LSANN can be viewed in **Table 1** but can be better visualized in **Figure 3**. Implementation of ChSANN and LSANN for *α* =0.75 with 10 equidistant training points and 6 NAC on the above equation gave the mean square error up to 1.66032×10<sup>−</sup><sup>7</sup> for ChSANN and 4.8089×10−<sup>7</sup> for LSANN. **Table 2** shows the numerical comparison for *α* =0.75 with 10 equidis‐ tant training points of ChSANN and LSANN with the methods in [13, 14], while **Tables 3** and **4** demonstrate the numerical comparison of the proposed methods with the methods in [7, 13, 14] for *α* =0.5and *α* =0.9 correspondingly. Numerical values of ChSANN for *α* =1 at *t* =1 are presented in **Table 5**.


**Table 1.** Numerical comparisons of ChSANN and LSANN values with exact values for fractional Riccati differential equation.

Numerical Simulation Using Artificial Neural Network on Fractional Differential Equations http://dx.doi.org/10.5772/64151 105

**Figure 3.** Absolute error of ChSANN and LSANN at *α* =1 for test example 1.


**Table 2.** Numerical comparison for *α* =0.75.

values of *α*. **Table 1** depicts the comparison of results obtained from both the methods with exact solution and the absolute error values for both the methods. Absolute error (AE) values for ChSANN and LSANN can be viewed in **Table 1** but can be better visualized in **Figure 3**. Implementation of ChSANN and LSANN for *α* =0.75 with 10 equidistant training points and

tant training points of ChSANN and LSANN with the methods in [13, 14], while **Tables 3** and **4** demonstrate the numerical comparison of the proposed methods with the methods in [7, 13, 14] for *α* =0.5and *α* =0.9 correspondingly. Numerical values of ChSANN for *α* =1 at *t* =1 are

*x* **Exact ChSANN LSANN AE of ChSANN AE of LSANN** 0.05 0.049884 0.0499572 0.0499441 1.20167 × 10−6 1.49267 × 10−5 0.10 0.099668 0.0996676 0.0996552 4.32269 × 10−6 1.27675 × 10−5 0.15 0.148885 0.148884 0.148876 1.13944 × 10−6 9.07723 × 10−6 0.20 0.197375 0.197372 0.197367 2.91448 × 10−6 7.92946 × 10−6 0.25 0.244919 0.244915 0.244909 4.13612 × 10−6 9.23796 × 10−6 0.30 0.291313 0.291309 0.291301 3.63699 × 10−6 1.12070 × 10−5 0.35 0.336376 0.336374 0.336363 1.41863 × 10−6 1.22430 × 10−5 0.40 0.379949 0.379949 0.379937 1.42762 × 10−6 1.17881 × 10−5 0.45 0.421899 0.421902 0.421889 3.33093 × 10−6 1.03088 × 10−5 0.50 0.462117 0.462117 0.462108 3.05535 × 10−6 8.76900 × 10−6 0.55 0.500520 0.500521 0.500512 3.81254 × 10−6 7.95700 × 10−6 0.60 0.537055 0.537046 0.537042 3.62984 × 10−6 8.00729 × 10−6 0.65 0.571670 0.571663 0.571662 6.95133 × 10−6 8.33268 × 10−6 0.70 0.604368 0.604360 0.604360 7.47479 × 10−6 8.04113 × 10−6 0.75 0.635149 0.635145 0.635142 4.22458 × 10−6 6.62579 × 10−6 0.80 0.664037 0.664038 0.664032 1.57001 × 10−6 4.48924 × 10−6 0.85 0.691069 0.691076 0.691067 6.23989 × 10−6 2.58549 × 10−6 0.90 0.716298 0.716303 0.716298 5.13615 × 10−6 2.66632 × 10−7 0.95 0.739783 0.739780 0.739793 3.29187 × 10−6 9.67625 × 10−6 1.00 0.761594 0.761584 0.761644 9.94216 × 10−6 4.97369 × 10−5

**Table 1.** Numerical comparisons of ChSANN and LSANN values with exact values for fractional Riccati differential

for LSANN. **Table 2** shows the numerical comparison for *α* =0.75 with 10 equidis‐

for ChSANN and

6 NAC on the above equation gave the mean square error up to 1.66032×10<sup>−</sup><sup>7</sup>

4.8089×10−<sup>7</sup>

equation.

presented in **Table 5**.

104 Numerical Simulation - From Brain Imaging to Turbulent Flows


**Table 3.** Numerical comparison for *α* =0.5.


**Table 4.** Numerical comparison for *α* =0.9.


**Table 5.** Numerical values of ChSANN at *t* =1 and for *α* =1.

#### *Example 2:*

Consider the nonlinear Riccati differential equation along with the following initial condition:

$$\frac{d^a \mathcal{Y}(t)}{dt^a} + \mathcal{Y}^2(t) - 2\mathcal{Y}(t) - 1 = 0, \quad \mathcal{Y}(0) = 0 \quad , \quad 0 < a \le 1$$

The exact solution when *α* =1 is given by [7]:

$$\mathcal{N}(t) = 1 + \sqrt{2\pi} \tanh\left(\sqrt{2\pi}\, t + \frac{1}{2}\log\left(\frac{\sqrt{2\pi}\, -1}{\sqrt{2\pi}\, +1}\right)\right)$$

ChSANN and LSANN algorithms are executed on the above test experiment with 6NAC, *α* =1 and 20 equidistant points that gave the mean square error up to 1.6127×10<sup>−</sup><sup>7</sup> and 4.68641×10<sup>−</sup><sup>6</sup> for ChSANN and LSANN, respectively. **Table 6** shows the absolute errors and the numerical comparison with exact values for both the methods, while graphical comparison can be better envisioned through **Figure 4**. **Tables 7** and **8** display the numerical comparison of the proposed methods with the results obtained in [7] for *α* =0.75and [13] for *α* =0.9, respectively, whereas the mean square error, number of training points, and NAC for different values of *α* are presented in **Table 9**. The effects on accuracy of results with variable NAC and training points can be understood through **Table 10**.

Numerical Simulation Using Artificial Neural Network on Fractional Differential Equations http://dx.doi.org/10.5772/64151 107


*x* **ChSANN LSANN IABMM [14] EHPM [14] MHPM [7] Bernstein [13]**

**No of NAC No of training points Mean square error** *y***(***t***) Absolute error** 10 9.7679 × 10−5 0.760078 1.51570 × 10−3 20 2.3504 × 10−7 0.761644 5.02121 × 10−5 20 5.5016 × 10−9 0.761584 9.94216 × 10−6

Consider the nonlinear Riccati differential equation along with the following initial condition:

( ) ( ) ( ) ( ) <sup>2</sup> 2 1 0, 0 0 , 0 1 *d yt y t yt y dt*

( ) 1 2 1 1 2 tanh 2 log <sup>2</sup> 2 1 *y t <sup>t</sup>* æ ö æ ö - <sup>=</sup> + + ç ÷ ç ÷

ChSANN and LSANN algorithms are executed on the above test experiment with 6NAC, *α* =1

for ChSANN and LSANN, respectively. **Table 6** shows the absolute errors and the numerical comparison with exact values for both the methods, while graphical comparison can be better envisioned through **Figure 4**. **Tables 7** and **8** display the numerical comparison of the proposed methods with the results obtained in [7] for *α* =0.75and [13] for *α* =0.9, respectively, whereas the mean square error, number of training points, and NAC for different values of *α* are presented in **Table 9**. The effects on accuracy of results with variable NAC and training points

and 20 equidistant points that gave the mean square error up to 1.6127×10<sup>−</sup><sup>7</sup>

è ø + è ø

+ - -= = < £

a

and 4.68641×10<sup>−</sup><sup>6</sup>

0.2 0.234602 0.236053 0.2393 0.2647 0.2391 0.23878 0.4 0.419229 0.419898 0.4234 0.4591 0.4229 0.42258 0.6 0.563627 0.564474 0.5679 0.6031 0.5653 0.56617 0.8 0.672722 0.673241 0.6774 0.7068 0.6740 0.67462 1.0 0.753188 0.755002 0.7584 0.7806 0.7569 0.75458

0 0 0

*Example 2:*

**Table 4.** Numerical comparison for *α* =0.9.

106 Numerical Simulation - From Brain Imaging to Turbulent Flows

**Table 5.** Numerical values of ChSANN at *t* =1 and for *α* =1.

a

The exact solution when *α* =1 is given by [7]:

can be understood through **Table 10**.

a

**Table 6.** Numerical comparison of ChSANN and LSANN values with exact values at *α* =1 for fractional Riccati differential equation test example 2.

**Figure 4.** Absolute error of ChSANN and LSANN at *α* =1 for test example 2.


**Table 7.** Numerical comparison for *α* =0.75.


**Table 8.** Numerical comparison for *α* =0.9.


**Table 9.** Value of mean square error at different values of *α*.


**Table 10.** Numerical values of ChSANN at *t* =1 and for *α* =1.
