Resolução de um Problema Inverso Diferencial Fracionário no Processo de Fermentação Batelada Usando o Método da Colocação Ortogonal e o Algoritmo de Busca Fractal Estocástica

LOBATO, FRAN SÉRGIO

doi:10.5540/tema.2020.021.03.0537

FRAN SÉRGIO LOBATO

Faculdade de Engenharia Química, Universidade Federal de Uberlândia, Av. João Naves de Ávila, 2121, 38400-902, Bairro Santa Mônica, Uberlândia-MG, Brasil. - E-mail: fslobato@ufu.br

https://orcid.org/0000-0002-7401-4718

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Figura 1:
Perfis de concentração de biomassa, substrato e produto experimentais e preditos.

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Figura 2:
Perfis de concentração de biomassa, substrato e produto em função da ordem fracionária α.

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Tabela 1:
Dados experimentais para o processo de fermentação da enzima lacase.

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Tabela 2:
Erros absolutos obtidos para a primeira EDOF considerando

α = 3 / 2

e diferentes abordagens numéricas.

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Tabela 3:
Erros absolutos obtidos para a segunda EDOF considerando

t = 1

e diferentes valores para h e NPC .

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Tabela 4:
Função objetivo estimada em um problema puramente matemático considerando

N P C = 10

.

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Tabela 5:
Parâmetros estimados considerando diferentes metodologias numéricas e os modelos inteiro e fracionário.

(2.1)

(2.2)

(2.3)

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

(4.6)

(5.1)

(5.2)

(5.3)

(5.4)

(6.1)

(6.2)

(6.3)

(6.4)

t (Dia)	X (g/L)	S (g/L)	P (U/L)
1	0,1	2,0	0
2	0,1	2,0	125,3
3	0,3	1,9	1233,8
4	0,4	1,6	2699,1
5	0,7	1,3	5656,6
6	0,8	1,0	7089,1
7	1,1	0,8	5984,7
8	1,1	0,5	2897,9
9	1,2	0,4	1175,5
10	1,2	0,2	112,2
11	1,2	0,1	32,5
12	1,1	0,1	1,1

t	M p -H ¹³13 R. Lin & F. Liu. Fractional high order methods for the nonlinear fractional ordinary differential equation. Nonlinear Analysis, 1(66) (2007), 856-869.	MAO ¹²12 C. Li & F. Zeng. The Finite Difference Methods for Fractional Ordinary Differential Equations. Numerical Functional Analysis and Optimization, 2(34) (2013), 149-179.	$N P C = 5$	$N P C = 10$	$N P C = 15$	$N P C = 20$
0,2	3,22×10^-6	3,22×10^-6	1,03×10^-5	3,19×10^-8	7,28×10^-10	9,08×10^-14
0,4	2,91×10^-5	2,91×10^-5	1,11×10^-5	3,20×10^-8	6,24×10^-10	6,98×10^-14
0,6	3,97×10^-5	3,97×10^-5	1,15×10^-5	3,11×10^-8	6,76×10^-10	6,11×10^-14
0,8	3,48×10^-5	3,49×10^-5	1,10×10^-5	2,55×10^-8	5,34×10^-10	5,63×10^-14
1,0	1,54×10^-5	1,54×10^-5	1,31×10^-5	3,45×10^-8	4,98×10^-10	4,65×10^-14
1,2	1,81×10^-5	1,82×10^-5	1,47×10^-5	3,25×10^-8	3,34×10^-10	3,25×10^-14
1,4	6,83×10^-5	6,87×10^-5	1,55×10^-5	2,99×10^-7	2,24×10^-10	6,22×10^-13
1,6	1,43×10^-4	1,44×10^-4	1,67×10^-5	3,05×10^-7	4,41×10^-9	6,98×10^-13
1,8	2,57×10^-4	2,58×10^-4	1,23×10^-4	3,28×10^-7	7,89×10^-9	7,23×10^-13
2,0	4,05×10^-4	4,07×10^-4	3,38×10^-4	7,99×10^-7	9,33×10^-9	8,66×10^-13

Método	$α = 0, 1$	$α = 0, 3$	$α = 0, 5$	$α = 0, 7$	$α = 0, 9$
$h = 1 / 10$ ¹²12 C. Li & F. Zeng. The Finite Difference Methods for Fractional Ordinary Differential Equations. Numerical Functional Analysis and Optimization, 2(34) (2013), 149-179.	2,43×10^-5	1,91×10^-5	3,49×10^-5	5,38×10^-5	7,72×10^-5
$h = 1 / 20$ ¹²12 C. Li & F. Zeng. The Finite Difference Methods for Fractional Ordinary Differential Equations. Numerical Functional Analysis and Optimization, 2(34) (2013), 149-179.	3,19×10^-6	2,50×10^-6	4,52×10^-6	6,88×10^-6	9,76×10^-6
$h = 1 / 40$ ¹²12 C. Li & F. Zeng. The Finite Difference Methods for Fractional Ordinary Differential Equations. Numerical Functional Analysis and Optimization, 2(34) (2013), 149-179.	4,05×10^-7	3,20×10^-7	5,75×10^-7	8,69×10^-7	1,22×10^-6
$h = 1 / 80$ ¹²12 C. Li & F. Zeng. The Finite Difference Methods for Fractional Ordinary Differential Equations. Numerical Functional Analysis and Optimization, 2(34) (2013), 149-179.	5,06×10^-8	4,05×10^-8	7,24×10^-8	1,09×10^-7	1,53×10^-7
$h = 1 / 160$ ¹²12 C. Li & F. Zeng. The Finite Difference Methods for Fractional Ordinary Differential Equations. Numerical Functional Analysis and Optimization, 2(34) (2013), 149-179.	6,25×10^-9	5,09×10^-9	9,09×10^-9	1,36×10^-8	1,91×10^-8
$h = 1 / 320$ ¹²12 C. Li & F. Zeng. The Finite Difference Methods for Fractional Ordinary Differential Equations. Numerical Functional Analysis and Optimization, 2(34) (2013), 149-179.	7,71×10^-10	6,38×10^-10	1,13×10^-9	1,71×10^-9	2,39×10^-9
$h = 1 / 640$ ¹²12 C. Li & F. Zeng. The Finite Difference Methods for Fractional Ordinary Differential Equations. Numerical Functional Analysis and Optimization, 2(34) (2013), 149-179.	9,50×10^-11	7,99×10^-11	1,42×10^-10	2,14×10^-10	3,00×10^-10
$N P C = 5$	1,33×10^-5	1,45×10^-5	2,65×10^-5	1,45×10^-5	1,02×10^-5
$N P C = 10$	2,01×10^-8	1,15×10^-8	5,66×10^-8	2,88×10^-8	1,34×10^-8
$N P C = 15$	2,11×10^-10	1,77×10^-10	1,09×10^-9	3,44×10^-10	1,76×10^-10
$N P C = 20$	5,66×10^-14	3,14×10^-14	2,15×10^-14	3,33×10^-14	4,45×10^-14

		$α = 0, 1$		$α = 0, 5$		$α = 0, 9$
		$ξ = 0 %$	$ξ = 5 %$	$ξ = 0 %$	$ξ = 5 %$	$ξ = 0 %$	$ξ = 5 %$
ED	α	0,0999	0,0801	0,5000	0,4781	0,9000	0,8245
ED	FO ₁	9,2344x10^-7	0,24434	2,9442x10^-7	0,42323	2,4656x10^-7	0,86543
BFE¹	a	0,1000	0,0812	0,4999	0,4744	0,9000	0,8222
BFE¹	FO ₁	3,9834x10^-8	0,23444	6,8877x10^-8	0,20984	2,8887x10^-8	0,71232
BFE²	a	0,1000	0,0812	0,5000	0,4786	0,9000	0,8244
BFE²	FO ₁	1,4098x10^-8	0,17431	3,9878x10^-8	0,41554	1,9332x10^-8	0,61112

	ED		BFE¹		BFE²
α	1	0,9225	1	0,9227	1	0,9226
µ_max (dia^-α )	2,1860	2,3065	2,1857	2,1877	2,1865	2,1966
K_s (g/L)	7,6441	7,6156	7,6265	7,6245	7,6243	7,5960
Y_X/S (g/g)	0,6321	0,6410	0,6333	0,6408	0,6324	0,6304
K₁(Ug − 1dia^-α )	3743,4397	3915,1765	3743,4933	3914,3694	3743,4744	3915,5343
K₂(Ug − 1dia^-α )	7046,0178	7292,5605	7046,0123	7293,7587	7046,0822	7291,5644
K_p (dia^-α )	0,9999	0,9341	0,9999	0,9259	0,9999	0,9349
t_s (dia)	7,8050	7,7149	7,8065	7,7944	7,8054	7,7832
FO ₂	0,6172	0,5896	0,6173	0,5863	0,6173	0,5898

\frac{d P}{d t} = \{\begin{cases} K_{1} (1 - e x p (- (t - 0, 1) / 0, 1)) S - K_{p} P s e t \leq t_{s} \\ (K_{1} - K_{2}) (1 - e x p (- (t - 0, 1) / 0, 1)) S s e t > t_{s} \end{cases} (P (0) = P_{0})

P_{i}^{n e w} (j) = \{\begin{matrix} P_{1 r} (j) - r a n d (0, 1) \times (P_{2 r} (j) - P_{i} (j)), & se γ_{i} \leq r a n d (0, 1) \\ P_{j} (j), & se γ_{i} > r a n d (0, 1) \end{matrix}

P_{i}^{n e w} (j) = \{\begin{cases} P_{i} (j) - r a n d (0, 1) \times (Q_{1 r} (j) - P_{b e s t} (j)), & se r a n d (0, 1) \leq 0.5 \\ P_{i} (j) + r a n d (0, 1) \times (Q_{1 r} (j) - Q_{2 r} (j)), & se r a n d (0, 1) > 0 . 5 \end{cases}

F O_{2} \equiv \sum_{i = 1}^{M} \frac{{(X_{i}^{\exp} - X_{i}^{c a l})}^{2}}{{(X_{\max})}^{2}} + \sum_{i = 1}^{M} \frac{{(S_{i}^{\exp} - S_{i}^{c a l})}^{2}}{{(S_{\max})}^{2}} + \sum_{i = 1}^{M} \frac{{(P_{i}^{\exp} - P_{i}^{c a l})}^{2}}{{(P_{\max})}^{2}}

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Resolução de um Problema Inverso Diferencial Fracionário no Processo de Fermentação Batelada Usando o Método da Colocação Ortogonal e o Algoritmo de Busca Fractal Estocástica

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