Initial value problem for fractional Volterra integro-differential equations with Caputo derivative

Huy Tuan Nguyen; Huu Can Nguyen; Renhai Wang; Yong Zhou

doi:10.3934/dcdsb.2021030

Article Contents

2021, Volume 26, Issue 12: 6483-6510. Doi: 10.3934/dcdsb.2021030 open access

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Initial value problem for fractional Volterra integro-differential equations with Caputo derivative

1.
Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam, Vietnam National University, Ho Chi Minh City, Vietnam
2.
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3.
Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China
4.
Faculty of Information Technology, Macau University of Science and Technology, Macau 999078, China
5.
Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China

^* Corresponding author: Huu Can Nguyen (nguyenhuucan@tdtu.edu.vn)
^* Corresponding author: Huu Can Nguyen (nguyenhuucan@tdtu.edu.vn)

Dedicated to Tomás Caraballo on his 60th birthday.

Received: March 31, 2020

Revised: November 30, 2020

Early access: February 2021

Published: December 2021

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Abstract

In this paper, we consider the time-fractional Volterra integro-differential equations with Caputo derivative. For globally Lispchitz source term, we investigate the global existence for a mild solution. The main tool is to apply the Banach fixed point theorem on some new weighted spaces combining some techniques on the Wright functions. For the locally Lipschitz case, we study the existence of local mild solutions to the problem and provide a blow-up alternative for mild solutions. We also establish the problem of continuous dependence with respect to initial data. Finally, we present some examples to illustrate the theoretical results.

Keywords:

Volterra integro-differential equations,
Caputo derivative,
blow-up,
global and local existence.

Mathematics Subject Classification: 35K55, 35K70, 35K92, 47A52, 47J06.

Citation:

FullText(HTML)

Figure 1. Ex1. The solutions $ u(x,t) $ at $ t \in \{0.1, 0.5, 0.9\} $ for $ \alpha = 0.2 $ and $ \epsilon \in \{0.1, 0.01, 0.001\} $

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Figure 2. Ex1. The solutions $ u(x,t) $ on $ (x,t) \in (0,\pi) \times (0,1) $ for $ \alpha = 0.2 $ and $ \epsilon \in \{0.1, 0.01, 0.001\} $

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Figure 3. Ex2. The solutions $ u(x,t) $ at $ t \in \{0.1, 0.5, 0.9\} $ for $ \alpha = 0.9 $ and $ \epsilon \in \{0.1, 0.01, 0.001\} $

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Figure 4. Ex2. The solutions $ u(x,t) $ on $ (x,t) \in (0,\pi) \times (0,1) $ for $ \alpha = 0.9 $ and $ \epsilon \in \{0.1, 0.01, 0.001\} $

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Table 1. Ex1. The error estimation for $ \alpha = 0.2 $ and $ t \in \{0.1, 0.5, 0.9\} $

	$ \epsilon = \|\alpha^* - \alpha\| $	$ N(j) = 10,\; P =10,\; M = N =50 $
	$ \epsilon = \|\alpha^* - \alpha\| $	Calculative error	Percent error $ \delta^{ \alpha'}_ \alpha $
$ \mathrm{Error}^{ \alpha^*}_ \alpha $(0.1)	0.1	0.011036875514009	15.46 %
	0.01	0.004874547920700	6.83 %
	0.001	0.002176754401323	3.05 %
$ \mathrm{Error}^{ \alpha^*}_ \alpha $(0.5)	0.1	0.051353021961229	14.38 %
	0.01	0.027169280169359	7.61 %
	0.001	0.012084128051511	3.38 %
$ \mathrm{Error}^{ \alpha^*}_ \alpha $(0.9)	0.1	0.074877070037197	11.65 %
	0.01	0.042748723217464	6.65 %
	0.001	0.028729381706881	4.47 %

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Table 2. Ex2. The error estimation for $ \alpha = 0.9 $ and $ t \in \{0.1, 0.5, 0.9\} $

	$ \epsilon = \|\alpha^* - \alpha\| $	$ N(j) = 10,\; P =10,\; M = N =50 $
	$ \epsilon = \|\alpha^* - \alpha\| $	Calculative error	Percent error $ \delta^{ \alpha'}_ \alpha $
$ \mathrm{Error}^{ \alpha^*}_ \alpha $(0.1)	0.1	0.601407040658915	83.81 %
	0.01	0.143648723675101	20.02 %
	0.001	0.027371563705550	3.81 %
$ \mathrm{Error}^{ \alpha^*}_ \alpha $(0.5)	0.1	0.643179867112674	79.89 %
	0.01	0.167904683423887	20.85 %
	0.001	0.025589340748129	3.18 %
$ \mathrm{Error}^{ \alpha^*}_ \alpha $(0.9)	0.1	0.629790962460797	61.54 %
	0.01	0.219601251416672	21.46 %
	0.001	0.040291675260527	3.94 %

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