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

Previous Article
Global stability of HIV/HTLV co-infection model with CTL-mediated immunity
DCDS-B Home
This Issue
Next Article
Feedback synchronization of FHN cellular neural networks

doi: 10.3934/dcdsb.2021030

Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

Huy Tuan Nguyen ^1, , Huu Can Nguyen ^2,, , Renhai Wang ^3, and Yong Zhou ^4,5,

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)

Dedicated to Tomás Caraballo on his 60th birthday.

Received April 2020 Revised December 2020 Early access February 2021

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: Huy Tuan Nguyen, Huu Can Nguyen, Renhai Wang, Yong Zhou. Initial value problem for fractional Volterra integro-differential equations with Caputo derivative. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021030

References:

[1]	B. Andrade and A. Viana, Abstract Volterra integro-differential equations with applications to parabolic models with memory, Math. Ann., 369 (2017), 1131-1175. doi: 10.1007/s00208-016-1469-z. Google Scholar
[2]	B. Andrade and A. Viana, Integrodifferential equations with applications to a plate equation with memory, Mathematische Nachrichten, 289 (17–18), 2159-2172. doi: 10.1002/mana.201500205. Google Scholar
[3]	B. Andrade, A. N. Carvalho, P. M. Carvalho-Neto and P. Marin-Rubio, Semilinear fractional differential equations: Global solutions, critical nonlinearities and comparison results, Topological Methods in Nonlinear Analysis, 45 (2015), 439-467. doi: 10.12775/TMNA.2015.022. Google Scholar
[4]	K. Balachandran and J. J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integro-differential equations in Banach spaces, Nonlinear Anal., 72 (2010), 4587-4593. doi: 10.1016/j.na.2010.02.035. Google Scholar
[5]	M. Bonforte, Y. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767. doi: 10.3934/dcds.2015.35.5725. Google Scholar
[6]	L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 767-807. doi: 10.1016/j.anihpc.2015.01.004. Google Scholar
[7]	T. Caraballo and J. Real, Attractors for 2D-Navier–Stokes models with delays, J. Differ. Equ., 205 (2004), 271-297. doi: 10.1016/j.jde.2004.04.012. Google Scholar
[8]	T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194. doi: 10.1098/rspa.2003.1166. Google Scholar
[9]	Y. Chen, H. Gao, M. Garrido-Atienza and B. Schmalfuß, Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 79-98. doi: 10.3934/dcds.2014.34.79. Google Scholar
[10]	B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors,, Z. Angew. Math. Phys., 18 (1967), 199-208. doi: 10.1007/BF01596912. Google Scholar
[11]	B. D. Coleman and V. J. Mizel, Norms and semigroups in the theory of fading memory, Arch. Rational Mech. Anal., 28 (1966), 87-123. doi: 10.1007/BF00251727. Google Scholar
[12]	M. Conti, E. Marchini and V. Pata, A well posedness result for nonlinear viscoelastic equations with memory, Nonlinear Anal., 94 (2014), 206-216. doi: 10.1016/j.na.2013.08.015. Google Scholar
[13]	M. Conti, F. Dell'Oro and V. Pata, Nonclassical diffusion with memory lacking instantaneous damping, Communications on Pure and Applied Analysis, 19 (2020). doi: 10.3934/cpaa.2020090. Google Scholar
[14]	M. Conti, E. Marchini and V. Pata, Reaction-diffusion with memory in the minimal state framework, Trans. Amer. Math. Soc., 366 (2014), 4969-4986. doi: 10.1090/S0002-9947-2013-06097-7. Google Scholar
[15]	M. D'Abbico, The influence of a nonlinear memory on the damped wave equation, Nonlinear Anal, 95 (2014), 130-145. doi: 10.1016/j.na.2013.09.006. Google Scholar
[16]	M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Asymp. Anal., 81 (2002), 1245-1264. doi: 10.1080/0003681021000035588. Google Scholar
[17]	H. Gou and B. Li, Local and global existence of mild solution to impulsive fractional semilinear integro-differential equation with noncompact semigroup, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 204-214. doi: 10.1016/j.cnsns.2016.05.021. Google Scholar
[18]	M. L. Heard and S. M. Rankin III, A semilinear parabolic Volterra integro-differential equation, J. Differential Equations, 71 (1988), 201-233. doi: 10.1016/0022-0396(88)90023-X. Google Scholar
[19]	V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics, 301, Longman, Harlow, 1994. doi: 978-0582219779. Google Scholar
[20]	L. Li and G. J. Liu, A generalized definition of Caputo derivatives and its application to fractional ODEs, SIAM J. Math. Anal., 50 (2018), 2867-2900. doi: 10.1137/17M1160318. Google Scholar
[21]	A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224. doi: 10.1137/0521066. Google Scholar
[22]	R. C. MacCamy, An integro-differential equation with application in heat flow, Quart. Appl. Math., 35 (1977), 1-19. doi: 10.1090/qam/452184. Google Scholar
[23]	F. Mainardi, A. Mura and G. Pagnini, The $M$-Wright function in time-fractional diffusion processes: A tutorial survey, Int. J. Differ. Equ., (2010), 104505, 29 pp. doi: 10.1155/2010/104505. Google Scholar
[24]	L. Peng, A. Debbouche and Y. Zhou, Existence and approximation of solutions for time-fractional Navier-stokes equations, Math. Methods Appl. Sci., 41 (2018), 8973-8984. doi: 10.1002/mma.4779. Google Scholar
[25]	I. Podlubny, Fractional Differential Equations, Academic press, California, 1999. doi: 10.1016/978-0-12-558840-9. Google Scholar
[26]	M. H. M. Rashid and Y. E. Qaderi, Semilinear fractional integro-differential equations with compact semigroup, Nonlinear Anal., 71 (2009), 6276-6282. doi: 10.1016/j.na.2009.06.035. Google Scholar
[27]	M. H. M. Rashid and A. Al-Omari, Local and global existence of mild solutions for impulsive fractional semi-linear integro-differential equation, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3493-3503. doi: 10.1016/j.cnsns.2010.12.043. Google Scholar
[28]	K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058. Google Scholar
[29]	J. V. C. Sousa, F. G. Rodrigues and E. C. Oliveira, Stability of the fractional Volterra integro-differential equation by means of $\psi$-Hilfer operator, Math. Methods Appl. Sci., 42 (2019), 3033-3043. doi: 10.1002/mma.5563. Google Scholar
[30]	H. G. Sun, Y. Zhang, D. Baleanu, W. Chen and Y. Q. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213-231. doi: 10.1515/fca-2020-0012. Google Scholar
[31]	A. Viana, Local well-posedness for a Lotka-Volterra system in Besov spaces, Comput. Math. Appl., 69 (2015), 667-674. doi: 10.1016/j.camwa.2015.02.013. Google Scholar
[32]	A. Viana, A local theory for a fractional reaction-diffusion equation, Commun. Contemp. Math, 21 (2019), 1850033, 26 pp. doi: 10.1142/S0219199718500335. Google Scholar
[33]	G. Webb, An abstract semilinear Volterra integro-differential equation, Proc. Amer. Math. Soc., 69 (1978), 255-260. doi: 10.1090/S0002-9939-1978-0467214-4. Google Scholar
[34]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Different. Equ., 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008. Google Scholar
[35]	Y. Zhou, J. Manimaran, L. Shangerganesh and A. Debbouche, Weakness and Mittag-Leffler stability of solutions for time-fractional Keller-Segel models, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 753-761. doi: 10.1515/ijnsns-2018-0035. Google Scholar

show all references

References:

[1]	B. Andrade and A. Viana, Abstract Volterra integro-differential equations with applications to parabolic models with memory, Math. Ann., 369 (2017), 1131-1175. doi: 10.1007/s00208-016-1469-z. Google Scholar
[2]	B. Andrade and A. Viana, Integrodifferential equations with applications to a plate equation with memory, Mathematische Nachrichten, 289 (17–18), 2159-2172. doi: 10.1002/mana.201500205. Google Scholar
[3]	B. Andrade, A. N. Carvalho, P. M. Carvalho-Neto and P. Marin-Rubio, Semilinear fractional differential equations: Global solutions, critical nonlinearities and comparison results, Topological Methods in Nonlinear Analysis, 45 (2015), 439-467. doi: 10.12775/TMNA.2015.022. Google Scholar
[4]	K. Balachandran and J. J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integro-differential equations in Banach spaces, Nonlinear Anal., 72 (2010), 4587-4593. doi: 10.1016/j.na.2010.02.035. Google Scholar
[5]	M. Bonforte, Y. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767. doi: 10.3934/dcds.2015.35.5725. Google Scholar
[6]	L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 767-807. doi: 10.1016/j.anihpc.2015.01.004. Google Scholar
[7]	T. Caraballo and J. Real, Attractors for 2D-Navier–Stokes models with delays, J. Differ. Equ., 205 (2004), 271-297. doi: 10.1016/j.jde.2004.04.012. Google Scholar
[8]	T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194. doi: 10.1098/rspa.2003.1166. Google Scholar
[9]	Y. Chen, H. Gao, M. Garrido-Atienza and B. Schmalfuß, Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 79-98. doi: 10.3934/dcds.2014.34.79. Google Scholar
[10]	B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors,, Z. Angew. Math. Phys., 18 (1967), 199-208. doi: 10.1007/BF01596912. Google Scholar
[11]	B. D. Coleman and V. J. Mizel, Norms and semigroups in the theory of fading memory, Arch. Rational Mech. Anal., 28 (1966), 87-123. doi: 10.1007/BF00251727. Google Scholar
[12]	M. Conti, E. Marchini and V. Pata, A well posedness result for nonlinear viscoelastic equations with memory, Nonlinear Anal., 94 (2014), 206-216. doi: 10.1016/j.na.2013.08.015. Google Scholar
[13]	M. Conti, F. Dell'Oro and V. Pata, Nonclassical diffusion with memory lacking instantaneous damping, Communications on Pure and Applied Analysis, 19 (2020). doi: 10.3934/cpaa.2020090. Google Scholar
[14]	M. Conti, E. Marchini and V. Pata, Reaction-diffusion with memory in the minimal state framework, Trans. Amer. Math. Soc., 366 (2014), 4969-4986. doi: 10.1090/S0002-9947-2013-06097-7. Google Scholar
[15]	M. D'Abbico, The influence of a nonlinear memory on the damped wave equation, Nonlinear Anal, 95 (2014), 130-145. doi: 10.1016/j.na.2013.09.006. Google Scholar
[16]	M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Asymp. Anal., 81 (2002), 1245-1264. doi: 10.1080/0003681021000035588. Google Scholar
[17]	H. Gou and B. Li, Local and global existence of mild solution to impulsive fractional semilinear integro-differential equation with noncompact semigroup, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 204-214. doi: 10.1016/j.cnsns.2016.05.021. Google Scholar
[18]	M. L. Heard and S. M. Rankin III, A semilinear parabolic Volterra integro-differential equation, J. Differential Equations, 71 (1988), 201-233. doi: 10.1016/0022-0396(88)90023-X. Google Scholar
[19]	V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics, 301, Longman, Harlow, 1994. doi: 978-0582219779. Google Scholar
[20]	L. Li and G. J. Liu, A generalized definition of Caputo derivatives and its application to fractional ODEs, SIAM J. Math. Anal., 50 (2018), 2867-2900. doi: 10.1137/17M1160318. Google Scholar
[21]	A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224. doi: 10.1137/0521066. Google Scholar
[22]	R. C. MacCamy, An integro-differential equation with application in heat flow, Quart. Appl. Math., 35 (1977), 1-19. doi: 10.1090/qam/452184. Google Scholar
[23]	F. Mainardi, A. Mura and G. Pagnini, The $M$-Wright function in time-fractional diffusion processes: A tutorial survey, Int. J. Differ. Equ., (2010), 104505, 29 pp. doi: 10.1155/2010/104505. Google Scholar
[24]	L. Peng, A. Debbouche and Y. Zhou, Existence and approximation of solutions for time-fractional Navier-stokes equations, Math. Methods Appl. Sci., 41 (2018), 8973-8984. doi: 10.1002/mma.4779. Google Scholar
[25]	I. Podlubny, Fractional Differential Equations, Academic press, California, 1999. doi: 10.1016/978-0-12-558840-9. Google Scholar
[26]	M. H. M. Rashid and Y. E. Qaderi, Semilinear fractional integro-differential equations with compact semigroup, Nonlinear Anal., 71 (2009), 6276-6282. doi: 10.1016/j.na.2009.06.035. Google Scholar
[27]	M. H. M. Rashid and A. Al-Omari, Local and global existence of mild solutions for impulsive fractional semi-linear integro-differential equation, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3493-3503. doi: 10.1016/j.cnsns.2010.12.043. Google Scholar
[28]	K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058. Google Scholar
[29]	J. V. C. Sousa, F. G. Rodrigues and E. C. Oliveira, Stability of the fractional Volterra integro-differential equation by means of $\psi$-Hilfer operator, Math. Methods Appl. Sci., 42 (2019), 3033-3043. doi: 10.1002/mma.5563. Google Scholar
[30]	H. G. Sun, Y. Zhang, D. Baleanu, W. Chen and Y. Q. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213-231. doi: 10.1515/fca-2020-0012. Google Scholar
[31]	A. Viana, Local well-posedness for a Lotka-Volterra system in Besov spaces, Comput. Math. Appl., 69 (2015), 667-674. doi: 10.1016/j.camwa.2015.02.013. Google Scholar
[32]	A. Viana, A local theory for a fractional reaction-diffusion equation, Commun. Contemp. Math, 21 (2019), 1850033, 26 pp. doi: 10.1142/S0219199718500335. Google Scholar
[33]	G. Webb, An abstract semilinear Volterra integro-differential equation, Proc. Amer. Math. Soc., 69 (1978), 255-260. doi: 10.1090/S0002-9939-1978-0467214-4. Google Scholar
[34]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Different. Equ., 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008. Google Scholar
[35]	Y. Zhou, J. Manimaran, L. Shangerganesh and A. Debbouche, Weakness and Mittag-Leffler stability of solutions for time-fractional Keller-Segel models, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 753-761. doi: 10.1515/ijnsns-2018-0035. Google Scholar

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\} $

Figure Options

Download full-size image

Download as PowerPoint slide

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\} $

Figure Options

Download full-size image

Download as PowerPoint slide

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\} $

Figure Options

Download full-size image

Download as PowerPoint slide

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\} $

Figure Options

Download full-size image

Download as PowerPoint slide

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 %

Table Options

Download as excel

	$ \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 %

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 %

Table Options

Download as excel

	$ \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 %

[1]	Seda İğret Araz. New class of volterra integro-differential equations with fractal-fractional operators: Existence, uniqueness and numerical scheme. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2297-2309. doi: 10.3934/dcdss.2021053
[2]	Miloud Moussai. Application of the bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with caputo fractional derivatives. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021021
[3]	Yoshikazu Giga. Interior derivative blow-up for quasilinear parabolic equations. Discrete & Continuous Dynamical Systems, 1995, 1 (3) : 449-461. doi: 10.3934/dcds.1995.1.449
[4]	John A. D. Appleby, Denis D. Patterson. Blow-up and superexponential growth in superlinear Volterra equations. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3993-4017. doi: 10.3934/dcds.2018174
[5]	Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053
[6]	Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191
[7]	Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333
[8]	Shiming Li, Yongsheng Li, Wei Yan. A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1899-1912. doi: 10.3934/dcdss.2016077
[9]	Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[10]	Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051
[11]	Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827
[12]	Wenjun Liu, Jiangyong Yu, Gang Li. Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021121
[13]	Hermann Brunner. The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays. Communications on Pure & Applied Analysis, 2006, 5 (2) : 261-276. doi: 10.3934/cpaa.2006.5.261
[14]	Faranak Rabiei, Fatin Abd Hamid, Zanariah Abd Majid, Fudziah Ismail. Numerical solutions of Volterra integro-differential equations using General Linear Method. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 433-444. doi: 10.3934/naco.2019042
[15]	Jens Lorenz, Wilberclay G. Melo, Natã Firmino Rocha. The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3819-3841. doi: 10.3934/dcdsb.2018332
[16]	Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072
[17]	Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677
[18]	Kyudong Choi. Persistence of Hölder continuity for non-local integro-differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1741-1771. doi: 10.3934/dcds.2013.33.1741
[19]	Elena Bonetti, Elisabetta Rocca. Global existence and long-time behaviour for a singular integro-differential phase-field system. Communications on Pure & Applied Analysis, 2007, 6 (2) : 367-387. doi: 10.3934/cpaa.2007.6.367
[20]	Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057

2020 Impact Factor: 1.327

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.327

Tools

Figures and Tables

2020 Impact Factor: 1.327

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors