• Previous Article
    Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient
  • DCDS-B Home
  • This Issue
  • Next Article
    Erratum: Existence and uniqueness of solutions of free boundary problems in heterogeneous environments
doi: 10.3934/dcdsb.2021225
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.

Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation

School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan 411105, China

* Corresponding author: Aiguo Xiao

Received  May 2020 Revised  April 2021 Early access September 2021

Fund Project: This research is supported by the National Natural Science Foundation of China (Nos. 12071403, 11671343), the Research Foundation of Education Commission of Hunan Province of China (No. 19B565) and the Postgraduate Innovation Fund of Hunan Province in China (No. CX20190420)

This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler–Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler–Maruyama method can reach strong first-order superconvergence when $ \alpha = 1 $. Finally, several numerical tests are reported for verification of the theoretical findings.

Citation: Xinjie Dai, Aiguo Xiao, Weiping Bu. Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021225
References:
[1]

A. AghajaniY. Jalilian and J. J. Trujillo, On the existence of solutions of fractional integro-differential equations, Fract. Calc. Appl. Anal., 15 (2012), 44-69.  doi: 10.2478/s13540-012-0005-4.  Google Scholar

[2]

P. T. AnhT. S. Doan and P. T. Huong, A variation of constant formula for Caputo fractional stochastic differential equations, Statist. Probab. Lett., 145 (2019), 351-358.  doi: 10.1016/j.spl.2018.10.010.  Google Scholar

[3]

M. Asgari, Block pulse approximation of fractional stochastic integro-differential equation, Commun. Numer. Anal., 2014 (2014), 1-7.   Google Scholar

[4]

A. A. Badr and H. S. El-Hoety, Monte–Carlo Galerkin approximation of fractional stochastic integro-differential equation, Math. Probl. Eng., 2012 (2012), 709106. doi: 10.1155/2012/709106.  Google Scholar

[5]

P. Balasubramaniam and P. Tamilalagan, The solvability and optimal controls for impulsive fractional stochastic integro-differential equations via resolvent operators, J. Optim. Theory Appl., 174 (2017), 139-155.  doi: 10.1007/s10957-016-0865-6.  Google Scholar

[6]

W. Cao, Z. Zhang and G. E. Karniadakis, Numerical methods for stochastic delay differential equations via the Wong–Zakai approximation, SIAM J. Sci. Comput., 37 (2015), A295–A318. doi: 10.1137/130942024.  Google Scholar

[7]

Z.-Q. ChenK.-H. Kim and P. Kim, Fractional time stochastic partial differential equations, Stochastic Process. Appl., 125 (2015), 1470-1499.  doi: 10.1016/j.spa.2014.11.005.  Google Scholar

[8]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC, 2004. doi: 10.1201/9780203485217.  Google Scholar

[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[10]

X. DaiW. Bu and A. Xiao, Well-posedness and EM approximations for non-Lipschitz stochastic fractional integro-differential equations, J. Comput. Appl. Math., 356 (2019), 377-390.  doi: 10.1016/j.cam.2019.02.002.  Google Scholar

[11]

K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

[12]

T. S. Doan, P. T. Huong, P. E. Kloeden and A. M. Vu, Euler–Maruyama scheme for Caputo stochastic fractional differential equations, J. Comput. Appl. Math., 380 (2020), 112989. doi: 10.1016/j.cam.2020.112989.  Google Scholar

[13]

N. T. Dung, Fractional stochastic differential equations with applications to finance, J. Math. Anal. Appl., 397 (2013), 334-348.  doi: 10.1016/j.jmaa.2012.07.062.  Google Scholar

[14]

D. J. HighamX. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), 1041-1063.  doi: 10.1137/S0036142901389530.  Google Scholar

[15]

M. HutzenthalerA. Jentzen and P. E. Kloeden, Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 1563-1576.  doi: 10.1098/rspa.2010.0348.  Google Scholar

[16]

M. HutzenthalerA. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab., 22 (2012), 1611-1641.  doi: 10.1214/11-AAP803.  Google Scholar

[17]

G. Izzo, E. Messina and A. Vecchio, Stability of numerical solutions for Abel–Volterra integral equations of the second kind, Mediterr. J. Math., 15 (2018), Paper No. 113. doi: 10.1007/s00009-018-1149-1.  Google Scholar

[18]

B. JinY. Yan and Z. Zhou, Numerical approximation of stochastic time-fractional diffusion, ESAIM Math. Model. Numer. Anal., 53 (2019), 1245-1268.  doi: 10.1051/m2an/2019025.  Google Scholar

[19]

M. Kamrani, Numerical solution of stochastic fractional differential equations, Numer. Algorithms, 68 (2015), 81-93.  doi: 10.1007/s11075-014-9839-7.  Google Scholar

[20]

M. Kamrani, Convergence of Galerkin method for the solution of stochastic fractional integro differential equations, Optik Int. J. Light Electron Opt., 127 (2016), 10049-10057.   Google Scholar

[21]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[22] V. Lakshmikantham and M. Rama Mohana Rao, Theory of Integro-Differential Equations, CRC Press, 1995.   Google Scholar
[23]

J. J. Levin and J. A. Nohel, On a system of integrodifferential equations occuring in reactor dynamics, J. Math. Mech., 9 (1960), 347-368.  doi: 10.1512/iumj.1960.9.59020.  Google Scholar

[24]

L. LiJ.-G. Liu and J. Lu, Fractional stochastic differential equations satisfying fluctuation-dissipation theorem, J. Stat. Phys., 169 (2017), 316-339.  doi: 10.1007/s10955-017-1866-z.  Google Scholar

[25]

Y. Li and Y. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differential Equations, 266 (2019), 3514-3558.  doi: 10.1016/j.jde.2018.09.009.  Google Scholar

[26]

H. LiangZ. Yang and J. Gao, Strong superconvergence of the Euler–Maruyama method for linear stochastic Volterra integral equations, J. Comput. Appl. Math., 317 (2017), 447-457.  doi: 10.1016/j.cam.2016.11.005.  Google Scholar

[27]

M. Maleki and M. Tavassoli Kajani, Numerical approximations for Volterra's population growth model with fractional order via a multi-domain pseudospectral method, Appl. Math. Model., 39 (2015), 4300-4308.  doi: 10.1016/j.apm.2014.12.045.  Google Scholar

[28]

X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2008. doi: 10.1533/9780857099402.  Google Scholar

[29]

S. A. McKinley and H. D. Nguyen, Anomalous diffusion and the generalized Langevin equation, SIAM J. Math. Anal., 50 (2018), 5119-5160.  doi: 10.1137/17M115517X.  Google Scholar

[30]

F. Mirzaee and N. Samadyar, Application of orthonormal Bernstein polynomials to construct a efficient scheme for solving fractional stochastic integro-differential equation, Optik Int. J. Light Electron Opt., 132 (2017), 262-273.  doi: 10.1016/j.ijleo.2016.12.029.  Google Scholar

[31]

F. Mirzaee and N. Samadyar, On the numerical solution of fractional stochastic integro-differential equations via meshless discrete collocation method based on radial basis functions, Eng. Anal. Bound. Elem., 100 (2019), 246-255.  doi: 10.1016/j.enganabound.2018.05.006.  Google Scholar

[32]

F. Mohammadi, Efficient Galerkin solution of stochastic fractional differential equations using second kind Chebyshev wavelets, Bol. Soc. Parana. Mat., 35 (2017), 195-215.  doi: 10.5269/bspm.v35i1.28262.  Google Scholar

[33]

S. M. Momani, Local and global existence theorems on fractional integro-differential equations, J. Fract. Calc., 18 (2000), 81-86.   Google Scholar

[34]

J.-C. Pedjeu and G. S. Ladde, Stochastic fractional differential equations: Modeling, method and analysis, Chaos Solitons Fractals, 45 (2012), 279-293.  doi: 10.1016/j.chaos.2011.12.009.  Google Scholar

[35]

A. N. V. Rao and C. P. Tsokos, On the existence, uniqueness, and stability behavior of a random solution to a nonlinear perturbed stochastic integro-differential equation, Information and Control, 27 (1975), 61-74.  doi: 10.1016/S0019-9958(75)90074-1.  Google Scholar

[36]

F. M. Scudo, Vito Volterra and theoretical ecology, Theoret. Population Biol., 2 (1971), 1-23.  doi: 10.1016/0040-5809(71)90002-5.  Google Scholar

[37]

D. T. SonP. T. HuongP. E. Kloeden and H. T. Tuan, Asymptotic separation between solutions of Caputo fractional stochastic differential equations, Stoch. Anal. Appl., 36 (2018), 654-664.  doi: 10.1080/07362994.2018.1440243.  Google Scholar

[38]

Z. TaheriS. Javadi and E. Babolian, Numerical solution of stochastic fractional integro-differential equation by the spectral collocation method, J. Comput. Appl. Math., 321 (2017), 336-347.  doi: 10.1016/j.cam.2017.02.027.  Google Scholar

[39]

V. E. Tarasov, Fractional integro-differential equations for electromagnetic waves in dielectric media, Theoret. and Math. Phys., 158 (2009), 355-359.  doi: 10.1007/s11232-009-0029-z.  Google Scholar

[40]

K. G. TeBeest, Classroom Note: Numerical and analytical solutions of Volterra's population model, SIAM Rev., 39 (1997), 484-493.  doi: 10.1137/S0036144595294850.  Google Scholar

[41]

H. T. Tuan, On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1749-1762.  doi: 10.3934/dcdsb.2020318.  Google Scholar

[42]

Y. WangJ. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Anal., 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020.  Google Scholar

[43]

Z. Wang, Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients, Statist. Probab. Lett., 78 (2008), 1062-1071.  doi: 10.1016/j.spl.2007.10.007.  Google Scholar

[44]

Z. Yang, H. Yang and Z. Yao, Strong convergence analysis for Volterra integro-differential equations with fractional Brownian motions, J. Comput. Appl. Math., 383 (2021), 113156. doi: 10.1016/j.cam.2020.113156.  Google Scholar

[45]

H. YeJ. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.  Google Scholar

[46]

Ş. Yüzbaşı, A numerical approximation for Volterra's population growth model with fractional order, Appl. Math. Model., 37 (2013), 3216-3227.  doi: 10.1016/j.apm.2012.07.041.  Google Scholar

[47]

G. Zhang and R. Zhu, Runge–Kutta convolution quadrature methods with convergence and stability analysis for nonlinear singular fractional integro-differential equations, Commun. Nonlinear Sci. Numer. Simul., 84 (2020), 105132. doi: 10.1016/j.cnsns.2019.105132.  Google Scholar

[48]

G. Zou, Numerical solutions to time-fractional stochastic partial differential equations, Numer. Algorithms, 82 (2019), 553-571.  doi: 10.1007/s11075-018-0613-0.  Google Scholar

show all references

References:
[1]

A. AghajaniY. Jalilian and J. J. Trujillo, On the existence of solutions of fractional integro-differential equations, Fract. Calc. Appl. Anal., 15 (2012), 44-69.  doi: 10.2478/s13540-012-0005-4.  Google Scholar

[2]

P. T. AnhT. S. Doan and P. T. Huong, A variation of constant formula for Caputo fractional stochastic differential equations, Statist. Probab. Lett., 145 (2019), 351-358.  doi: 10.1016/j.spl.2018.10.010.  Google Scholar

[3]

M. Asgari, Block pulse approximation of fractional stochastic integro-differential equation, Commun. Numer. Anal., 2014 (2014), 1-7.   Google Scholar

[4]

A. A. Badr and H. S. El-Hoety, Monte–Carlo Galerkin approximation of fractional stochastic integro-differential equation, Math. Probl. Eng., 2012 (2012), 709106. doi: 10.1155/2012/709106.  Google Scholar

[5]

P. Balasubramaniam and P. Tamilalagan, The solvability and optimal controls for impulsive fractional stochastic integro-differential equations via resolvent operators, J. Optim. Theory Appl., 174 (2017), 139-155.  doi: 10.1007/s10957-016-0865-6.  Google Scholar

[6]

W. Cao, Z. Zhang and G. E. Karniadakis, Numerical methods for stochastic delay differential equations via the Wong–Zakai approximation, SIAM J. Sci. Comput., 37 (2015), A295–A318. doi: 10.1137/130942024.  Google Scholar

[7]

Z.-Q. ChenK.-H. Kim and P. Kim, Fractional time stochastic partial differential equations, Stochastic Process. Appl., 125 (2015), 1470-1499.  doi: 10.1016/j.spa.2014.11.005.  Google Scholar

[8]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC, 2004. doi: 10.1201/9780203485217.  Google Scholar

[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[10]

X. DaiW. Bu and A. Xiao, Well-posedness and EM approximations for non-Lipschitz stochastic fractional integro-differential equations, J. Comput. Appl. Math., 356 (2019), 377-390.  doi: 10.1016/j.cam.2019.02.002.  Google Scholar

[11]

K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

[12]

T. S. Doan, P. T. Huong, P. E. Kloeden and A. M. Vu, Euler–Maruyama scheme for Caputo stochastic fractional differential equations, J. Comput. Appl. Math., 380 (2020), 112989. doi: 10.1016/j.cam.2020.112989.  Google Scholar

[13]

N. T. Dung, Fractional stochastic differential equations with applications to finance, J. Math. Anal. Appl., 397 (2013), 334-348.  doi: 10.1016/j.jmaa.2012.07.062.  Google Scholar

[14]

D. J. HighamX. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), 1041-1063.  doi: 10.1137/S0036142901389530.  Google Scholar

[15]

M. HutzenthalerA. Jentzen and P. E. Kloeden, Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 1563-1576.  doi: 10.1098/rspa.2010.0348.  Google Scholar

[16]

M. HutzenthalerA. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab., 22 (2012), 1611-1641.  doi: 10.1214/11-AAP803.  Google Scholar

[17]

G. Izzo, E. Messina and A. Vecchio, Stability of numerical solutions for Abel–Volterra integral equations of the second kind, Mediterr. J. Math., 15 (2018), Paper No. 113. doi: 10.1007/s00009-018-1149-1.  Google Scholar

[18]

B. JinY. Yan and Z. Zhou, Numerical approximation of stochastic time-fractional diffusion, ESAIM Math. Model. Numer. Anal., 53 (2019), 1245-1268.  doi: 10.1051/m2an/2019025.  Google Scholar

[19]

M. Kamrani, Numerical solution of stochastic fractional differential equations, Numer. Algorithms, 68 (2015), 81-93.  doi: 10.1007/s11075-014-9839-7.  Google Scholar

[20]

M. Kamrani, Convergence of Galerkin method for the solution of stochastic fractional integro differential equations, Optik Int. J. Light Electron Opt., 127 (2016), 10049-10057.   Google Scholar

[21]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[22] V. Lakshmikantham and M. Rama Mohana Rao, Theory of Integro-Differential Equations, CRC Press, 1995.   Google Scholar
[23]

J. J. Levin and J. A. Nohel, On a system of integrodifferential equations occuring in reactor dynamics, J. Math. Mech., 9 (1960), 347-368.  doi: 10.1512/iumj.1960.9.59020.  Google Scholar

[24]

L. LiJ.-G. Liu and J. Lu, Fractional stochastic differential equations satisfying fluctuation-dissipation theorem, J. Stat. Phys., 169 (2017), 316-339.  doi: 10.1007/s10955-017-1866-z.  Google Scholar

[25]

Y. Li and Y. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differential Equations, 266 (2019), 3514-3558.  doi: 10.1016/j.jde.2018.09.009.  Google Scholar

[26]

H. LiangZ. Yang and J. Gao, Strong superconvergence of the Euler–Maruyama method for linear stochastic Volterra integral equations, J. Comput. Appl. Math., 317 (2017), 447-457.  doi: 10.1016/j.cam.2016.11.005.  Google Scholar

[27]

M. Maleki and M. Tavassoli Kajani, Numerical approximations for Volterra's population growth model with fractional order via a multi-domain pseudospectral method, Appl. Math. Model., 39 (2015), 4300-4308.  doi: 10.1016/j.apm.2014.12.045.  Google Scholar

[28]

X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2008. doi: 10.1533/9780857099402.  Google Scholar

[29]

S. A. McKinley and H. D. Nguyen, Anomalous diffusion and the generalized Langevin equation, SIAM J. Math. Anal., 50 (2018), 5119-5160.  doi: 10.1137/17M115517X.  Google Scholar

[30]

F. Mirzaee and N. Samadyar, Application of orthonormal Bernstein polynomials to construct a efficient scheme for solving fractional stochastic integro-differential equation, Optik Int. J. Light Electron Opt., 132 (2017), 262-273.  doi: 10.1016/j.ijleo.2016.12.029.  Google Scholar

[31]

F. Mirzaee and N. Samadyar, On the numerical solution of fractional stochastic integro-differential equations via meshless discrete collocation method based on radial basis functions, Eng. Anal. Bound. Elem., 100 (2019), 246-255.  doi: 10.1016/j.enganabound.2018.05.006.  Google Scholar

[32]

F. Mohammadi, Efficient Galerkin solution of stochastic fractional differential equations using second kind Chebyshev wavelets, Bol. Soc. Parana. Mat., 35 (2017), 195-215.  doi: 10.5269/bspm.v35i1.28262.  Google Scholar

[33]

S. M. Momani, Local and global existence theorems on fractional integro-differential equations, J. Fract. Calc., 18 (2000), 81-86.   Google Scholar

[34]

J.-C. Pedjeu and G. S. Ladde, Stochastic fractional differential equations: Modeling, method and analysis, Chaos Solitons Fractals, 45 (2012), 279-293.  doi: 10.1016/j.chaos.2011.12.009.  Google Scholar

[35]

A. N. V. Rao and C. P. Tsokos, On the existence, uniqueness, and stability behavior of a random solution to a nonlinear perturbed stochastic integro-differential equation, Information and Control, 27 (1975), 61-74.  doi: 10.1016/S0019-9958(75)90074-1.  Google Scholar

[36]

F. M. Scudo, Vito Volterra and theoretical ecology, Theoret. Population Biol., 2 (1971), 1-23.  doi: 10.1016/0040-5809(71)90002-5.  Google Scholar

[37]

D. T. SonP. T. HuongP. E. Kloeden and H. T. Tuan, Asymptotic separation between solutions of Caputo fractional stochastic differential equations, Stoch. Anal. Appl., 36 (2018), 654-664.  doi: 10.1080/07362994.2018.1440243.  Google Scholar

[38]

Z. TaheriS. Javadi and E. Babolian, Numerical solution of stochastic fractional integro-differential equation by the spectral collocation method, J. Comput. Appl. Math., 321 (2017), 336-347.  doi: 10.1016/j.cam.2017.02.027.  Google Scholar

[39]

V. E. Tarasov, Fractional integro-differential equations for electromagnetic waves in dielectric media, Theoret. and Math. Phys., 158 (2009), 355-359.  doi: 10.1007/s11232-009-0029-z.  Google Scholar

[40]

K. G. TeBeest, Classroom Note: Numerical and analytical solutions of Volterra's population model, SIAM Rev., 39 (1997), 484-493.  doi: 10.1137/S0036144595294850.  Google Scholar

[41]

H. T. Tuan, On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1749-1762.  doi: 10.3934/dcdsb.2020318.  Google Scholar

[42]

Y. WangJ. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Anal., 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020.  Google Scholar

[43]

Z. Wang, Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients, Statist. Probab. Lett., 78 (2008), 1062-1071.  doi: 10.1016/j.spl.2007.10.007.  Google Scholar

[44]

Z. Yang, H. Yang and Z. Yao, Strong convergence analysis for Volterra integro-differential equations with fractional Brownian motions, J. Comput. Appl. Math., 383 (2021), 113156. doi: 10.1016/j.cam.2020.113156.  Google Scholar

[45]

H. YeJ. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.  Google Scholar

[46]

Ş. Yüzbaşı, A numerical approximation for Volterra's population growth model with fractional order, Appl. Math. Model., 37 (2013), 3216-3227.  doi: 10.1016/j.apm.2012.07.041.  Google Scholar

[47]

G. Zhang and R. Zhu, Runge–Kutta convolution quadrature methods with convergence and stability analysis for nonlinear singular fractional integro-differential equations, Commun. Nonlinear Sci. Numer. Simul., 84 (2020), 105132. doi: 10.1016/j.cnsns.2019.105132.  Google Scholar

[48]

G. Zou, Numerical solutions to time-fractional stochastic partial differential equations, Numer. Algorithms, 82 (2019), 553-571.  doi: 10.1007/s11075-018-0613-0.  Google Scholar

Figure 1.  The mean square errors of the EM scheme (11) for Example 5.1
Figure 2.  The mean square errors of the EM scheme (11) for Example 5.2
[1]

Wei Mao, Liangjian Hu, Xuerong Mao. Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 587-613. doi: 10.3934/dcdsb.2018198

[2]

Weiyin Fei, Liangjian Hu, Xuerong Mao, Dengfeng Xia. Advances in the truncated Euler–Maruyama method for stochastic differential delay equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2081-2100. doi: 10.3934/cpaa.2020092

[3]

Can Li, Weihua Deng, Lijing Zhao. Well-posedness and numerical algorithm for the tempered fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1989-2015. doi: 10.3934/dcdsb.2019026

[4]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809

[5]

Fuke Wu, Xuerong Mao, Peter E. Kloeden. Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 885-903. doi: 10.3934/dcds.2013.33.885

[6]

Caochuan Ma, Zaihong Jiang, Renhui Wan. Local well-posedness for the tropical climate model with fractional velocity diffusion. Kinetic & Related Models, 2016, 9 (3) : 551-570. doi: 10.3934/krm.2016006

[7]

Luciano Abadías, Carlos Lizama, Pedro J. Miana, M. Pilar Velasco. On well-posedness of vector-valued fractional differential-difference equations. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2679-2708. doi: 10.3934/dcds.2019112

[8]

Yong Zhou, Jishan Fan. Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 813-818. doi: 10.3934/cpaa.2010.9.813

[9]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195

[10]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635

[11]

Jean-Daniel Djida, Arran Fernandez, Iván Area. Well-posedness results for fractional semi-linear wave equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 569-597. doi: 10.3934/dcdsb.2019255

[12]

Junxiong Jia, Jigen Peng, Kexue Li. Well-posedness of abstract distributed-order fractional diffusion equations. Communications on Pure & Applied Analysis, 2014, 13 (2) : 605-621. doi: 10.3934/cpaa.2014.13.605

[13]

Hongjun Gao, Chengfeng Sun. Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3053-3073. doi: 10.3934/dcdsb.2016087

[14]

Tomás Caraballo, P.E. Kloeden, Pedro Marín-Rubio. Numerical and finite delay approximations of attractors for logistic differential-integral equations with infinite delay. Discrete & Continuous Dynamical Systems, 2007, 19 (1) : 177-196. doi: 10.3934/dcds.2007.19.177

[15]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007

[16]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382

[17]

Elaine Cozzi, James P. Kelliher. Well-posedness of the 2D Euler equations when velocity grows at infinity. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2361-2392. doi: 10.3934/dcds.2019100

[18]

Jiali Lian. Global well-posedness of the free-interface incompressible Euler equations with damping. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2061-2087. doi: 10.3934/dcds.2020106

[19]

Daniel Coutand, Steve Shkoller. A simple proof of well-posedness for the free-surface incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 429-449. doi: 10.3934/dcdss.2010.3.429

[20]

Xumin Gu. Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition. Communications on Pure & Applied Analysis, 2019, 18 (2) : 569-602. doi: 10.3934/cpaa.2019029

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (157)
  • HTML views (226)
  • Cited by (0)

Other articles
by authors

[Back to Top]