    September  2018, 23(7): 2679-2694. doi: 10.3934/dcdsb.2017188

## Effect of perturbation in the numerical solution of fractional differential equations

 1 Dipartimento di Matematica Università degli Studi di Bari Via E. Orabona 4,70125 Bari, Italy 2 Dipartimento di Matematica e Applicazioni Università degli Studi di Napoli "Federico Ⅱ" Via Cintia, I-80126 Napoli, Italy 3 C.N.R. National Research Council of Italy Institute for Computational Application "Mauro Picone" Via P. Castellino, 111 -80131 Napoli -Italy

* Corresponding author: E. Messina

Received  October 2016 Revised  May 2017 Published  September 2018 Early access  July 2017

Fund Project: This work is supported under the INdAM-GNCS project 2016 "Metodi numerici per operatori non-locali nella simulazione di fenomeni complessi".

The equations describing engineering and real-life models are usually derived in an approximated way. Thus, in most cases it is necessary to deal with equations containing some kind of perturbation. In this paper we consider fractional differential equations and study the effects on the continuous and numerical solution, of perturbations on the given function, over long-time intervals. Some bounds on the global error are also determined.

Citation: Roberto Garrappa, Eleonora Messina, Antonia Vecchio. Effect of perturbation in the numerical solution of fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2679-2694. doi: 10.3934/dcdsb.2017188
##### References:
  T. M. Atanacković, S. Pilipović, B. Stanković and D. Zorica, Fractional Calculus with Applications in Mechanics Mechanical Engineering and Solid Mechanics Series. ISTE, London; John Wiley & Sons, Inc. , Hoboken, NJ, 2014. Wave propagation, impact and variational principles. Google Scholar  L. C. Becker, Resolvents and solutions of weakly singular linear Volterra integral equations, Nonlinear Anal., 74 (2011), 1892-1912.  doi: 10.1016/j.na.2010.10.060.  Google Scholar  H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations volume 15 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.  Google Scholar  H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations volume 3 of CWI Monographs, North-Holland Publishing Co. , Amsterdam, 1986. Google Scholar  E. Capelas de Oliveira and J. A. T. Machado, A review of definitions for fractional derivatives and integral Math. Probl. Eng. , 2014 (2014), Art. ID 238459, 6 pp. doi: 10.1155/2014/238459.  Google Scholar  R. Caponetto, G. Dongola, L. Fortuna and I. Petráš, Fractional Order Systems: Modeling and Control Applications volume 72 of Series on Nonlinear Science, Series A, World Scientific, Singapore, 2010. Google Scholar  M. Concezzi, R. Garra and R. Spigler, Fractional relaxation and fractional oscillation models involving Erdélyi-Kober integrals, Fract. Calc. Appl. Anal., 18 (2015), 1212-1231.  doi: 10.1515/fca-2015-0070.  Google Scholar  F. R. de Hoog and R. S. Anderssen, Kernel perturbations for a class of second-kind convolution Volterra equations with non-negative kernels, Appl. Math. Lett., 25 (2012), 1222-1225.  doi: 10.1016/j.aml.2012.02.058.  Google Scholar  K. Diethelm, The Analysis of Fractional Differential Equations Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar  K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3-22.  doi: 10.1023/A:1016592219341.  Google Scholar  K. Diethelm, N. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52.  doi: 10.1023/B:NUMA.0000027736.85078.be.  Google Scholar  J. Dixon, On the order of the error in discretization methods for weakly singular second kind Volterra integral equations with nonsmooth solutions, BIT, 25 (1985), 624-634.  doi: 10.1007/BF01936141.  Google Scholar  R. Garra, R. Gorenflo, F. Polito and Ž. Tomovski, Hilfer-Prabhakar derivatives and some applications, Appl. Math. Comput., 242 (2014), 576-589.  doi: 10.1016/j.amc.2014.05.129.  Google Scholar  R. Garrappa, Trapezoidal methods for fractional differential equations: Theoretical and computational aspects, Math. Comput. Simulation, 110 (2015), 96-112.  doi: 10.1016/j.matcom.2013.09.012.  Google Scholar  R. Garrappa, Grünwald-Letnikov operators for fractional relaxation in Havriliak-Negami models, Commun. Nonlinear Sci. Numer. Simul., 38 (2016), 178-191.  doi: 10.1016/j.cnsns.2016.02.015.  Google Scholar  R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag–Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics. Springer, New York, 2014. Google Scholar  A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations volume 204 of {North-Holland Mathematics Studies}, Elsevier Science B. V. , Amsterdam, 2006. Google Scholar  F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity Imperial College Press, London, 2010. An introduction to mathematical models. doi: 10.1142/9781848163300.  Google Scholar  G. Pagnini, Erdélyi-Kober fractional diffusion, Fract. Calc. Appl. Anal., 15 (2012), 117-127.  doi: 10.2478/s13540-012-0008-1.  Google Scholar  I. Podlubny, Fractional Differential Equations volume 198 of Mathematics in Science and Engineering, Academic Press, Inc. , San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Google Scholar  V. E. Tarasov, Fractional Dynamics Nonlinear Physical Science. Springer, Heidelberg; Higher Education Press, Beijing, 2010. Applications of fractional calculus to dynamics of particles, fields and media. doi: 10.1007/978-3-642-14003-7.  Google Scholar  G. Teschl, Ordinary Differential Equations and Dynamical Systems volume 140 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 2012. doi: 10.1090/gsm/140.  Google Scholar  V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Volume Ⅱ Nonlinear Physical Science. Higher Education Press, Beijing; Springer, Heidelberg, 2013. Applications. doi: 10.1007/978-3-642-33911-0.  Google Scholar  A. Young, Approximate product-integration, Proc. Roy. Soc. London Ser. A., 224 (1954), 552-561.  doi: 10.1098/rspa.1954.0179.  Google Scholar

show all references

##### References:
  T. M. Atanacković, S. Pilipović, B. Stanković and D. Zorica, Fractional Calculus with Applications in Mechanics Mechanical Engineering and Solid Mechanics Series. ISTE, London; John Wiley & Sons, Inc. , Hoboken, NJ, 2014. Wave propagation, impact and variational principles. Google Scholar  L. C. Becker, Resolvents and solutions of weakly singular linear Volterra integral equations, Nonlinear Anal., 74 (2011), 1892-1912.  doi: 10.1016/j.na.2010.10.060.  Google Scholar  H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations volume 15 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.  Google Scholar  H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations volume 3 of CWI Monographs, North-Holland Publishing Co. , Amsterdam, 1986. Google Scholar  E. Capelas de Oliveira and J. A. T. Machado, A review of definitions for fractional derivatives and integral Math. Probl. Eng. , 2014 (2014), Art. ID 238459, 6 pp. doi: 10.1155/2014/238459.  Google Scholar  R. Caponetto, G. Dongola, L. Fortuna and I. Petráš, Fractional Order Systems: Modeling and Control Applications volume 72 of Series on Nonlinear Science, Series A, World Scientific, Singapore, 2010. Google Scholar  M. Concezzi, R. Garra and R. Spigler, Fractional relaxation and fractional oscillation models involving Erdélyi-Kober integrals, Fract. Calc. Appl. Anal., 18 (2015), 1212-1231.  doi: 10.1515/fca-2015-0070.  Google Scholar  F. R. de Hoog and R. S. Anderssen, Kernel perturbations for a class of second-kind convolution Volterra equations with non-negative kernels, Appl. Math. Lett., 25 (2012), 1222-1225.  doi: 10.1016/j.aml.2012.02.058.  Google Scholar  K. Diethelm, The Analysis of Fractional Differential Equations Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar  K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3-22.  doi: 10.1023/A:1016592219341.  Google Scholar  K. Diethelm, N. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52.  doi: 10.1023/B:NUMA.0000027736.85078.be.  Google Scholar  J. Dixon, On the order of the error in discretization methods for weakly singular second kind Volterra integral equations with nonsmooth solutions, BIT, 25 (1985), 624-634.  doi: 10.1007/BF01936141.  Google Scholar  R. Garra, R. Gorenflo, F. Polito and Ž. Tomovski, Hilfer-Prabhakar derivatives and some applications, Appl. Math. Comput., 242 (2014), 576-589.  doi: 10.1016/j.amc.2014.05.129.  Google Scholar  R. Garrappa, Trapezoidal methods for fractional differential equations: Theoretical and computational aspects, Math. Comput. Simulation, 110 (2015), 96-112.  doi: 10.1016/j.matcom.2013.09.012.  Google Scholar  R. Garrappa, Grünwald-Letnikov operators for fractional relaxation in Havriliak-Negami models, Commun. Nonlinear Sci. Numer. Simul., 38 (2016), 178-191.  doi: 10.1016/j.cnsns.2016.02.015.  Google Scholar  R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag–Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics. Springer, New York, 2014. Google Scholar  A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations volume 204 of {North-Holland Mathematics Studies}, Elsevier Science B. V. , Amsterdam, 2006. Google Scholar  F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity Imperial College Press, London, 2010. An introduction to mathematical models. doi: 10.1142/9781848163300.  Google Scholar  G. Pagnini, Erdélyi-Kober fractional diffusion, Fract. Calc. Appl. Anal., 15 (2012), 117-127.  doi: 10.2478/s13540-012-0008-1.  Google Scholar  I. Podlubny, Fractional Differential Equations volume 198 of Mathematics in Science and Engineering, Academic Press, Inc. , San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Google Scholar  V. E. Tarasov, Fractional Dynamics Nonlinear Physical Science. Springer, Heidelberg; Higher Education Press, Beijing, 2010. Applications of fractional calculus to dynamics of particles, fields and media. doi: 10.1007/978-3-642-14003-7.  Google Scholar  G. Teschl, Ordinary Differential Equations and Dynamical Systems volume 140 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 2012. doi: 10.1090/gsm/140.  Google Scholar  V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Volume Ⅱ Nonlinear Physical Science. Higher Education Press, Beijing; Springer, Heidelberg, 2013. Applications. doi: 10.1007/978-3-642-33911-0.  Google Scholar  A. Young, Approximate product-integration, Proc. Roy. Soc. London Ser. A., 224 (1954), 552-561.  doi: 10.1098/rspa.1954.0179.  Google Scholar Plot of $t^{\alpha} E_{1, \alpha+1}(-\lambda t)$ when $\lambda=1$ (left plot) and $\lambda=10$ (right plot) Comparison of the difference $\delta y(t)$ between the exact and perturbed solutions and the bound (9)
Relationship between global and truncation errors with respect to Theorem 4.2 for $\alpha=0.4$ (here $\eta\approx0.320$, $K=1.0$, $A\approx1.578$)
 $h$ $\eta + h^{\alpha}A$ $\displaystyle\sup_{n=0, N}|e_n|$ $\displaystyle\frac{\sup_{n=0, N} |T_n(h)|}{1 - (\eta + h^{\alpha}A)}$ $2^{-4}$ $0.840$ $2.5692(-2)$ $1.2938(-1)$ $2^{-5}$ $0.714$ $1.3885(-2)$ $4.0188(-2)$ $2^{-6}$ $0.618$ $7.4352(-3)$ $1.6738(-2)$ $2^{-7}$ $0.546$ $4.0198(-3)$ $7.8711(-3)$ $2^{-8}$ $0.491$ $2.1891(-3)$ $3.9330(-3)$ $2^{-9}$ $0.450$ $1.1840(-3)$ $2.0099(-3)$ $2^{-10}$ $0.418$ $6.1270(-4)$ $1.0002(-3)$
 $h$ $\eta + h^{\alpha}A$ $\displaystyle\sup_{n=0, N}|e_n|$ $\displaystyle\frac{\sup_{n=0, N} |T_n(h)|}{1 - (\eta + h^{\alpha}A)}$ $2^{-4}$ $0.840$ $2.5692(-2)$ $1.2938(-1)$ $2^{-5}$ $0.714$ $1.3885(-2)$ $4.0188(-2)$ $2^{-6}$ $0.618$ $7.4352(-3)$ $1.6738(-2)$ $2^{-7}$ $0.546$ $4.0198(-3)$ $7.8711(-3)$ $2^{-8}$ $0.491$ $2.1891(-3)$ $3.9330(-3)$ $2^{-9}$ $0.450$ $1.1840(-3)$ $2.0099(-3)$ $2^{-10}$ $0.418$ $6.1270(-4)$ $1.0002(-3)$
Relationship between global and truncation errors with respect to Theorem 4.2 for $\alpha=0.6$ (here $\eta\approx0.240$, $K=1.0$, $A\approx1.791$)
 $h$ $\eta + h^{\alpha}A$ $\displaystyle\sup_{n=0, N}|e_n|$ $\displaystyle\frac{\sup_{n=0, N} |T_n(h)|}{1 - (\eta + h^{\alpha}A)}$ $2^{-3}$ $0.755$ $8.5380(-3)$ $3.1044(-2)$ $2^{-4}$ $0.580$ $4.5485(-3)$ $9.7699(-3)$ $2^{-5}$ $0.464$ $2.0622(-3)$ $3.5605(-3)$ $2^{-6}$ $0.388$ $8.9133(-4)$ $1.3787(-3)$ $2^{-7}$ $0.338$ $3.8127(-4)$ $5.5464(-4)$ $2^{-8}$ $0.305$ $1.6256(-4)$ $2.2801(-4)$ $2^{-9}$ $0.283$ $6.8446(-5)$ $9.3858(-5)$ $2^{-10}$ $0.268$ $2.7444(-5)$ $3.7099(-5)$
 $h$ $\eta + h^{\alpha}A$ $\displaystyle\sup_{n=0, N}|e_n|$ $\displaystyle\frac{\sup_{n=0, N} |T_n(h)|}{1 - (\eta + h^{\alpha}A)}$ $2^{-3}$ $0.755$ $8.5380(-3)$ $3.1044(-2)$ $2^{-4}$ $0.580$ $4.5485(-3)$ $9.7699(-3)$ $2^{-5}$ $0.464$ $2.0622(-3)$ $3.5605(-3)$ $2^{-6}$ $0.388$ $8.9133(-4)$ $1.3787(-3)$ $2^{-7}$ $0.338$ $3.8127(-4)$ $5.5464(-4)$ $2^{-8}$ $0.305$ $1.6256(-4)$ $2.2801(-4)$ $2^{-9}$ $0.283$ $6.8446(-5)$ $9.3858(-5)$ $2^{-10}$ $0.268$ $2.7444(-5)$ $3.7099(-5)$
Relationship between global and truncation errors with respect to Theorem 4.2 for $\alpha=0.8$ (here $\eta\approx0.203$, $K=1.0$, $A\approx1.933$)
 $h$ $\eta + h^{\alpha}A$ $\displaystyle\sup_{n=0, N}|e_n|$ $\displaystyle\frac{\sup_{n=0, N} |T_n(h)|}{1 - (\eta + h^{\alpha}A)}$ $2^{-2}$ $0.840$ $1.6121(-2)$ $9.4144(-2)$ $2^{-3}$ $0.569$ $3.6570(-3)$ $7.9159(-3)$ $2^{-4}$ $0.413$ $7.7545(-4)$ $1.2719(-3)$ $2^{-5}$ $0.324$ $1.7330(-4)$ $2.4809(-4)$ $2^{-6}$ $0.272$ $6.8838(-5)$ $9.2726(-5)$ $2^{-7}$ $0.243$ $2.4179(-5)$ $3.1554(-5)$ $2^{-8}$ $0.226$ $8.1096(-6)$ $1.0403(-5)$ $2^{-9}$ $0.216$ $2.6392(-6)$ $3.3532(-6)$ $2^{-10}$ $0.210$ $8.1173(-7)$ $1.0258(-6)$
 $h$ $\eta + h^{\alpha}A$ $\displaystyle\sup_{n=0, N}|e_n|$ $\displaystyle\frac{\sup_{n=0, N} |T_n(h)|}{1 - (\eta + h^{\alpha}A)}$ $2^{-2}$ $0.840$ $1.6121(-2)$ $9.4144(-2)$ $2^{-3}$ $0.569$ $3.6570(-3)$ $7.9159(-3)$ $2^{-4}$ $0.413$ $7.7545(-4)$ $1.2719(-3)$ $2^{-5}$ $0.324$ $1.7330(-4)$ $2.4809(-4)$ $2^{-6}$ $0.272$ $6.8838(-5)$ $9.2726(-5)$ $2^{-7}$ $0.243$ $2.4179(-5)$ $3.1554(-5)$ $2^{-8}$ $0.226$ $8.1096(-6)$ $1.0403(-5)$ $2^{-9}$ $0.216$ $2.6392(-6)$ $3.3532(-6)$ $2^{-10}$ $0.210$ $8.1173(-7)$ $1.0258(-6)$
Perturbed problem: relationship between global error and bounds from Theorem 4.5 for $\alpha=0.4$ (here $\tilde{\eta}\approx0.318$, $\tilde{K}=1.0$, $\tilde{A}\approx1.578$)
 $h$ $\tilde{\eta} + h^{\alpha}\tilde{A}$ $\displaystyle\sup_{n=0, N}|\tilde{e}_n|$ $\textrm{Bound (23)}$ $2^{-4}$ 0.838 9.8267(-2) 2.5169(-1) $2^{-5}$ 0.712 9.4799(-2) 1.6371(-1) $2^{-6}$ 0.617 9.3258(-2) 1.4043(-1) $2^{-7}$ 0.544 9.2595(-2) 1.3162(-1) $2^{-8}$ 0.489 9.2311(-2) 1.2770(-1) $2^{-9}$ 0.448 9.2198(-2) 1.2578(-1) $2^{-10}$ 0.416 9.2154(-2) 1.2477(-1)
 $h$ $\tilde{\eta} + h^{\alpha}\tilde{A}$ $\displaystyle\sup_{n=0, N}|\tilde{e}_n|$ $\textrm{Bound (23)}$ $2^{-4}$ 0.838 9.8267(-2) 2.5169(-1) $2^{-5}$ 0.712 9.4799(-2) 1.6371(-1) $2^{-6}$ 0.617 9.3258(-2) 1.4043(-1) $2^{-7}$ 0.544 9.2595(-2) 1.3162(-1) $2^{-8}$ 0.489 9.2311(-2) 1.2770(-1) $2^{-9}$ 0.448 9.2198(-2) 1.2578(-1) $2^{-10}$ 0.416 9.2154(-2) 1.2477(-1)
Perturbed problem: relationship between global error and bounds from Theorem 4.5 for $\alpha=0.6$ (here $\tilde{\eta}\approx0.238$, $\tilde{K}=1.0$, $\tilde{A}\approx1.791$)
 $h$ $\tilde{\eta} + h^{\alpha}\tilde{A}$ $\displaystyle\sup_{n=0, N}|\tilde{e}_n|$ $\textrm{Bound (23)}$ $2^{-3}$ 0.753 6.1846(-2) 1.1423(-1) $2^{-4}$ 0.578 6.3482(-2) 9.3146(-2) $2^{-5}$ 0.462 6.3552(-2) 8.6966(-2) $2^{-6}$ 0.386 6.3352(-2) 8.4792(-2) $2^{-7}$ 0.336 6.3249(-2) 8.3971(-2) $2^{-8}$ 0.303 6.3206(-2) 8.3645(-2) $2^{-9}$ 0.281 6.3189(-2) 8.3511(-2) $2^{-10}$ 0.266 6.3183(-2) 8.3455(-2)
 $h$ $\tilde{\eta} + h^{\alpha}\tilde{A}$ $\displaystyle\sup_{n=0, N}|\tilde{e}_n|$ $\textrm{Bound (23)}$ $2^{-3}$ 0.753 6.1846(-2) 1.1423(-1) $2^{-4}$ 0.578 6.3482(-2) 9.3146(-2) $2^{-5}$ 0.462 6.3552(-2) 8.6966(-2) $2^{-6}$ 0.386 6.3352(-2) 8.4792(-2) $2^{-7}$ 0.336 6.3249(-2) 8.3971(-2) $2^{-8}$ 0.303 6.3206(-2) 8.3645(-2) $2^{-9}$ 0.281 6.3189(-2) 8.3511(-2) $2^{-10}$ 0.266 6.3183(-2) 8.3455(-2)
Perturbed problem: relationship between global error and bounds from Theorem 4.5 for $\alpha=0.8$ (here $\tilde{\eta}\approx0.201$, $\tilde{K}=1.0$, $\tilde{A}\approx1.933$)
 $h$ $\tilde{\eta} + h^{\alpha}\tilde{A}$ $\displaystyle\sup_{n=0, N}|\tilde{e}_n|$ $\textrm{Bound (23)}$ $2^{-2}$ 0.839 2.7785(-2) 1.5982(-1) $2^{-3}$ 0.568 4.4303(-2) 7.4435(-2) $2^{-4}$ 0.412 4.8468(-2) 6.7814(-2) $2^{-5}$ 0.322 4.9372(-2) 6.6793(-2) $2^{-6}$ 0.271 4.9568(-2) 6.6638(-2) $2^{-7}$ 0.241 4.9612(-2) 6.6577(-2) $2^{-8}$ 0.224 4.9620(-2) 6.6556(-2) $2^{-9}$ 0.215 4.9622(-2) 6.6549(-2) $2^{-10}$ 0.209 4.9622(-2) 6.6547(-2)
 $h$ $\tilde{\eta} + h^{\alpha}\tilde{A}$ $\displaystyle\sup_{n=0, N}|\tilde{e}_n|$ $\textrm{Bound (23)}$ $2^{-2}$ 0.839 2.7785(-2) 1.5982(-1) $2^{-3}$ 0.568 4.4303(-2) 7.4435(-2) $2^{-4}$ 0.412 4.8468(-2) 6.7814(-2) $2^{-5}$ 0.322 4.9372(-2) 6.6793(-2) $2^{-6}$ 0.271 4.9568(-2) 6.6638(-2) $2^{-7}$ 0.241 4.9612(-2) 6.6577(-2) $2^{-8}$ 0.224 4.9620(-2) 6.6556(-2) $2^{-9}$ 0.215 4.9622(-2) 6.6549(-2) $2^{-10}$ 0.209 4.9622(-2) 6.6547(-2)
  T. Diogo, N. B. Franco, P. Lima. High order product integration methods for a Volterra integral equation with logarithmic singular kernel. Communications on Pure & Applied Analysis, 2004, 3 (2) : 217-235. doi: 10.3934/cpaa.2004.3.217  Wen Li, Song Wang, Volker Rehbock. A 2nd-order one-point numerical integration scheme for fractional ordinary differential equations. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 273-287. doi: 10.3934/naco.2017018  Valentina Casarino, Paolo Ciatti, Silvia Secco. Product structures and fractional integration along curves in the space. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 619-635. doi: 10.3934/dcdss.2013.6.619  Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3529-3539. doi: 10.3934/dcdss.2020432  M. R. Arias, R. Benítez. Properties of solutions for nonlinear Volterra integral equations. Conference Publications, 2003, 2003 (Special) : 42-47. doi: 10.3934/proc.2003.2003.42  Dajana Conte, Raffaele D'Ambrosio, Beatrice Paternoster. On the stability of $\vartheta$-methods for stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2695-2708. doi: 10.3934/dcdsb.2018087  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  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, 2021, 26 (12) : 6483-6510. doi: 10.3934/dcdsb.2021030  Ndolane Sene. Mittag-Leffler input stability of fractional differential equations and its applications. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 867-880. doi: 10.3934/dcdss.2020050  Leonid Shaikhet. Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3651-3657. doi: 10.3934/dcdsb.2020077  Anna Karczewska, Carlos Lizama. On stochastic fractional Volterra equations in Hilbert space. Conference Publications, 2007, 2007 (Special) : 541-550. doi: 10.3934/proc.2007.2007.541  Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075  Paola Buttazzoni, Alessandro Fonda. Periodic perturbations of scalar second order differential equations. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 451-455. doi: 10.3934/dcds.1997.3.451  Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929  Onur Alp İlhan. Solvability of some volterra type integral equations in hilbert space. Conference Publications, 2007, 2007 (Special) : 28-34. doi: 10.3934/proc.2007.2007.28  Tianxiao Wang, Yufeng Shi. Symmetrical solutions of backward stochastic Volterra integral equations and their applications. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 251-274. doi: 10.3934/dcdsb.2010.14.251  Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613  Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib, N. M. A. Nik Long. Effective approximation method for solving linear Fredholm-Volterra integral equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004  Tomás Caraballo, Leonid Shaikhet. Stability of delay evolution equations with stochastic perturbations. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2095-2113. doi: 10.3934/cpaa.2014.13.2095  Hermann Brunner, Chunhua Ou. On the asymptotic stability of Volterra functional equations with vanishing delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 397-406. doi: 10.3934/cpaa.2015.14.397

2020 Impact Factor: 1.327