July  2021, 14(7): 2297-2309. doi: 10.3934/dcdss.2021053

New class of volterra integro-differential equations with fractal-fractional operators: Existence, uniqueness and numerical scheme

Siirt University, Department of Mathematics Education, Siirt, Turkey

Received  July 2019 Revised  December 2020 Published  July 2021 Early access  May 2021

In this paper, we introduce a new fractional integro-differential equation involving newly introduced differential and integral operators so-called fractal-fractional derivatives and integrals. We present a numerical scheme that is convenient for obtaining solution of such equations. We give the general conditions for the existence and uniqueness of the solution of the considered equation using Banach fixed-point theorem. Both the suggested new equation and new numerical scheme will considerably contribute for our readers in theory and applications.

Citation: Seda İğret Araz. New class of volterra integro-differential equations with fractal-fractional operators: Existence, uniqueness and numerical scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2297-2309. doi: 10.3934/dcdss.2021053
References:
[1]

K. A. AbroM. M. RashidiI. KhanI. A. Abro and A. Tassaddiq, Analysis of stokes' second problem for nanofluids using modern approach of Atangana-Baleanu fractional derivative, Journal of Nanofluids, 7 (2018), 738-747.  doi: 10.1166/jon.2018.1486.

[2]

J. F. G. Aguilar and A. Atangana, New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132 12017, 13.

[3]

H. M. Ahmed, Controllability for sobolev type fractional integro-differential systems in a banach space, Advance in Difference Equations, 2012 (2012), 167. doi: 10.1186/1687-1847-2012-167.

[4]

H. M. Ahmed, M. M. El-Borai, H. M. Ei-Owaidy and A. S. Ghanem, Existence solution and controllability of Sobolev type delay nonlinear fractional integro-differential system, Mathematics, 7 (2019), 79. doi: 10.3390/math7010079.

[5]

E. Atangana, New insight kinetic modeling: Models above classical chemical mechanic, Chaos, Solitons & Fractals, 128 (2019), 16-24.  doi: 10.1016/j.chaos.2019.07.013.

[6]

A. Atangana and J. F. Gómez-Aguilar, Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws, Chaos, Solitons & Fractals, 102 (2017), 285-294.  doi: 10.1016/j.chaos.2017.03.022.

[7]

A. Atangana and S. İğret Araz, Fractional stochastic modelling illustration with modified Chua attractor, The European Physical Journal Plus, 134 (2019), 160. doi: 10.1140/epjp/i2019-12565-6.

[8]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.

[9]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.

[10]

A. Atangana and S. Jain, The role of power decay, exponential decay and Mittag-Leffler function's waiting time distribution: Application of cancer spread, Physica A, 512 (2018), 330-351.  doi: 10.1016/j.physa.2018.08.033.

[11]

A. Atangana and S. İğret Araz, Analysis of a new partial integro-differential equation with mixed fractional operators, Chaos, Solitons & Fractals, 127 (2019), 257-271.  doi: 10.1016/j.chaos.2019.06.005.

[12]

A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos, Solitons Fractals, 102 (2017), 396-406.  doi: 10.1016/j.chaos.2017.04.027.

[13]

K. BalachandranE. R. Anandhi and J. P. Dauer, Boundary controllability of Sobolev-type abstract nonlinear integro-differential systems, Journal of Mathematical Analysis and Applications, 277 (2003), 446-464.  doi: 10.1016/S0022-247X(02)00522-X.

[14]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 73-85. 

[15]

B. Cuahutenango-Barro, M. A. Taneco-Hernández, J. F. Gó mez-Aguilar, On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel, Chaos, Solitons & Fractals, 115 (2018), 283-299. doi: 10.1016/j.chaos.2018.09.002.

[16]

E. F. D. Goufo, A biomathematical view on the fractional dynamics of cellulose degradation, Fractional Calculus and Applied Analysis, 18 (2015), 554-564.  doi: 10.1515/fca-2015-0034.

[17]

J. Hristov, Derivatives with non-singular kernels from the Caputo–Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, Frontiers in fractional calculus, Sharjah: Bentham Science Publishers, (2017), 269–341. doi: 10.2174/9781681085999118010013.

[18]

S. İğret Araz, Analysis of a Covid-19 model: Optimal control, stability and simulations, Alexandria Engineering Journal, 60 (2020), 647-658.  doi: 10.1016/j.aej.2020.09.058.

[19]

S. İğret Araz, Numerical analysis of a new Volterra integro-differential equation involving fractal-fractional operators, Chaos, Solitons & Fractals, 130 (2020), 109396. doi: 10.1016/j.chaos.2019.109396.

[20]

K. Kavitha, V. Vijayakumar and R. Udhayakumar, Results on controllability of Hilfer fractional neutral differential equations with infinite delay via measures of noncompactness, Chaos, Solitons & Fractals, 139 (2020), 110035. doi: 10.1016/j.chaos.2020.110035.

[21]

A. M. S. Mahdy, Numerical studies for solving fractional integro-differential equations, Journal of Ocean Engineering and Science, 3 (2018), 127-132.  doi: 10.1016/j.joes.2018.05.004.

[22]

T. Mekkaoui and A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, The European Physical Journal Plus, 132 (2017), 444.

[23]

F. Mohammadi and C. Cattani, A generalized fractional-order Legendre wavelet Tau method for solving fractional differential equations, Journal of Computational and Applied Mathematics, 339 (2018), 306-316.  doi: 10.1016/j.cam.2017.09.031.

[24]

K. M. Owolabi and Z. Hammouch, Mathematical modeling and analysis of two-variable system with noninteger-order derivative, Chaos, 29 (2019), 013145, 15 pp. doi: 10.1063/1.5086909.

[25]

C. Ravichandran, K. Logeswari, S. K. Panda and K. S. Nisar, On new approach of fractional derivative by Mittag-Leffler kernel to neutral integro-differential systems with impulsive conditions, Chaos, Solitons & Fractals, 139 (2020), 110012, 9 pp. doi: 10.1016/j.chaos.2020.110012.

[26]

R. SakthivelQ. H. Choi and S. M. Anthoni, Controllability of nonlinear neutral evolution integro-differential systems, Journal of Mathematical Analysis and Applications, 275 (2002), 402-417.  doi: 10.1016/S0022-247X(02)00375-X.

[27]

S. T. Sutar and K. D. Kucche, On fractional volterra integro differential equations with fractional integrable impulses, Math Model Anal, 24 (2019), 457-477.  doi: 10.3846/mma.2019.028.

[28]

S. Tate, V. V. Kharat and H. T. Dinde, On Nonlinear Fractional Integro–Differential Equations with Positive Constant Coefficient, Mediterranean Journal of Mathematics, 16 (2019), 41. doi: 10.1007/s00009-019-1325-y.

show all references

References:
[1]

K. A. AbroM. M. RashidiI. KhanI. A. Abro and A. Tassaddiq, Analysis of stokes' second problem for nanofluids using modern approach of Atangana-Baleanu fractional derivative, Journal of Nanofluids, 7 (2018), 738-747.  doi: 10.1166/jon.2018.1486.

[2]

J. F. G. Aguilar and A. Atangana, New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132 12017, 13.

[3]

H. M. Ahmed, Controllability for sobolev type fractional integro-differential systems in a banach space, Advance in Difference Equations, 2012 (2012), 167. doi: 10.1186/1687-1847-2012-167.

[4]

H. M. Ahmed, M. M. El-Borai, H. M. Ei-Owaidy and A. S. Ghanem, Existence solution and controllability of Sobolev type delay nonlinear fractional integro-differential system, Mathematics, 7 (2019), 79. doi: 10.3390/math7010079.

[5]

E. Atangana, New insight kinetic modeling: Models above classical chemical mechanic, Chaos, Solitons & Fractals, 128 (2019), 16-24.  doi: 10.1016/j.chaos.2019.07.013.

[6]

A. Atangana and J. F. Gómez-Aguilar, Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws, Chaos, Solitons & Fractals, 102 (2017), 285-294.  doi: 10.1016/j.chaos.2017.03.022.

[7]

A. Atangana and S. İğret Araz, Fractional stochastic modelling illustration with modified Chua attractor, The European Physical Journal Plus, 134 (2019), 160. doi: 10.1140/epjp/i2019-12565-6.

[8]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.

[9]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.

[10]

A. Atangana and S. Jain, The role of power decay, exponential decay and Mittag-Leffler function's waiting time distribution: Application of cancer spread, Physica A, 512 (2018), 330-351.  doi: 10.1016/j.physa.2018.08.033.

[11]

A. Atangana and S. İğret Araz, Analysis of a new partial integro-differential equation with mixed fractional operators, Chaos, Solitons & Fractals, 127 (2019), 257-271.  doi: 10.1016/j.chaos.2019.06.005.

[12]

A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos, Solitons Fractals, 102 (2017), 396-406.  doi: 10.1016/j.chaos.2017.04.027.

[13]

K. BalachandranE. R. Anandhi and J. P. Dauer, Boundary controllability of Sobolev-type abstract nonlinear integro-differential systems, Journal of Mathematical Analysis and Applications, 277 (2003), 446-464.  doi: 10.1016/S0022-247X(02)00522-X.

[14]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 73-85. 

[15]

B. Cuahutenango-Barro, M. A. Taneco-Hernández, J. F. Gó mez-Aguilar, On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel, Chaos, Solitons & Fractals, 115 (2018), 283-299. doi: 10.1016/j.chaos.2018.09.002.

[16]

E. F. D. Goufo, A biomathematical view on the fractional dynamics of cellulose degradation, Fractional Calculus and Applied Analysis, 18 (2015), 554-564.  doi: 10.1515/fca-2015-0034.

[17]

J. Hristov, Derivatives with non-singular kernels from the Caputo–Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, Frontiers in fractional calculus, Sharjah: Bentham Science Publishers, (2017), 269–341. doi: 10.2174/9781681085999118010013.

[18]

S. İğret Araz, Analysis of a Covid-19 model: Optimal control, stability and simulations, Alexandria Engineering Journal, 60 (2020), 647-658.  doi: 10.1016/j.aej.2020.09.058.

[19]

S. İğret Araz, Numerical analysis of a new Volterra integro-differential equation involving fractal-fractional operators, Chaos, Solitons & Fractals, 130 (2020), 109396. doi: 10.1016/j.chaos.2019.109396.

[20]

K. Kavitha, V. Vijayakumar and R. Udhayakumar, Results on controllability of Hilfer fractional neutral differential equations with infinite delay via measures of noncompactness, Chaos, Solitons & Fractals, 139 (2020), 110035. doi: 10.1016/j.chaos.2020.110035.

[21]

A. M. S. Mahdy, Numerical studies for solving fractional integro-differential equations, Journal of Ocean Engineering and Science, 3 (2018), 127-132.  doi: 10.1016/j.joes.2018.05.004.

[22]

T. Mekkaoui and A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, The European Physical Journal Plus, 132 (2017), 444.

[23]

F. Mohammadi and C. Cattani, A generalized fractional-order Legendre wavelet Tau method for solving fractional differential equations, Journal of Computational and Applied Mathematics, 339 (2018), 306-316.  doi: 10.1016/j.cam.2017.09.031.

[24]

K. M. Owolabi and Z. Hammouch, Mathematical modeling and analysis of two-variable system with noninteger-order derivative, Chaos, 29 (2019), 013145, 15 pp. doi: 10.1063/1.5086909.

[25]

C. Ravichandran, K. Logeswari, S. K. Panda and K. S. Nisar, On new approach of fractional derivative by Mittag-Leffler kernel to neutral integro-differential systems with impulsive conditions, Chaos, Solitons & Fractals, 139 (2020), 110012, 9 pp. doi: 10.1016/j.chaos.2020.110012.

[26]

R. SakthivelQ. H. Choi and S. M. Anthoni, Controllability of nonlinear neutral evolution integro-differential systems, Journal of Mathematical Analysis and Applications, 275 (2002), 402-417.  doi: 10.1016/S0022-247X(02)00375-X.

[27]

S. T. Sutar and K. D. Kucche, On fractional volterra integro differential equations with fractional integrable impulses, Math Model Anal, 24 (2019), 457-477.  doi: 10.3846/mma.2019.028.

[28]

S. Tate, V. V. Kharat and H. T. Dinde, On Nonlinear Fractional Integro–Differential Equations with Positive Constant Coefficient, Mediterranean Journal of Mathematics, 16 (2019), 41. doi: 10.1007/s00009-019-1325-y.

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