On Caputo–Katugampola Fractional Stochastic Differential Equation
Abstract
:1. Introduction
- : susceptible class,
- : infected class,
- recovered class,
- : death class,
- is the average number of contacts per person per time t,
- is the recovery rate,
- k is the death rate,
- is the vaccine of suspected population.
1.1. Preliminaries
- When one obtains the left and right Caputo fractional derivatives.
- When , applying L’Hospital’s rule as follows
- A complete normed space is called a Banach space.
- Let X be a norm space and . Then, is called a contraction mapping if there exists a positive real number such that for all ,
- A fixed point of a mapping is a point such that .
1.2. Formulation of the Solution
- 1.
- Let be a probability space with , -Borel σ-algebra and a probability measure . We define to be a class of random variables on with finite pth moments.
- 2.
- The symbol is known as stochastic differential. It has a representation in terms of Itô integral given by , with the following property: Take second moment on the integral and use Itô isometry to obtainFrom the left hand side,
- 3.
- The space is a complete inner product space (Hilbert space).
2. Main Results
2.1. Some Auxilliary Results
2.2. Existence and Uniqueness Result
2.3. Energy Growth-Bound
2.4. Asymptotic Behaviour
3. Examples
- (1)
- To illustrate Theorem 4, we choose , and define by with Lipschitz constant . Then, the equation
- (2)
- Additionally, we give examples to illustrate the result in Theorem 5, which represent plots (graphs) for the upper bound growth of our energy solution. For convenience, we set the positive constants to be equal to one. That is, to have
- In Figure 1, we consider and ;
- Next, in Figure 2, we consider and ;
- Lastly, in Figure 3, we consider and :
We observe that the values of and have little or no significant effect on the growth of the upper bound, however, as time t becomes large, the speed (rate) of growth becomes very sharp and fast. See the figures below.
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Foondun, M.; Liu, W.; Tian, K. Moment bounds for a class of Fractional Stochastic Heat Equations. Ann. Probab. 2017, 45, 2131–2153. [Google Scholar] [CrossRef] [Green Version]
- Foondun, M.; Liu, W.; Omaba, M.E. On Some Properties of a class of Fractional Stochastic Heat Equations. J. Theoret. Probab. 2017, 30, 1310–1333. [Google Scholar]
- Foondun, M.; Nane, E. Asymptotic properties of some space-time fractional stochastic equations. Math. Z. 2017, 287, 493–519. [Google Scholar] [CrossRef] [Green Version]
- Foondun, M.; Liu, W.; Nane, E. Some non-existence results for a class of stochastic partial differential equations. J. Differ. Equ. 2019, 266, 2575–2596. [Google Scholar] [CrossRef] [Green Version]
- Mijena, J.; Nane, E. Space-time fractional stochastic partial differential equations. Stoch. Process. Appl. 2015, 159, 3301–3326. [Google Scholar] [CrossRef]
- Nane, E.; Nwaeze, E.R.; Omaba, M.E. Asymptotic behavior and non-existence of global solution to a class of conformable time-fractional stochastic differential equation. Stat. Probab. Lett. 2020, 163, 108792. [Google Scholar] [CrossRef] [Green Version]
- Omaba, M.E.; Nwaeze, E.R. Moment Bound of Solution to a Class of Conformable Time-Fractional Stochastic Equation. Fractal Fract. 2019, 3, 18. [Google Scholar] [CrossRef] [Green Version]
- Omaba, M.E. On a mild solution to hilfer time-fractional stochastic differetial equation. J. Fract. Calc. Appl. 2021, 12, 1–10. [Google Scholar]
- Omaba, M.E.; Enyi, C.D. Atangana-Baleanu time-fractional stochastic integro-differential equation. Partial. Differ. Equ. Appl. Math. 2021, 4, 100100. [Google Scholar] [CrossRef]
- Sakthivel, R.; Revathi, P.; Ren, Y. Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal. 2013, 81, 70–86. [Google Scholar] [CrossRef]
- Almeida, R.; Malinowska, A.B.; Odzijewicz, T. Fractional differential equations with dependence on the Caputo–Katugampola derivative. J. Comput. Nonlinear Dynam. 2016, 11, 061017. [Google Scholar] [CrossRef] [Green Version]
- Katugampola, U.N. New approach to a generalized fractional integral. Appl. Math. Comput. 2011, 218, 860–865. [Google Scholar] [CrossRef] [Green Version]
- Katugampola, U.N. A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl. 2014, 6, 1–15. [Google Scholar]
- Jarad, F.; Ugurlu, E.; Abdeljawad, T.; Baleanu, D. On a new class of fractional operators. Adv. Differ. Equ. 2017, 2017, 247. [Google Scholar] [CrossRef]
- Jarad, F.; Abdeljawad, T.; Baleanu, D. On the generalized fractional derivatives and their Caputo modification. J. Nonlinear Sci. Appl. 2017, 10, 2607–2619. [Google Scholar] [CrossRef] [Green Version]
- Katugampola, U.N. Existence and uniqueness results for a class of generalized fractional differential equations. arXiv 2014, arXiv:1411.5229. [Google Scholar]
- Basti, B.; Arioua, Y.; Benhamidouche, N. Existence and uniqueness of solutions for nonlinear Katugampola fractional differential equations. J. Math. Appl. 2019, 42, 35–61. [Google Scholar] [CrossRef]
- Basti, B.; Hammami, N.; Berrabah, I.; Nouioua, F.; Djemiat, R.; Benhamidouche, N. Stability Analysis and Existence of solutions for a Modified SIRD Model of COVID-19 with fractional derivatives. Symmetry 2021, 13, 1431. [Google Scholar] [CrossRef]
- Rezapour, S.; Deressa, C.T.; Hussain, A.; Etemad, S.; George, R.; Ahmad, B. A Theoretical Analysis of a Fractional Multi-dimensional system of boundary value problems on the Methylpropane Graph via Fixed Point Technique. Mathematics 2022, 10, 568. [Google Scholar] [CrossRef]
- Rashid, S.; Ashraf, R.; Bonyah, E. On Analytical solution of Time-fractional Biological population model by means of Generalized Integral Transform with their Uniqueness and Convergence Analysis. Adv. Nonlinear Anal. Appl. 2022, 2022, 7021288. [Google Scholar] [CrossRef]
- Sintunavarat, W.; Turab, A. Mathematical analysis of an extended SEIR model of COVID-19 using the ABC-fractional operator. Math. Comput. Simul. 2022, 198, 65–84. [Google Scholar] [CrossRef] [PubMed]
- Gambo, Y.Y.; Jarad, F.; Baleanu, D.; Abdeljawad, T. On Caputo modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2014, 2014, 10. [Google Scholar] [CrossRef] [Green Version]
- Jarad, F.; Abdeljawad, T.; Baleanu, D. Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012, 2012, 142. [Google Scholar] [CrossRef] [Green Version]
- Lipovan, O. A retarded Gronwall-Like Inequality and Its Applications. J. Math. Anal. Appl. 2000, 252, 389–401. [Google Scholar] [CrossRef] [Green Version]
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Omaba, M.E.; Sulaimani, H.A. On Caputo–Katugampola Fractional Stochastic Differential Equation. Mathematics 2022, 10, 2086. https://doi.org/10.3390/math10122086
Omaba ME, Sulaimani HA. On Caputo–Katugampola Fractional Stochastic Differential Equation. Mathematics. 2022; 10(12):2086. https://doi.org/10.3390/math10122086
Chicago/Turabian StyleOmaba, McSylvester Ejighikeme, and Hamdan Al Sulaimani. 2022. "On Caputo–Katugampola Fractional Stochastic Differential Equation" Mathematics 10, no. 12: 2086. https://doi.org/10.3390/math10122086
APA StyleOmaba, M. E., & Sulaimani, H. A. (2022). On Caputo–Katugampola Fractional Stochastic Differential Equation. Mathematics, 10(12), 2086. https://doi.org/10.3390/math10122086