Nonlinear equations with global differential and integral operators: Existence, uniqueness with application to epidemiology
Very recently, the concept of instantaneous change was extended with the aim to accommodate prediction of more complex real world problems that could not be predicted or depicted by the existing rate of change. The extension gave birth to a more general differential operator that to be a derivative...
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| Format: | Article |
| Language: | English |
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Elsevier
2021-01-01
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| Series: | Results in Physics |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2211379720320337 |
| _version_ | 1829520585119498240 |
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| author | Abdon Atangana Seda İğret Araz |
| author_facet | Abdon Atangana Seda İğret Araz |
| author_sort | Abdon Atangana |
| collection | DOAJ |
| description | Very recently, the concept of instantaneous change was extended with the aim to accommodate prediction of more complex real world problems that could not be predicted or depicted by the existing rate of change. The extension gave birth to a more general differential operator that to be a derivative associate to the well-known Riemann-Stieltjes integral. In addition to this, using specific functions, one is able to recover all existing local differential operators defined as rate of change. This extended concept is still at its genesis and more works need to be done to establish a Riemann-Stieltjes calculus. In this paper, we aim to present a detailed analysis of an important class of differential equations called stochastic equations with the new classes of differential operators with the global derivative with integer and non-integer orders. We considered many classes as nonlinear Cauchy problems, then we presented existence and the uniqueness of their solutions using the linear growth and the Lipchitz conditions. We derived numerical solutions for each class and presented the error analysis. To show the applicability of these operators, we considered three epidemiological problems, including the zombie virus spread model, the zika virus spread model and Ebola model. We solved each model using the suggested numerical scheme and presented the numerical solutions for different values of fractional order and the global function gt. Our results showed that, more complex real world problems could be depicted using these classes of differential equations. |
| first_indexed | 2024-12-16T15:00:39Z |
| format | Article |
| id | doaj.art-f13aaee70106404d9fbd8b3d8b7e1af7 |
| institution | Directory Open Access Journal |
| issn | 2211-3797 |
| language | English |
| last_indexed | 2024-12-16T15:00:39Z |
| publishDate | 2021-01-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Results in Physics |
| spelling | doaj.art-f13aaee70106404d9fbd8b3d8b7e1af72022-12-21T22:27:19ZengElsevierResults in Physics2211-37972021-01-0120103593Nonlinear equations with global differential and integral operators: Existence, uniqueness with application to epidemiologyAbdon Atangana0Seda İğret Araz1Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, South Africa; Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, TaiwanDepartment of Mathematic Education, Faculty of Education, Siirt University, Siirt 56100, TurkeyVery recently, the concept of instantaneous change was extended with the aim to accommodate prediction of more complex real world problems that could not be predicted or depicted by the existing rate of change. The extension gave birth to a more general differential operator that to be a derivative associate to the well-known Riemann-Stieltjes integral. In addition to this, using specific functions, one is able to recover all existing local differential operators defined as rate of change. This extended concept is still at its genesis and more works need to be done to establish a Riemann-Stieltjes calculus. In this paper, we aim to present a detailed analysis of an important class of differential equations called stochastic equations with the new classes of differential operators with the global derivative with integer and non-integer orders. We considered many classes as nonlinear Cauchy problems, then we presented existence and the uniqueness of their solutions using the linear growth and the Lipchitz conditions. We derived numerical solutions for each class and presented the error analysis. To show the applicability of these operators, we considered three epidemiological problems, including the zombie virus spread model, the zika virus spread model and Ebola model. We solved each model using the suggested numerical scheme and presented the numerical solutions for different values of fractional order and the global function gt. Our results showed that, more complex real world problems could be depicted using these classes of differential equations.http://www.sciencedirect.com/science/article/pii/S2211379720320337Global rate of changeGlobal derivativeFractional kernelsZombie virusZika virusEbola virus |
| spellingShingle | Abdon Atangana Seda İğret Araz Nonlinear equations with global differential and integral operators: Existence, uniqueness with application to epidemiology Results in Physics Global rate of change Global derivative Fractional kernels Zombie virus Zika virus Ebola virus |
| title | Nonlinear equations with global differential and integral operators: Existence, uniqueness with application to epidemiology |
| title_full | Nonlinear equations with global differential and integral operators: Existence, uniqueness with application to epidemiology |
| title_fullStr | Nonlinear equations with global differential and integral operators: Existence, uniqueness with application to epidemiology |
| title_full_unstemmed | Nonlinear equations with global differential and integral operators: Existence, uniqueness with application to epidemiology |
| title_short | Nonlinear equations with global differential and integral operators: Existence, uniqueness with application to epidemiology |
| title_sort | nonlinear equations with global differential and integral operators existence uniqueness with application to epidemiology |
| topic | Global rate of change Global derivative Fractional kernels Zombie virus Zika virus Ebola virus |
| url | http://www.sciencedirect.com/science/article/pii/S2211379720320337 |
| work_keys_str_mv | AT abdonatangana nonlinearequationswithglobaldifferentialandintegraloperatorsexistenceuniquenesswithapplicationtoepidemiology AT sedaigretaraz nonlinearequationswithglobaldifferentialandintegraloperatorsexistenceuniquenesswithapplicationtoepidemiology |