On the modeling of the interaction between tumor growth and the immune system using some new fractional and fractional-fractal operators

Ghanbari, Behzad

doi:10.1186/s13662-020-03040-x

Cited by 69 publications

(18 citation statements)

References 38 publications

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“…During the past decades, several mathematical models have been investigated to model prey-predator systems [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The understanding of the relationship between herbivores and plants is extremely important in the behavior of the ecosystems.…”

Section: Introductionmentioning

confidence: 99%

A predator–prey model involving variable-order fractional differential equations with Mittag-Leffler kernel

et al. 2021

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This paper is about to formulate a design of predator–prey model with constant and time fractional variable order. The predator and prey act as agents in an ecosystem in this simulation. We focus on a time fractional order Atangana–Baleanu operator in the sense of Liouville–Caputo. Due to the nonlocality of the method, the predator–prey model is generated by using another FO derivative developed as a kernel based on the generalized Mittag-Leffler function. Two fractional-order systems are assumed, with and without delay. For the numerical solution of the models, we not only employ the Adams–Bashforth–Moulton method but also explore the existence and uniqueness of these schemes. We use the fixed point theorem which is useful in describing the existence of a new approach with a particular set of solutions. For the illustration, several numerical examples are added to the paper to show the effectiveness of the numerical method.

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Section: Introductionmentioning

confidence: 99%

A predator–prey model involving variable-order fractional differential equations with Mittag-Leffler kernel

et al. 2021

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“…By considering the homogenous balance principle in (7), the value of M can be determined. Equation ( 9) possesses the following solutions:…”

Section: The Extended Sinh-gordon Equation Expansion Methodsmentioning

confidence: 99%

“…Despite the recent extensive advances in the theory of differential equations, it can generally be said that it is still a complex task to determine an analytical solution for many ordinary and partial differential equations [1][2][3][4][5][6][7][8][9]. One of the events that led to the introduction of a wide range of new methods was the emergence and use of computers.…”

Section: Introductionmentioning

confidence: 99%

Analytical solitons for the space-time conformable differential equations using two efficient techniques

Neirameh

Parvaneh

2021

Adv Differ Equ

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Exact solutions to nonlinear differential equations play an undeniable role in various branches of science. These solutions are often used as reliable tools in describing the various quantitative and qualitative features of nonlinear phenomena observed in many fields of mathematical physics and nonlinear sciences. In this paper, the generalized exponential rational function method and the extended sinh-Gordon equation expansion method are applied to obtain approximate analytical solutions to the space-time conformable coupled Cahn–Allen equation, the space-time conformable coupled Burgers equation, and the space-time conformable Fokas equation. Novel approximate exact solutions are obtained. The conformable derivative is considered to obtain the approximate analytical solutions under constraint conditions. Numerical simulations obtained by the proposed methods indicate that the approaches are very effective. Both techniques employed in this paper have the potential to be used in solving other models in mathematics and physics.

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“…Using the numerical method presented in formula (21) to solve problem (1), we obtain the following iterative schemes:…”

Section: A Numerical Schemementioning

confidence: 99%

Dynamical behavior of two predators–one prey model with generalized functional response and time-fractional derivative

Djilali

Ghanbari

2021

Adv Differ Equ

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The behavior of any complex dynamic system is a natural result of the interaction between the components of that system. Important examples of these systems are biological models that describe the characteristics of complex interactions between certain organisms in a biological environment. The study of these systems requires the use of precise and advanced computational methods in mathematics. In this paper, we discuss a prey–predator interaction model that includes two competitive predators and one prey with a generalized interaction functional. The primary presumption in the model construction is the competition between two predators on the only prey, which gives a strong implication of the real-world situation. We successfully establish the existence and stability of the equilibria. Further, we investigate the impact of the memory measured by fractional time derivative on the temporal behavior. We test the obtained mathematical results numerically by a proper numerical scheme built using the Caputo fractional-derivative operator and the trapezoidal product-integration rule.

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On the modeling of the interaction between tumor growth and the immune system using some new fractional and fractional-fractal operators

Cited by 69 publications

References 38 publications

A predator–prey model involving variable-order fractional differential equations with Mittag-Leffler kernel

A predator–prey model involving variable-order fractional differential equations with Mittag-Leffler kernel

Analytical solitons for the space-time conformable differential equations using two efficient techniques

Dynamical behavior of two predators–one prey model with generalized functional response and time-fractional derivative

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