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Game theory is an excellent mathematical tool to describe the interaction between the immune system and cancerous leukocytes c. l e u . The feature of cancerous leukocytes to differentiate and mutate to give rise to leukemia is in the domain of ecological models as well. In this work, the dynamic of leukemia is described and compared by two models: firstly by a simple probabilistic mathematical model using the zero-sum two player game of Hawk and Dove, and secondly by Leslie Predator Prey model of ecology. The main goal of this study is to compare the results of both models and then discuss the treatment of leukemia i.e., Hematopoietic Stem cell transplant with the best model among them. Hawk and Dove model also describes the cell to cell interaction of cancerous leukocytes and healthy leukocytes l e u after diagnoses and the condition of the patient before and after treatments. In this work, Hematopoietic Stem cell transplant is discussed by using concepts of a zero-sum three player game. Also, both models will be characterized by determining the stability properties, identifying basins of attraction, and locating the equilibrium points to see, at what extent the patient's survival is possible with leukemia in its body. Results for both models will be presented graphically. [ABSTRACT FROM AUTHOR]

In this paper, DSEK model with fractional derivatives of the Atangana-Baleanu Caputo (ABC) is proposed. This paper gives a brief overview of the ABC fractional derivative and its attributes. Fixed point theory has been used to establish the uniqueness and existence of solutions for the fractional DSEK model. According to this theory, we will define two operators based on Lipschitzian and prove that they are contraction mapping and relatively compact. Ulam-Hyers stability theorem is implemented to prove the fractional DSEK model's stability in Banach space. Also, fractional Euler's numerical method is derived for initial value problems with ABC fractional derivative and implemented on fractional DSEK model. The symmetric properties contribute to determining the appropriate method for finding the correct solution to fractional differential equations. The numerical solutions generated using fractional Euler's method have been plotted for different values of α where α ∈ 0,1 and different step sizes h. Result discussion will be given, describing the changes that occur due to the step size h. [ABSTRACT FROM AUTHOR]

The fundamental idea of this work is to modify Descemet's Stripping Endothelial Keratoplasty DSEK mathematical model by incorporating the Zernike polynomial to examine Higher Order Aberrations (HOA). This model has been developed in a way that can identify the type of aberration occurring after a keratoplasty. Model variables are compared with the data available in already published literature. Surgically induced Higher Order Aberrations (HOA) and change in corneal power are the main outcome measures. Eye illness and their treatments have always been an interesting topic of discussion for researchers. Mechanical and software engineers have managed to helped ophthalmologists by developing advanced machines that can identify the patients' complaints in seconds. Moreover, several new and advanced surgical procedures have been adopted for eye problems. One amongst them is eye aberration. Only recently has it become possible to detect one type of aberration (Astigmatism & Higher Order Aberrations) with special equipments. However, it is still not easy to identify aberrations as Myopia and Hyperopia. This paper studies the induced eye aberrations caused by DSEK and their detection through a modified DSEK model. [ABSTRACT FROM AUTHOR]

The aim of this research was to relate two physical effects for partial differential equations on the time-coordinate, notably the multipledelay times and fractional-derivative. Time Fractional Delay Partial Differential Equations (TFDPDEs) usually interpret some complex physical phenomenon. This study works to solve TFDPDE with shrinking in x and proportional delays in t numerically by utilizing the fractional derivative of Caputo sense in the numerical method known as Perturbation Iteration Algorithm (PIA). A few famous numerical examples have been solved using PIA and their comparison with an exact solutions is illustrated for α = 1. Also, different values of ® have been depicted in graphical form to show their fractional behavior. The delay term k is also discussed extensively in this TFDPDE study. Numerical results show that this technique is reliable, convenient, and attractive for computational use in modern times. [ABSTRACT FROM AUTHOR]

This work is based on the implementation of an iterative perturbation method to attain the series solutions of nonlinear fractional differential difference equation (NFDΔE). Perturbation‐iteration algorithm (PIA) assigns a perturbation parameter ϵ to all nonlinear terms and converts it into a simple fractional differential difference equation (FDΔE). By simply solving this FDΔE, series solutions can be obtained. To show the efficacy and accuracy of this method, three famous NFDΔE, i.e, fractional Lotka‐Volterra equation, fractional discrete KdV equation and discretized fractional mKdV lattice equation, will be solved numerically via PIA. Also, comparison of numerical results for α=1 will be done with exact solutions, and their absolute error will also be provided. Graphical illustrations for different values of α will be given to establish the certainty of results. Also, to prove the proficiency among other methods, comparison with different numerical methods is given. Advantage of PIA is that nonlinear terms get vanished during the calculations of Taylor series expansion; therefore, less calculation effort can obtain comprehensive accurate solutions. [ABSTRACT FROM AUTHOR]

The main objective of this work is to numerically solve three types of Fourth order Emden-Fowler equations by using Successive Differentiation Method (SDM) and to inspect its shape factor 'k' in detail. Numerical solutions and graphical illustrations give a detailed insight into the variation of numerical solution due to shape factor and evince the reason why 'k' is always kept positive. The singularity term in Emden-Fowler equations makes it difficult to find its analytical solution. It is usually arduous to obtain the numerical solutions due to its singularity. Successive Differentiation Method is easy to understand and give accurate results as compared to its contemporary methods. Graphical illustrations have been shown to establish our argument. [ABSTRACT FROM AUTHOR]

This work comprises of development and analysis of a new mathematical model based on Descemet's Stripping Endothelial Keratoplasty (DSEK). Formulating the nonlinear system of ordinary differential equations to describe changes occuring in ocular parameters during DSEK for scarred cornea, is a unique perspective. In this paper, the formation of the model and the existence of its solution is proved. The stability of DSEK model is discussed by the Jacobian matrix and its eigen values are examined. Also this DSEK model is proved to be uniformly and Lipschitz continuous. [ABSTRACT FROM AUTHOR]

In this work, four well known time Fractional Partial Differential Equations (FPDEs) namely, time Fractional Fornberg-Whitham Equation (FFWE), time Fractional KdV Equation (FKdVE), time Fractional Convection-Diffusion Equation (FCDE) and time Fractional BBMBurger Equation (FBBMBE) are solved numerically through Fractional Wave Variable Transformation (FWVT) and Successive Differentiation Method (SDM). By using the FWVT, = ,x ¡ vt® ¡(® + 1), these FPDEs are converted into an Ordinary Differential Equation (ODE). Then SDM is applied on thus formed ODE to produce their Taylor series. A comparison of obtained numerical series at ® = 1 with the exact solution is presented to prove the accuracy of this technique as well as graphical illustrations for different values of ®. SDM along with fractional wave variable transformation proves to be an adequate and accurate method for obtaining numerical solutions of FPDEs. [ABSTRACT FROM AUTHOR]

This universe has veiled many secrets for mankind. On exploring them mathematically several differential equations have been created. Lane-Emden Equation (LEE) is among the most famous equations concerned with the study of stellar structure. The main purpose of this work is to show the effectiveness of a numerical method named Successive Differentiation Method (SDM) which can obtain analytical solutions for nonlinear singular differential equations. To establish the claim, the nonlinear singular boundary and initial value problem of Lane-Emden that governs the polytropic and isothermal gas spheres is considered here. Different illustrative types of Lane-Emden Equation have been solved numerically in this paper and these numerical results are compared with the numerical results of formerly solved numerical techniques. SDM proved to be the easiest method not only in accuracy but also in tranquil implementation of numerical techniques over the entire finite domain. [ABSTRACT FROM AUTHOR]

In this work successive differentiation method is applied to solve highly nonlinear partial differential equations (PDEs) such as Benjamin-Bona-Mahony equation, Burger's equation, Fornberg-Whitham equation, and Gardner equation. To show the efficacy of this new technique, figures have been incorporated to compare exact solution and results of this method. Wave variable is used to convert the highly nonlinear PDE into ordinary differential equation with order reduction. Then successive differentiation method is utilized to obtain the numerical solution of considered PDEs in this paper. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]