Research proposal For PHD admission

Achieving high accuracy of Mathematical models using fractional differential techniques. Abstract: In the study, focus will be on fractional derivatives and it`s Applications. Calculations and solutions will be obtained by analytical, semi-numerical and numerical ways for mathematical models constructed on fractional Partial differential equations. For validation of results a comparison will be drawn between fractional and non-fractional models and between analytical and numeral results. As an application a mathematical model for \"calculating Erythrocyte sedimentation rate in human body [1][2]\" is selected in this proposal. Introduction: On September 30, 1695, an audacious and prophetic response began to non-integer order calculation, popularized as fractional calculus [3, 4]. After all, what was Leibniz’s thought when his friend l-Hospital sends a letter, formulating a question involving a possible generalization of the order derivative whole to a fractional order? I believe there have been a lot of ideas, thoughts and even questions. Leibniz’s response to his friend was courageous and clear, one day ”this fraction” will have several important consequences. In addition to more, other important mathematicians, like Euler, Lagrange, Fourier, Abel, among others, were fundamental to the development of such a study.In the world recently renowned researchers also contribute to the continuity and expansion of calculus among which we mention: Delfin, Almeida, Trujillo, Baleanu, Ortigueira, Caputo, Mainardi, Tenreiro Machado, among others [5, 6]. Although the fractional calculation arose at the same time as the calculation of whole order, as proposed by Newton and Leibniz, a priori there was not much in the scientific community and for many years remained hidden. Only from an international congress in 1974 [7, 8], fractional calculation becomes exposed and consolidates in numerous applications from diverse fields, in mathematics, physics, biology, in engineering, among others. The most significant advantage is its non local ownership, not only depending on your current condition, but on all previous conditions [9, 10, 11], making applications more realistic, and thus better describing the phenomena physicists. Although much of the theory was developed before the twentieth century, it has been in recent years that it has made a significant leap, particularly in applications. Currently, fractional calculus branches into three different approaches, derived from fractions with non-singular nucleus, singular nucleus and also, through limits, being that all so-called fractional derivatives, within a convenient limit, must regain their whole order fractional derivative. Use differential equations to model a particular physical phenomenon, It was not always a simple task and much less easy. The fact is that by the amount of parameters involved in the system, it is extremely difficult to obtain results each closer to reality

way to apply fractional calculation is to use local or non-local fractional derivatives, that is, either: M -Fractional, Riemann-Liouville,Caputo, Hilfer,Riemann-Liouville, of Gunwald- Letnikov, among others; and replace the whole order derivative, and then propose a fractional model. Motivation: The main motivation for the study is based on nutrient concentration C(x,t), equation via fractional calculation, comes from the mathematical model proposed by Sharma et al. [1] and Sousa et al.[2]. The purpose of this study will be to use the fractional derivative in the sense of analytical and numerical ways and propose a fractional model,to obtain an analytical solution of the time-fractional diffusion equation in terms of the special functions like Mittag-Leer and Wright. More precisely, input from the observations will be made in the fractional model to establish some comparisons so that the development of the work becomes clear. Background Of the study: Differential equations are studied mathematically from different mainly concerned with their solutions and the set of functions that satisfy a given equation. Only the simplest differential equations admit explicit solutions; however, some properties of the solutions of a given differential equation can be determined without finding its exact shape. Discuss and analyze the behavior of solutions of an equation system differentials is, in fact, one of the great goals of mathematical analysis, in particular, when it comes to studying the existence, uniqueness, attractiveness, stability of solutions of linear and nonlinear Cauchy problems [12, 13, 14]. On the other hand, by through a differential equation, it is possible to propose mathematical models that describe physical behaviors. Among the many equations we mention: Hamilton equation, equation for radioactive decay, wave equation, Laplace equation, equation of heat, Schrodinger equation, Navier-Stokes equations, among others [15, 16, 17]. Differential equations arise when one knows or postulates a relationship which involves some continuously varying quantities by functions and their rates of change in space and / or time (expressed as derivatives). Because these relationships are extremely common, differential equations play a prominent role in many areas including engineering, physics, economics and biology. Here,the goal will be to present fractional model for sedimentation rate. red blood cells also known as erythrocyte sedimentation importance in a clinical examination test, as well as revealing the inflammatory activity in the body and its contribution to monitoring a treatment. Statement of the Research: Sharma et al. [1] presented the concentration of nutrients in blood by means of a non-homogeneous linear convection-diffusion PDE. Then Sousa et al. [2] introduced the fractional version of that linear PDE for same model. We will try to solve the model presented by Sousa et al. [2] using different variances of fractional derivatives in analytic as well as numerical ways. Analytic Methods:

1:Hilfer derivative[18] 2:Caputo Fabrizio derivative[19][20] 3:Atangana–Baleanu derivative[19][21] Numerical Methods: 1:Finite difference methods [22] 2:Finite element methods.[22] Research Objectives: The objective of proposing a fractional model is to provide an accurate description of natural phenomena and thus obtain more accurate results with the reality. The study of differential equations has always been prominent in the community and with the prominence, evidenced by the fractional derivatives, became one more useful and important tool in particular just to mention: equation convection-diffusion, heat equation, wave equation, advection-diffusion equation, among others. The so-called erythrocyte sedimentation rate test can be used to obtain various clinical diagnoses and can be studied as a particular type of transport phenomenon [23]. In the study by Sharma et al. [1] the authors work with the study of the concentration of nutrients in the blood that affect the sedimentation rate of erythrocytes. The study of this analysis will be obtained and discussed through the analytical and numerical solution of a differential convection-diffusion equation, given by the application of the transform of Laplace in the time variable t. There are several transport phenomena whose respective fractional versions best describe its classic model. The objective of our research will be to 1:Finding analytic, semi-analytic and numeric solutions for fractional models like (ESR model) 2:analyzing different scenario which affects the model. 3:analyzing time fractional model and spatial-fractional model 4:comparison of different results like solutions obtained by different methods and fractional and non-fractional models. Reference: [1] G. C. Sharma, M. Jain, R. N. Saral, A mathematical model for concentration of blood affecting erythrocyte sedimentation, Comput. Biol. Med., 26 (1996) 1–7. [2] J. Vanterler da C. Sousa, E. Capelas de Oliveira, L. A. Magna, Fractional calculus and the ESR test, AIMS Mathematics 2 (4) (2017) 692–705. [3] S. Nayha, Normal variation in erythrocyte sedimentation rate in males over 50 years old, Scand. J. Primary Health Care, 5 (1987), no. 1, 5–8. [4] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffer function in the kernel, Yokohama Math. J., 19 (1971), 7–15.

[5] M. Caputo and M. Fabrizio, A new de?nition of fractional derivative without singular kernel, Progr. Fract. Dier. Appl 1 (2015), no. 2, 1–13. [6] P. Chaturani, S. Narasimbham, R. R. Puniyani, and D. A. Kale, A comparative study of erythrocyte sedimentation rate of hypertension and normal controls, In: Physiol. Fluid Dynamics II: Tata McGraw Hill New Delhi, 1987, pp. 265–280 [7] S. Nayha, Normal variation in erythrocyte sedimentation rate in males over 50 years old, Scand. J. Primary Health Care, 5 (1987), no. 1, 5–8. [8] I. Talstad, P. Scheie, H. Dalen, and J. R¨oli, In?uence of plasma proteins on erythrocyte morphology and sedimentation, Scand. J. Haematology, 31 (1983), no. 5, 478–484. [9] G. W. Leibniz, Letter from Hanover, Germany to G.F.A L’Hospital, September 30, 1695, Leibniz Mathematische Schriften, Olms-Verlag, Hildesheim, Germany, 1962, (First published in 1849), 301– 302. [10] J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Advances in fractional calculus, vol. 4, Springer, 2007. [11] G. S. Teodoro, D. S. Oliveira, and E. Capelas de Oliveira, On fractional derivatives, Revista Brasileira de Ensino de F´ısica 40 (2018), no. 2. [12] A Grönwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator, submetido à publicação (2017). [13] On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equation by means of the ψ-Hilfer operator, submetido à publicação (2017). [14] Global attractivity for nonlinear fractional differential equations with ψ-Hilfer operator, em preparação (2018). [15] M. Braun and M. Golubitsky, Differential Equations and Their Applications, vol. 4, Springer, New York, 1983 [16] S. J. Farlow, An Introduction to Differential Equations and Their Applications, Courier Corporation, New York, 2006. [17]G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press, 1995. [18] de Oliveira, Edmundo Capelas; Tenreiro Machado, José António \"A Review of Definitions for Fractional Derivatives and Integral\". Mathematical Problems in Engineering. 2014: 1–6 [19] Algahtani, Obaid Jefain Julaighim. \"Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model\". Chaos, Solitons & Fractals. Nonlinear Dynamics and Complexity. 89: 552–559. [20] Caputo, Michele; Fabrizio, Mauro. \"Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels\". Progress in Fractional Differentiation and Applications. 2 (1): 1–11 [21] Atangana, Abdon; Baleanu, Dumitru. \"New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model\". Thermal Science. 20 (2): 763–769 [22]Li, Changpin & Zeng, Fanhai. (2015). Numerical methods for fractional calculus. 10.1201/b18503. [23] E. N. Lightfoot, Transport Phenomena and Living Systems, John Wiley, New York (1974).