Elsevier

Optik

Volume 261, July 2022, 169086
Optik

Original research article
Some novel mathematical analysis on a corneal shape model by using Caputo fractional derivative

https://doi.org/10.1016/j.ijleo.2022.169086Get rights and content

Abstract

In this article, we solve a fractional boundary value problem (FBVP) for modeling the human corneal shape dynamics. We use the Caputo fractional derivative having singular type kernel. We propose some novel simulations to prove the existence of a unique solution of the given FBVP. The numerical solution of the proposed problem is derived by using a polynomial least squares method. We do a number of graphical observations at various values of the given model parameters along with the orders of considered fractional derivative. The main motivation of this research article is to specify the possibilities that the corneal shape may slightly differ to the shapes which were investigated in various past studies at any fixed set of available parameters. The given model was not generalized before by using any fractional derivative which is the main reason for the proposal of this study as well as the novelty of the work.

Introduction

One of the most delicate elements of the human body is the cornea. Many vision problems are caused by irregularities in corneal geometry. Common illnesses like myopia, hyperopia, and astigmatism, for example, are caused by incorrect corneal geometry. The dynamics of corneal topography and models described in [1] is critical for the success of refractive surgery and the accurate fitting of contact lenses. As a result, the correct description of corneal shape (Fig. 1) is critical for optical and opthalmologic purposes.

There are various different types of cornea mathematical models in use today. Surfaces of revolution are the most prevalent, with meridians being conic sections (mainly parabolas and ellipses) [2]. Regrettably, they are often considered without much physical reason or justification. Some researchers have been proposed that this conics can be extended by applying variable eccentricity [3]. Aside from these models, there is a collection of shell theory-based models. They are typically pretty complicated, but they go into great detail into physics [4]. There have also been several computational models of corneal biomechanics defined via finite element methods developed [5]. There are additional models that use Zernike polynomial approximations to characterize cornea [6]. With regard to a certain scalar product, they are orthogonal polynomials. They are used to describe corneal or lens aberrations. Further models store and compress information about the corneal surface gathered during videokeratoscopy using rational functions created using Zernike polynomials [7].

The cornea is thought to be responsible for around two-thirds of the eye’s refractive power. “The anterior surface of the human cornea is a key refractive element”, according to [8]. The rigid gas permeable (RGP) contact lens and corneal laser refractive surgery require a thorough understanding of the corneal shape (LASIK).

We are primarily directed to a model of the topography of the human cornea in our research. This physically based model was first proposed in [9], and it accurately described the curvature of the cornea. Authors in [9] was proposed the following boundary value problem (BVP) to define the topography of the human cornea: y(t)+ay(t)=b1+(y(t))2,y(0)=0,y(1)=0. Here the curve y(t) is a meridian of a surface of revolution describing corneal geometry, t is a distance from the center of symmetry and a and b are the positive constants significantly defined in Ref. [9].

The existence and uniqueness results obtained in [9] for the given BVP (1)–(2) were generalized previously in [10]. An efficient semi-analytical solution of the given BVP model has been derived in Ref. [11]. Recently, in Ref. [12], an approximate analytical solution of the given BVP has been investigated using linear solution and Taylor series method.

In the all above mentioned works, the given BVP is considered in the sense of integer-order derivatives. Nowadays, fractional-order operators are also very useful tools to solve number of real-world problems, mathematically [13], [14]. Various types of fractional derivatives are available in the literature along with many recent developments [15], [16]. Also, number of numerical methods have been proposed by the researchers in the sense of fractional-order operators [17], [18]. Recently, fractional-order operators have been used in number of areas like, epidemiology [19], [20], ecology [21], [22], physics [23], psychology [24], etc. In [25], some novel analysis on the existence and uniqueness of a non-linear q-difference boundary value problem of non-integer order are obtained.

The motivation behind the generalization of the classical differential systems into non-integer order sense by using fractional time derivatives always be to incorporate a history dependence into the dynamics which is called memory effects. Fractional calculus is strongly associated to many adaptive systems with memory and hereditary characteristics, which largely exist in our real-life. In this concern, we propose the generalization of the above given integer-order BVP (1)–(2) into the fractional-order sense as follows: cD0μy(t)=ay(t)b1+(cD0νy(t))2,t(0,1),y(0)=0,y(1)=0. Here cD0μ is the Caputo fractional derivative of the function y(t) of order 1<μ2 and cD0ν is the Caputo fractional derivative of the function y(t) of order 0<ν1. The proposed Caputo fractional derivative used in the generalization is defined as follows:

Definition 1 [14]

The Caputo fractional derivative of a function LC1d is defined by DtϱLt=dqLtdζq,ϱ=qN1Γ(qϱ)0ttϑqϱ1L(q)ϑdϑ,q1<ϱ<q,qN.

The given study is formulated in number of sections. In Section 2, we prove the existence of a unique solution in the sense of given fractional operator. In Section 3, we derive the numerical solution of the given BVP (3)–(4) by using polynomial least squares scheme. In Section 4, we perform number of graphs to justify the novelty of the work experimentally. At the end, we conclude our findings.

Section snippets

Existence and uniqueness analysis

Firstly, we recall the main problem (3)–(4) as follows: cD0μy(t)=ay(t)b1+(cD0νy(t))2,t(0,1),1<μ2,0<ν1,y(0)=0,y(1)=0. We will prove that the solution of the fractional boundary value problem (FBVP) (6)–(7) exists and is unique. To do this, we need some facts about the function given in the right-side of Eq. (6). It is clear that the function given in the right-side of (6) is continuous. Now let C(J) be the space of continuous functions defined on J=[0,1]. One can easily show that the space B

Numerical solution of the model

Now we derive the solution of the given problem by using well-known polynomial least squares scheme. Some applications of this method can be seen from Ref. [26], [27], [28].

Simulation results

As we know that an early identification of certain eye disorders and monitoring the postoperative results of refractive surgery can both benefit from a precise assessment of the corneal shape and prediction of changes in the corneal geometry. The corneal Eqs. (6)–(7) contain two constant parameters, a and b, which are affected by physical and biological factors such as the elasticity coefficient, surface tension, and corneal radius, as well as intraocular pressure. These characteristics alter

Conclusion

In this paper, we have proposed a fractional boundary value problem for studying the dynamics of human corneal shape. We have used the Caputo fractional derivative to achieve our results. We have proved that the given FBVP model exists a unique solution under the applications of Arzela–Ascoli and Schauder fixed point theorems. The numerical simulations for the proposed problem have been made by using the polynomial least squares method. We have performed a number of graphs at various values of

CRediT authorship contribution statement

Vedat Suat Erturk: Conceptualization, Investigation, Software, Writing – review & editing. Asghar Ahmadkhanlu: Conceptualization, Investigation, Software, Writing – review & editing. Pushpendra Kumar: Investigation, Formal analysis, Resources, Visualization, Writing – original draft. V. Govindaraj: Conceptualization, Investigation, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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