A nonlinear mathematical model of the corneal shape

Okrasiński, W.; Płociniczak, Łukasz

doi:10.1016/j.nonrwa.2011.11.014

Cited by 35 publications

(44 citation statements)

References 16 publications

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“…For ∂h/∂x 1, the effect should be negligible, but as we will observe in the numerical and graphical outputs, this is not the case, so the nonlinear term should be included rather than neglected as in [1]. For ∂h/∂x 1, the effect should be negligible, but as we will observe in the numerical and graphical outputs, this is not the case, so the nonlinear term should be included rather than neglected as in [1].…”

Section: Ode Routinementioning

confidence: 91%

“…The BVODE is derived by a static force balance on the cornea as detailed in [1]. Here, we have renumbered the original equations;…”

Section: Model Equationsmentioning

confidence: 99%

“…and also, certain original equation numbers are repeated as a second number to facilitate reference to the original derivation and discussion, for example, (8.2; 4) with 8.2 as the new number and 4 as the original number in [1]. A force balance leads to the BVODE (see Fig.…”

Section: Model Equationsmentioning

confidence: 99%

“…The important difference is that for the Neumann BC, a term involving the first derivative, ux [1], is included in the calculation of uxx [1]. The important difference is that for the Neumann BC, a term involving the first derivative, ux [1], is included in the calculation of uxx [1].…”

Section: Library Routines For Differentiation In Spacementioning

confidence: 99%

“…In this case study, we analyze a boundary value ordinary differential equation (BVODE) model for the calculation of the cornea shape reported by Okrasiński and Płociniczak [1]. In this case study, we analyze a boundary value ordinary differential equation (BVODE) model for the calculation of the cornea shape reported by Okrasiński and Płociniczak [1].…”

Section: Introductionmentioning

confidence: 99%

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Corneal Curvature

Schiesser

2014

Differential Equation Analysis in Biomedical Science and Engineering

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Section: Ode Routinementioning

confidence: 91%

“…The BVODE is derived by a static force balance on the cornea as detailed in [1]. Here, we have renumbered the original equations;…”

Section: Model Equationsmentioning

confidence: 99%

Section: Model Equationsmentioning

confidence: 99%

Section: Library Routines For Differentiation In Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Corneal Curvature

Schiesser

2014

Differential Equation Analysis in Biomedical Science and Engineering

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A highly accurate and efficient Genocchi‐based spectral technique applied to singular fractional order boundary value problems

Izadi,

Ansari,

Srivastava

2024

Math Methods in App Sciences

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This article focuses on an efficient and highly accurate approximate solver for a class of generalized singular boundary value problems (SBVPs) having nonlinearity and with two‐term fractional derivatives. The involved fractional derivative operators are given in the form of Liouville–Caputo. The developed algorithm for solving the generalized SBVPs consists of two main stages. The first stage is devoted to an iterative quasilinearization method (QLM) to conquer the (strong) nonlinearity of the governing SBVPs. Secondly, we employ the generalized Genocchi polynomials (GGPs) to treat the resulting sequence of linearized SBVPs numerically. An upper error estimate for the Genocchi series solution in the norm is obtained via a rigorous error analysis. The main benefit of the presented QLM‐GGPs procedure is that the required number of iteration in the first stage is within a few steps, and an accurate polynomial solution is obtained through computer implementations in the second stage. Three widely applicable test cases are investigated to observe the efficacy as well as the high‐order accuracy of the QLM‐GGPs algorithm. The comparable accuracy and robustness of the presented algorithm are validated by doing comparisons with the results of some well‐established available computational methods. It is apparently shown that the QLM‐GGPs algorithm provides a promising tool to solve strongly nonlinear SBVPs with two‐term fractional derivatives accurately and efficiently.

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A highly accurate matrix method for solving a class of strongly nonlinear BVP arising in modeling of human shape corneal

Abbasi

Izadi

Hosseini

2022

Math Methods in App Sciences

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This research study presents a novel highly accurate matrix approach for numerical treatments of a strongly nonlinear boundary value problem occurring in the modeling of the corneal shape of the human eye. Using the technique of quasilinearization, the nonlinear model is reduced into a sequence of linearized problems. Then, a spectral collocation procedure based on novel shifted Vieta‐Fibonacci (SVF) is applied to transform each subproblem into a linear algebraic system of equations. An upper bound for the error is provided, and convergence analysis of the SVF series solution in the weighted L2$$ {L}_2 $$ norm is discussed. The technique of error correction is used to improve the SVF polynomial solutions by means of the residual error function. Various numerical tests are provided to demonstrate the accuracy and efficiency of the presented collocation algorithm. The validation of the proposed approach is shown by comparison with available existing numerical solutions.

show abstract

A nonlinear mathematical model of the corneal shape

Cited by 35 publications

References 16 publications

Corneal Curvature

Corneal Curvature

A highly accurate and efficient Genocchi‐based spectral technique applied to singular fractional order boundary value problems

A highly accurate matrix method for solving a class of strongly nonlinear BVP arising in modeling of human shape corneal

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