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

Abstract

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.

A novel Touchard polynomial-based spectral matrix collocation method for solving the Lotka-Volterra competition system with diffusion

“…The applications of the spectral collocation approach with exponential-order accuracy have been examined for various model problems in physical sciences. For example, we may draw your attention to the recently published works [23][24][25][26][27][28][29][30]. The Touchard polynomials, also known as Touchard-Riordan polynomials or exponential polynomials, constitute a family of functions prominent in combinatorics and partition theory [31].…”

Simulating accurate and effective solutions of some nonlinear nonlocal two-point BVPs: Clique and QLM-clique matrix methods

“…These methods have been successfully applied to a number of significant model problems with various (orthogonal) basis functions. Among these types of bases, we mention Morgan-Voyce [13] , Vieta-Lucas [14] , Bessel [15] , [16] , [17] , Fibonacci [18] , Jacobi [19] , Chebyshev [20] , [21] , [22] , [23] , [24] , and Vieta-Fibonacci [25] , to name a few.…”

Robust QLM-SCFTK matrix approach applied to a biological population model of fractional order considering the carrying capacity

“…In addition to Chebyshev polynomials [13,18], different (orthogonal) polynomial functions have been utilized inside the spectral collocation approaches in the literature. Among others, we mention Dickson [25], Bessel [14,36,48], Legendre [37], Benoulli [2], Vieta-Fibonacci [1], and Jacobi [47], to name a few.…”

“…Thus, the presented GSCFTK collocation technique is applied to the linearized submodels, see cf. [1,17,48]. We refer to this approach as QLM-GSCFTK afterwards.…”

“…We further use T " 1 and use L " 5. The approximated solutions based on the direct GSCFTK matrix technique on 0 ď τ ď 1 are u 1 5 pτ q "20.0 ´11.999999 τ `2.9999929 τ 2 ´0.21495854 τ 3 ´0.080138584 τ 4 `0.022209248 τ 5 `0.00021689006 τ 6 ´0.0010911112 τ 7 `0.00017626386 τ 8 , v 1 5 pτ q "15.0 `2.2499998 τ ´0.73124785 τ 2 `0.023425032 τ 3 `0.031670479 τ 4 ´0.0074522499 τ 5 ´0.000088254 τ 6 `0.00034927112 τ 7…”