A computational method for solving a class of singular boundary value problems arising in science and engineering

In this paper, we present a numerical collocation method for nonlinear singular two-point boundary value problems of second order based on septic B-spline function. This method depends on di erent physiological processes such as steady-state oxygen di usion in a spherical cell with the kinetics uptake of Michaelis-Menten and heat sources distribution in human head. The proposed method has uniform convergence for the exact solution. We will provide some physiological models proving that our method is very e ective with acceptable maximum absolute errors and absolute residual errors.

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“…+2416 s (5) (t j ) + 1191 s (5) (t j+1 ) +120 s (5) (t j+2 ) + s (5) ; (23) s (6) (t j 3 ) + 120 s (6) (t j 2 ) + 1191 s (6)…”

Section: Z (T J 3 ) 1680 Z (T J 2 )mentioning

confidence: 99%

Septic B-spline method for solving nonlinear singular boundary value problems arising in physiological models

2018

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“…Several studies have been conducted on this technique, which uses OM methods to solve many problems based on various polynomials. Many researchers have solved various problems using OM based on the Bernstein polynomial (BOM), such as: [4] solved odd boundary value problems, [5] studied third order equations ODE, [6] solved fractional integral equations. Moreover, there are many researchers who have used operational matrices based on different polynomials, such as [7] used the Legendre operational Novel Approximate Solutions for Nonlinear Blasius Equations matrix (LOM) method to solve the fractional-order two-dimensional integral equations.…”

Section: Introductionmentioning

confidence: 99%

Novel Approximate Solutions for Nonlinear Blasius Equations

Mahdi,

AL-Jawary,

Mustafa Turkyilmazoglu

2024

IHJPAS

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The method of operational matrices based on different types of polynomials such as Bernstein, shifted Legendre and Bernoulli polynomials will be presented and implemented to solve the nonlinear Blasius equations approximately. The nonlinear differential equation will be converted into a system of nonlinear algebraic equations that can be solved using Mathematica®12. The efficiency of these methods has been studied by calculating the maximum error remainder ( ), and it was found that their efficiency increases as the polynomial degree (n) increases, since the errors decrease. Moreover, the approximate solutions obtained by the proposed methods are compared with the solution of the 4th order Runge-Kutta method (RK4), which gives very good agreement. In addition, the convergence of the proposed approximate methods is given based on one of the Banach fixed point theorem results.

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“…Moreover, some operational matrix forms of the Bernstein polynomials have been generated for numerical methods. For example, operational matrix of shifted orthonormal Bernstein polynomials has been used for the numerical solution of pantograph equations [21], computational methods based on operational matrix of differentiation and integration via the Berntein polynomials has been presented for solving differential equations [3,22,28,29,39], Volterra integral equations [26] and fractional order differential equations [34].…”

Section: Introductionmentioning

confidence: 99%