A simple efficient method for solving sixth-order nonlinear boundary value problems
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A simple efficient method for solving sixth-order nonlinear boundary value problems Dang Quang A1 · Dang Quang Long2
Received: 31 January 2017 / Revised: 13 April 2018 / Accepted: 10 May 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract In this paper, we propose a simple efficient method for a sixth-order nonlinear boundary value problem. It is based on the reduction of the problem to an operator equation for the right-hand-side function. The existence and uniqueness of a solution and its positivity are established. An iterative method for finding the solution is investigated. A numerical realization of the iterative method with the use of a difference scheme of sixth-order accuracy shows the efficiency and advantages of the proposed method over some other methods. Keywords Sixth-order boundary value problem · Existence and uniqueness of solution · Positivity of solution · Iterative method Mathematics Subject Classification 65L10 · 65L20 · 34B15
1 Introduction The sixth-order boundary value problems (BVPs) are known to arise in astrophysics; the narrow convecting layers bounded by stable layers which are believed to surround A-type stars may be modeled by sixth-order boundary value problems (Chandrasekhar 1981; Glatzmaier 1985; Toomore et al. 1976; Twizell and Boutayeb 1990). One of the sixth-order BVPs that has attracted much attention is the following: u (6) (x) = f (x, u(x)), a < x < b,
(1)
Communicated by Elbert Macau, Antônio Fernando Bertachini de Almeida Prado and Othon Cabo Winter.
B
Dang Quang A [email protected] Dang Quang Long [email protected]
1
Centre for Informatics and Computing, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
2
Institute of Information Technology, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
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Q.A Dang, Q.L. Dang
u(a) = A1 , u (a) = A2 , u (4) (a) = A3 , u(b) = B1 , u (b) = B2 , u (4) (b) = B3 ,
(2)
where f : [a, b] × R is a continuous function. To our best knowledge, the authors Al-Hayani (2011), Boutayeb and Twizell (1992), Khalid et al. (2014), Lang and Xu (2015), Noor et al. (2009), Noor and Mohyud-Din (2008), Wazwaz (2001), He (2003), Mohyud-Din et al. (2009), Pandey (2013) and Ramadan et al. (2008) and others only concentrate their attention on finding the solution of the problem but not on the investigation of the existence, uniqueness and properties of solutions. They assume that the solution exists and it is unique or refer to the general results of the existence and uniqueness of solution for higher order differential equations in Agarwal (1986). The methods for solving the problem (1)and (2) are diverse, including the variational iteration method (Noor et al. 2009), the variation of parameters method (Mohyud-Din et al. 2009), the modified decomposition method (Wazwaz 2001), the variational approach (He 2003), spline method (Lang and Xu 2015) and neural network method (Khalid et al. 2014). In the case where the Eq. (1) is linear, the finite difference method (Pandey 2013) and the non-polynomial
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