Simple numerical methods of second- and third-order convergence for solving a fully third-order nonlinear boundary value
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Simple numerical methods of secondand third-order convergence for solving a fully third-order nonlinear boundary value problem Quang A Dang1,3
· Quang Long Dang2
Received: 14 July 2020 / Accepted: 14 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we consider a fully third-order nonlinear boundary value problem that is of great interest of many researchers. First, we establish the existence and uniqueness of solution. Next, we propose simple iterative methods on both continuous and discrete levels. We prove that the discrete methods are of second-order and thirdorder of accuracy due to the use of appropriate formulas for numerical integration and obtain estimate for total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative methods. Keywords Third-order nonlinear equation · Existence and uniqueness of solution · Iterative method · Third-order accuracy · Total error Mathematics Subject Classification (2010) MSC 34B15 · MSC 65L10
1 Introduction Boundary value problems (BVPs) for third-order nonlinear differential equations appear in many applied fields, such as flexibility mechanics, chemical engineering, and heat conduction. Many works are devoted to the qualitative aspects of the problems (see, e.g., [6, 7, 22, 23, 33, 35, 37]). There are also many methods concerning the solution of third-order BVPs including analytical methods [1, 28, 32] and numerical methods by using interpolation polynomials [3] Taylor series expansion [5], quartic * Quang A Dang
[email protected] Quang Long Dang [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
3
Lac Hong University, 10 Huynh Van Nghe, Bien Hoa, Dong Nai, Vietnam
Numerical Algorithms
splines [21, 31], quintic splines [26], non-polynomial splines [24, 25, 27, 34], and wavelets [19]. A majority of the mentioned above numerical methods are devoted to linear equations or special nonlinear third-order differential equations. In this paper, we consider the following BVP: u(3) (t) = f (t, u(t), u′ (t), u′′ (t)), 0 < t < 1, u(0) = c1 , u′ (0) = c2 , u′ (1) = c3 .
(1)
Some authors have studied the existence and positivity of solution for this problem, for example, by using the lower and upper solutions method and fixed point theorem on cones; in [36], Yao and Feng established the existence of solution and positive solution for the case f = f (t, u(t)). In [20], Feng and Liu obtained existence results by the use of the lower and upper solutions method and a new maximum principle for the case f = f (t, u(t), u′ (t)). It should be emphasized that the results of these two works are pure existence but are not methods for finding solutions. Many researchers are interested in numerical solution of the problem (1) without giving attention to qualitative aspects of it, or refer to the book [2]. Below, w
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