A Novel Method for Nonlinear Impulsive Differential Equations in Broken Reproducing Kernel Space
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
A NOVEL METHOD FOR NONLINEAR IMPULSIVE DIFFERENTIAL EQUATIONS IN BROKEN REPRODUCING KERNEL SPACE∗
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Liangcai MEI (
Zhuhai Campus, Beijing Institute of Technology, Zhuhai 519088, China E-mail : [email protected] Abstract In this article, a new algorithm is presented to solve the nonlinear impulsive differential equations. In the first time, this article combines the reproducing kernel method with the least squares method to solve the second-order nonlinear impulsive differential equations. Then, the uniform convergence of the numerical solution is proved, and the time consuming Schmidt orthogonalization process is avoided. The algorithm is employed successfully on some numerical examples. Key words
Nonlinear impulsive differential equations; Broken reproducing kernel space; numerical algorithm
2010 MR Subject Classification
1
97N40; 97R20
Introduction
In recent years, the impulsive differential equation model has been applied to many aspects of life: population dynamics [1], physics, chemistry [2], irregular geometries and interface problems [3–5], and signal processing [6, 7]. Many scholars studied the existence and numerical solution of the impulsive differential equations [8–13]. Y. Epshteyn [14] solved the high-order linear differential equations with interface conditions based on Difference Potentials approach for the variable coefficient. However, so far, no scholars have discussed the numerical solution of the second-order nonlinear impulsive differential equations. Only a few scholars studied the existence of solutions [15]. A. Sadollaha [16] suggested a least square algorithm to solve a wide variety of linear and nonlinear ordinary differential equations. R. Zhang [17] presented the reproducing kernel method and least square to nonlinear boundary value problems. These research work shows that the least square method plays a very good role in solving nonlinear problems. As known to all, the reproducing kernel method is a powerful tool to solve differential equations [17–21]. Al-Smadi M. [22–27] introduced a iterative reproducing kernel method and other methods for providing numerical approximate solutions of time-fractional boundary value problem. ∗ Received
December 11, 2018; revised April 11, 2019. This work is supported by a Young Innovative Talents Program in Universities and Colleges of Guangdong Province (2018KQNCX338), and two Scientific ResearchInnovation Team Projects at Zhuhai Campus, Beijing Institute of Technology (XK-2018-15, XK-2019-10).
724
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
In this article, we consider the following second-order nonlinear impulsive differential equations(NIDEs for short): u′′ (x) + a (x)u′ (x) + a (x)u(x) + N (u) = f (x), x ∈ [a, b] \ {c} 1 0 (1.1) u(a) = α1 , u(b) = α2 , ∆u′ (c) = α3 , ∆u(c) = α4 ,
where ∆u′ (c) = u′ (c+ ) − u′ (c− ), α3 and α4 are not at the same time as 0, ai (x) and f (x) are known function, N : R → R is a continuous function, a
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