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A new method of solving singular multi-pantograph delay differential equation in reproducing kernel space. (English) Zbl 1162.34346

From the introduction: We consider the following inhomogeneous second order multipantograph delay differential equation with singularities in reproducing kernel space
\[ \begin{cases} u''(x) + \sum^m_{i=1}\frac{1}{p_i(x)}\;u'(h_ix)+\frac{1}{q(x)}\;u(x) = g(x),\quad 0 < x,\;h_i < 1,\;i = 1,\dots,m\\ u(0)=0,\;u(1)=0,\end{cases}\tag{*} \]
where \(u(x)\in W_2^3[0,1]\), \(g(x)\in W^1_2 [0,1]\), \(p_i(x)\), \(q(x)\) are continuous and maybe equal to zero at 0 or 1.
These equations arise in a variety of applications, such as number theory, electrodynamics, astrophysics, nonlinear dynamical systems, probability theory on algebraic structure, quantum mechanics and cell growth. Therefore, the problem has attracted much attention and has been studied by many authors [M. Z. Liu and D. S. Li, Appl. Math. Comput. 155, No. 3, 853–871 (2004; Zbl 1059.65060); P. Kelevedjiev, Nonlinear Anal., Theory Methods Appl. 50, No. 8(A), 1107–1118 (2002; Zbl 1014.34013); F. H. Wong and W. C. Lian, Comput. Math. Appl. 32, No. 9, 41–49 (1996; Zbl 0868.34019); Y. S. Liu and H. M. Yu, Comput. Math. Appl. 50, No. 1–2, 133–143 (2005; Zbl 1094.34015); D. S. Li and M. Z. Liu, Exact solution’s property of multi-pantograph delay differential equation. Harbin Institute of Technology 3 (2000); G. A. Derfel and F. Vogl, Eur. J. Appl. Math. 7, No. 5, 511–518 (1996; Zbl 0859.34049); F. Z. Geng and M. G. Cui, Applied Mathematics and Computation 192, 389–398 (2007); M. Sezer, Internat. J. Math., 447–468; M. Sezer, S. Yalcinbas and N. Sahin, Computational and Mathematics 7, 1–11 (2007)]. In recent years, much work has been done in reproducing kernel space [M. G. Cui and F. Z. Geng, J. Comput. Appl. Math. 205, No. 1, 6–15 (2007; Zbl 1149.65057); F. Z. Geng and M. G. Cui, loc.cit.; C. L. Li and M. G. Cui, Appl. Math. Comput. 143, No. 2–3, 393–399 (2003; Zbl 1034.47030)]. The basic motivation of this work is to apply a new method to solve multi-pantograph delay differential equation in reproducing kernel space.
We give the representation of the exact solution to (*) and approximate solution in the reproducing kernel space under the assumption that the solution to (*) is unique.

MSC:

34K10 Boundary value problems for functional-differential equations
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
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