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The RKHSM for solving neutral functional-differential equations with proportional delays. (English) Zbl 1267.34133

Summary: We consider the following neutral functional-differential equations with proportional delays \[ (u(t)+a(t)u(p_mt))^{(m)}=\beta u(t)+\sum\limits^{m-1}_{k=0}b_k(t)u^{(k)}(p_kt)+f(t),\quad t\geq 0,\tag{1} \] with the initial conditions \[ \sum\limits_{k=0}^{m-1}c_{ik}u^{(k)}(0)=\lambda_i,\quad i=0,1,\dotsc,m-1.\tag{2} \]
The reproducing kernel Hilbert space method (RKHSM) is applied to (1), (2). Its approximate solution is obtained by truncating the \(n\)-term of exact solution. Some examples are displayed to demonstrate the computation efficiency of the method. We also compare the performance of the method with a particular Runge-Kutta method, a one-leg \(\theta \)-method and variational iteration method. Experimental results indicate that the RKHSM is an accurate and efficient method for the solution of neutral functional-differential equations with proportional delays.

MSC:

34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
34K06 Linear functional-differential equations
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