Reproducing kernel method for solving singularly perturbed differential-difference equations with boundary layer behavior in Hilbert space (Q2406289)

In this paper, the Reproducing Kernel Hilbert Space Method (RKHSM) is used to obtain a proper approximation of the solution to the linear singularly perturbed problem with a boundary layer \[ \varepsilon u''(x)+p(x)u'(x-\delta)+q(x)u(x)=f(x),\;x\in[0,1], \] \[ u(x)=\phi (x)\text{ for } x\in[-\delta,0],\;u(1)=\gamma, \] where \(\varepsilon\) and \(\delta\) are the small parameters, \(0<\varepsilon,\delta\ll 1\); \(p>0\), \(q<0\), \(f\), \(\phi\) are smooth functions and \(\gamma\) is a constant. If the boundary layer exists on the left side of the interval \([0,1]\) (analogously for the boundary layer at \(x=1\)), the approximate solution of the original problem is obtained by combining the functions \(u_{1n_1}(x)\) in the regular region \([d,1]\) and \(u_{2n_2}(x)\) in the boundary layer region \([0,d],\) \(0<d<1,\) where \(u_{1n_1}\) and \(u_{2n_2}\) are the approximate solutions of the boundary value problems \[ (\varepsilon -\delta p(x))u''(x)+p(x)u'(x)+q(x)u(x)=f(x), \] \[ u(1)=\gamma, \;x\in[d,1] \] and \[ (\varepsilon -\delta p(x))u''(x)+p(x)u'(x)+q(x)u(x)=f(x), \] \[ u(0)=\phi(0), \;u(d)\;\mathrm{is \;known},\;x\in[0,d], \] that have been obtained using RKHSM in the spaces \(W^3_2[d,1]\) and \(W^3_2[0,d]\), respectively. The efficacy of the proposed algorithm is confirmed by the error analysis and demonstrated by examples.