Piecewise shooting reproducing kernel method for linear singularly perturbed boundary value problems

Abstract

In this letter, a new numerical method is proposed for solving second order linear singularly perturbed boundary value problems with left layers. Firstly a piecewise reproducing kernel method is proposed for second order linear singularly perturbed initial value problems. By combining the method and the shooting method, an effective numerical method is then proposed for solving second order linear singularly perturbed boundary value problems. Two numerical examples are used to show the effectiveness of the present method.

Using Reproducing Kernel for Solving a Class of Fractional Order Integral Differential Equations

“…Assuming that fx i g ∞ i=1 is dense on the interval ½0, 1, put ϕ ijk = l * ij k x k ðxÞ, where l * ij is the adjoint operator of l ij . From [23][24][25], we have…”

Hypersingular nonlinear boundary-value problems with a small parameter

“…Similar problems differ fundamentally from the classical problems with a boundary layer (in which the derivative at the boundary is of the order of ε −1 or less) and no such problems have been encountered in the literature (see, for example, [1][2][3][4][5][6][7][8][9][10][11]). At the present time, there are no effective numerical methods that allow directly (without a preliminary nonlinear point transformation of the given equation) to consider such hypersingular problems even for moderately small ε ≃ 1 50 and, especially, for smaller values of ε.…”

“…An important qualitative feature of singular boundary-value problems is that for the zero value of a small parameter the order of the differential equation under consideration decreases and some parts of the boundary conditions cannot be satisfied. Various problems and solution methods for ODEs and PDEs with a small parameter at the highest derivative are described, for example, in [1][2][3][4][5][6][7][8][9][10][11][12][13].…”

New method for investigating the density-dependent diffusion Nagumo equation

“…Aronszajn [4] reduced the studies of the formers and presented a systematic reproducing kernel theory containing the Bergman kernel function. For more details see [5][6][7][8][9][10][11][12][13].…”