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A new high-accuracy difference method for a class of weakly nonlinear singular boundary-value problems. (English) Zbl 1114.65091

Summary: A fourth-order finite difference method based on a uniform mesh is described by using Chawla’s identity [cf. M. Sakai and R. A. Usmani, BIT 30, No. 3, 566–568 (1990; Zbl 0705.65059)] for a class of singular two point boundary-value problems \[ (p(x)y')'=p(x)f(x,y),\;0<x\leq 1;\;y(0)=A,\;\alpha y(1)+\beta y'(1)=\gamma \] with \(p(x)>0\) on \((0,1]\) and \(p(x)=x^{b_0}g(x)\), for all \(x\) in \((0,1]\), \(0\leq b_0< 1\). Fourth-order convergence is established under quite general conditions on \(f(x,y)\). The method reduces to Numerov’s method for \(p(x)=1\) and boundary conditions \(y(0)=A\), \(y(1)=B\). This method provides better results than some existing fourth-order methods ford \(p(x)=x^{b_0}\), \(0\leq b_0<1\) and boundary conditions \(y(0)=A\), \(y(1)=B\), which is verified by two examples and also the order of the method is verified for general functions \(p(x)\).

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations

Citations:

Zbl 0705.65059
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References:

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