A method for the approximate solution of the second-order linear differential equations in terms of Taylor polynomials. (English) Zbl 0887.65084

Given a second-order linear differential equation for \(y:x \mapsto y(x)\) \[ P(x)y'' +Q(x)y' +R(x)y= f(x) \tag{1} \] where \(P,Q,R,f\) are elements of \(C^n([a,b], \mathbb{R})\) with \(P(x)\neq 0\) \(\forall x\in[a,b]\) and associated conditions of the form \[ \sum^1_{i=0} \bigl[a_i y^{(i)} (a)+b_iy^{(i)} (b)+c_iy^{(i)} (c)\bigr]= \lambda \tag{2} \]
\[ \sum^1_{i=0} \bigl[\alpha_iy^{(i)} (a)+ \beta_iy^{(i)} (b)+ \gamma_i y^{(i)} (c) \bigr]= \mu \] where \(a\leq c\leq b\) and \(a_i,b_i,c_i, \alpha_i, \beta_i, \gamma_i, \lambda\) and \(\mu\) are suitable real coefficients.
The author is interested in an approximate solution of problem (1), (2) in terms of Taylor polynomials about any point \[ y(x)= \sum^N_{n=0} {1\over n!} y^{(n)} (c)(x-c)^n \] with \(a\leq c\leq b\) and \(y^{(n)} (c)\), \(n=0,1,\dots,N\) are the coefficients to be determined. A matrix method is introduced to determine the Taylor polynomial of order \(N\) of a solution to (1), (2). The method is illustrated by considering four examples.
Reviewer: H.Ade (Mainz)


65L10 Numerical solution of boundary value problems involving ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
Full Text: DOI


[1] DOI: 10.1080/0020739890200310 · Zbl 0683.45001
[2] DOI: 10.1080/0020739940250501 · Zbl 0823.45005
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[6] DOI: 10.1080/0020739890200101 · Zbl 0684.34005
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