On nonlinear singular BVPs with nonsmooth data. II: Convergence of collocation methods. (English) Zbl 1482.65116
Summary: We discuss numerical solution of boundary value problems for systems of nonlinear ordinary differential equations with a time singularity,
\[
x'(t)=\frac{M(t)}{t}x(t)+\frac{f(t,x(t))}{t},\quad t\in(0,1],\quad b(x(0),x(1))=0,
\]
where \(M:[0,1]\to\mathbb{R}^{n\times n}\) and \(f:[0,1]\times\mathbb{R}^n\to\mathbb{R}^n\) are continuous matrix-valued and vector-valued functions, respectively. Moreover, \(b:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n\) is a continuous nonlinear mapping which is specified according to a spectrum of the matrix \(M(0)\) to guarantee the BVP to be well-posed. For the case where \(M(0)\) has eigenvalues with nonzero real parts, we prove new convergence results for the collocation method and analytical results about the necessary smoothness of the solution to the problem required in the numerical analysis.
We illustrate the theory by means of numerical examples.
For Part I see [Appl. Numer. Math. 130, 23–50 (2018; Zbl 1397.34048)].
We illustrate the theory by means of numerical examples.
For Part I see [Appl. Numer. Math. 130, 23–50 (2018; Zbl 1397.34048)].
MSC:
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
Keywords:
boundary value problems; ordinary differential equations; time singularity; global existence and uniqueness; collocation method; convergence analysisCitations:
Zbl 1397.34048References:
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