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**A spectral regularization method for a Cauchy problem of the modified Helmholtz equation.**
*(English)*
Zbl 1198.65114

Summary: We consider the Cauchy problem for a modified Helmholtz equation, where the Cauchy data is given at \(x=1\) and the solution is sought in the interval \(0<x<1\). A spectral method together with the choice of the regularization parameter is presented and an error estimate is obtained. Combining the method of lines, we formulate regularized solutions which are stably convergent to the exact solutions.

### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

34A30 | Linear ordinary differential equations and systems |

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

65L70 | Error bounds for numerical methods for ordinary differential equations |

### Keywords:

stability; convergence; Cauchy problem; Helmholtz equation; spectral method; regularization; error estimate; method of lines
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\textit{A. Qian} et al., Bound. Value Probl. 2010, Article ID 212056, 13 p. (2010; Zbl 1198.65114)

### References:

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