Jacobi interpolation approximations and their applications to singular differential equations.

*(English)*Zbl 0984.41004The article is devoted to Jacobi-Gauss-type interpolation approximations in certain Hilbert spaces with their applications to numerical solutions of singular differential equations. A related problem is numerical simulation of differential equations in unbounded domains.

In actual calculations Jacobi interpolation approximations are preferable, since one only needs to evaluate the values of unknown functions at the interpolating points.

The article contains six sections. Section 1 is an introduction. In section 2 the authors recall some basic results related to Jacobi polynomials and Jacobi interpolations. Section 3 is devoted to several weighted embedding inequalities and inverse inequalities. Various orthogonal projections in certain Hilbert spaces are discussed. The main results of the article are presented in section 4. Jacobi-Gauss, Jacobi-Gauss-Radau and Jacobi-Gauss-Lobatto interpolation approximations are considered here. These results play an important role in the numerical analysis of the Jacobi pseudospectral method for singular differential equations and related problems in unbounded domains. The problem of the stability of interpolation is investigated. Section 5 contains some applications. As examples, the authors consider a linear steady singular problem and a nonlinear singular evolutionary problem. The corresponding pseudospectral schemes are constructed and their spectral accuracy are proved. In section 6 some numerical results, which coincide with the theoretical analysis, are presented. The obtained results are well-illustrated.

Finally, the authors make some remarks which may be very useful for a following investigation. In particular, the authors note that it is more interesting to develop the Jacobi interpolation in multiple dimensions and their applications to numerical solutions of differential equations.

The article contains an extended bibliography which reflects the modern state of the investigations in this area.

In actual calculations Jacobi interpolation approximations are preferable, since one only needs to evaluate the values of unknown functions at the interpolating points.

The article contains six sections. Section 1 is an introduction. In section 2 the authors recall some basic results related to Jacobi polynomials and Jacobi interpolations. Section 3 is devoted to several weighted embedding inequalities and inverse inequalities. Various orthogonal projections in certain Hilbert spaces are discussed. The main results of the article are presented in section 4. Jacobi-Gauss, Jacobi-Gauss-Radau and Jacobi-Gauss-Lobatto interpolation approximations are considered here. These results play an important role in the numerical analysis of the Jacobi pseudospectral method for singular differential equations and related problems in unbounded domains. The problem of the stability of interpolation is investigated. Section 5 contains some applications. As examples, the authors consider a linear steady singular problem and a nonlinear singular evolutionary problem. The corresponding pseudospectral schemes are constructed and their spectral accuracy are proved. In section 6 some numerical results, which coincide with the theoretical analysis, are presented. The obtained results are well-illustrated.

Finally, the authors make some remarks which may be very useful for a following investigation. In particular, the authors note that it is more interesting to develop the Jacobi interpolation in multiple dimensions and their applications to numerical solutions of differential equations.

The article contains an extended bibliography which reflects the modern state of the investigations in this area.

Reviewer: Alexander Tovstolis (Donetsk)

##### MSC:

41A05 | Interpolation in approximation theory |

41A10 | Approximation by polynomials |

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

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