Singular expansions and collocation methods for generalized Abel integral equations. (English) Zbl 1530.65188
Summary: In this paper, the generalized Abel integral equation with singular solution is studied. First, the finite-term psi-series expansion of the solution about the origin is derived using the method of undetermined coefficients, which gives the complete singular information of the solution. Second, this singular expansion is used to separate the singularity such that the Abel integral equation is converted into a perturbed one with smooth solution on a regular interval. Third, a product trapezoidal method is designed to solve the transformed equation on the regular interval and the convergence analysis is conducted to show that the scheme has second order accuracy combining with a controllable perturbation term. Fourth, a Chebyshev collocation method is further constructed on the regular interval to show that the global orthogonal polynomial interpolation is also suitable for solving the transformed Abel integral equation with high accuracy. Finally, two numerical examples confirm the correctness of the truncated psi-series solution and the effectiveness of the piecewise and global collocation methods with singularity separation for solving this kind of Abel integral equation.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
Keywords:
Abel integral equation; series expansion of the solution about the origin; product trapezoidal method; Chebyshev collocation method; singularity separation; convergence analysisReferences:
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