Approximate solution of some nonlinear classes of Abel integral equations using hybrid expansion. (English) Zbl 1471.65223
Summary: Numerical schemes for nonlinear weakly singular Volterra and Fredholm integral equations of the first kind are rarely investigated in the literature. In this paper, we present numerical solutions for these types of equations including Abel equations by hybrid block-pulse functions and Legendre polynomials. Hybrid functions give us the opportunity to attend a highly accurate solution by adjusting the orders of block-pulse functions and Legendre polynomials. The main idea of the scheme is based on using the precise forms of the known functions in the approximation procedure which yields the simplicity, reliability and high accuracy of the method. This simple scheme converts these types of equations into a linear system of equations. The focus of this paper is to investigate the convergence analysis and to show high convergence rate of the scheme. Numerical examples confirm the efficiency and superiority of the present approach in comparison with those already available in the literature.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
Keywords:
direct numerical scheme; singular kernel; first kind nonlinear integral equations; convergence analysis; hybrid functionsReferences:
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