A new efficient method for cases of the singular integral equation of the first kind. (English) Zbl 1342.65240
Summary: Various cases of Cauchy type singular integral equation of the first kind occur rather frequently in mathematical physics and possess very unusual properties. These equations are usually difficult to solve analytically, and it is required to obtain approximate solutions. This paper investigates the numerical solution of various cases of Cauchy type singular integral equations using reproducing kernel Hilbert space (RKHS) method. The solution \(u(x)\) is represented in the form of a series in the reproducing kernel space, afterwards the \(n\)-term approximate solution \(u_n(x)\) is obtained and it is proved to converge to the exact solution \(u(x)\). The major advantage of the method is that it can produce good globally smooth approximate solutions. Moreover, in this paper, an efficient error estimation of the RKHS method is introduced. Finally, numerical experiments show that our reproducing kernel method is efficient.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45E05 | Integral equations with kernels of Cauchy type |
Keywords:
Cauchy type integral equation; singular integral equation; reproducing kernel Hilbert space; error estimationReferences:
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