The Triangle Hermite Interpolation Wavelet Algorithm For Solving Singular Integral Equations With Hilbert Kernel

The numerical calculation on the solutions of integral equations is an important part inscientific calculation, it provides a powerful theoretical basis for studying on manymathematical and physics problems such as elastic theory and fluid mechanics problems, andwith the advance in scientific calculation method, it is gradually showing more and moreimportant role. Singular integral equation is a kind of important integral equation, and itssingularity increases the difficulty in solving the equation. How to improve the precision andstability of the numerical solutions of singular integral equations and reduce computation costhas been many scholars’ concern.In order to improve the convergence speed and stability in solving, we choose to donumerical calculation on the Quak triangle Hermite-type interpolation wavelet functionsspace. Wavelet is applied to the numerical computation of equations because wavelet can notonly depict Hilbert space and constitute a group of base of Hilbert space, but also waveletfunctions have good attenuating property, smoothness and vanishing moment, which mayeffectively improve numerical stability of the solution and reduce computation load. Manyexperts and scholars devote themselves to the research on this aspect, and have some goodresults. In this paper, we expand the discussed Hilbert nuclear to trigonometric series ingeneralized functions, and solve in the triangle Hermite-type interpolation wavelet functionsspace. The benefits of this treatment are that equation can be solved in the framework of thetriangle function system and by using the orthogonality of triangle function system; we canavoid the errors which were caused by human cutting off in infinite series and get the desiredresults.In this paper, we use scale functions space and wavelet functions space to solve thesingular integral equation with Hilbert nuclear respectively. At first, we expand the Hilbertsingular integral nuclear to trigonometric series under generalized functions, then we useGalerkin method to discrete integral equation on scale functions space and wavelet functionsspace respectively. At last, by solving the system of linear equations that we got, we get thesolution of the integral equation. We get the coefficient matrix is the block matrix, threesub-block matrices can be decomposed into the sum of anti-symmetric and symmetric, andthe last block matrix is anti-symmetric. For an order of2n22n2coefficient matrix, we needto calculate the number of elements is much less than half of the total matrix elements and itgreatly reduces the amount of computation. The result of the paper shows that the wavelet method applies to solve the singular integral equation with Hilbert nuclear respectively well.In the end, some numerical examples are given to verify the validity of the results.