Graduate Thesis Or Dissertation
 

Approximate solutions of Fredholm integral equations of the second kind with singular kernels

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  • The kernel subtraction method of Kantorovich and Krylov is studied in the setting of "Collectively Compact Operator Approximation Theory." Fredholm integral equations of the second kind have weakly singular kernels smoothened by a simple manipulation. This allows for accurate numerical estimates of low order. The method is first explained in detail, along with a general description of the kernel. The numerical integration technique is discussed and then basic results from operator theory are developed. The technique is formulated in an operator setting, and compactness, collective compactness, and convergence are proven. Error bounds are established, and examples given to compare the three techniques of kernel subtraction, kernel modification, and kernel factoring.
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