الفهرس 
Preliminary concept of the integral equationOnly 14 pages are availabe for public view
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Abstract
This M.Sc. thesis is concerned with the numerical solutions of a certain protentional-type weakly singular Fredholm integral equation of the first kind, whose unknown function is singular near the end-points of the integration domain and has a weakly singular logarithmic kernel. We consider the boundary integral equation which is equivalent to the solution of the Dirichlet boundary value problem for Laplace equation for an open arc in the plane. The aim key and motivation of this thesis is to establish straightforward and non-complicated methods for the numerical solution of such equations in such a manner that we can clearly simplify the computation, easily treat the singularities, ensure a superior accuracy of the obtained numerical solutions and the desire to better overcome the computational difficulties explained by the other methods during the solution’s procedures and the singularities’ treatments. An important motivation for establishing the presented three methods is to easy evaluating the functional values of the unknown function at the end-points of the integration domain despite the failure of the exact solution or other published methods to find these values. This M.Sc. thesis consists in three published papers in which three new numerical methods are given. These methods reduce the round-off errors of computations and give a high order of accuracy to the solutions and save the computations time. The first method is based on the approximation of the unknown function and the given potential data function by monic Chebyshev polynomials approximation, the second is based on Newton interpolation in matrix form and in the third method, the unknown function is approximated by the economized monic Chebyshev polynomials and the given potential data function is approximated by only monic Chebyshev polynomials. Moreover, we investigate a technique for the analytical treatments of the singularity of the kernel by creating two asymptotic expressions which are established by expanding the two parametric functions of the parameterized kernel about the singular parameter using Taylor polynomial of the first degree.
M.Sc. Thesis’s Abstract
IV
Furthermore, we establish an approach to treat the singularity of the unknown function by replacing it with a product of two functions; the first is a regular unknown function; while the second is a badly-behaved function. The badly-behaved function is factorized into two semi-singular functions. Thus, the singularities of the unknown function and the kernel are completely isolated. We also adapted Gauss-Legendre formula in matrix form for the computation of the obtained proper integrals. Thus, and by an accurate choice of the collocation points, the required numerical solution of the considered integral equation is found to be equivalent to the solution of a linear system of algebraic equations in matrix form. Numerical solutions are illustrated with tables and graphs; showing that the presented three methods clearly simplifying the computations
and ensure a superior precise of the obtained numerical solution as well giving the functional values of the unknown function at the end-points of the integration domain despite the failure of the exact solution to find these values.