In this paper, the existence and uniqueness of solution of the Fredholm-Volterra integral equation (F-VIE), with a generalized singular kernel, are discussed and proved in the space    L2(Ω) X C(O,T)  The Fredholm integral term (FIT) is considered in position while the Volterra integral term (VIT) is considered in time. Using a numerical technique we have a system of Fredholm integral equations (SFIEs). This system of integral equations can be reduced to a linear algebraic system (LAS) of equations by using two different methods. These methods are: Toeplitz matrix method and Product Nyström method. A numerical examples are considered when the generalized kernel takes the following forms:  Carleman function, logarithmic form, Cauchy kernel, and Hilbert kernel.

An integral method, complex variable method, is used to obtain exact and closed expressions for Goursat functions for the stretched infinite plate weakened by a hole having arbitrary shape. The inner of   the infinite plate is free from stresses. The plates considered are conformally mapped on the area of the right half – plane.

Complex variable methods are used to solve the thermo-elastic problem of the infinite  isotropic homogeneous plate with a hypitrochoidal hole with multi round corners conformally mapped on the domain outside a unit circle by means of a rational mapping function .The thermo-elastic problem is equivalent to finding two analytic functions(Goursat functions)at any point  z =x + iy  within the region of the plate . The problem is transformed to solve an integrodifferential equation, in the complex plane, with singular kernel.  Closed expressions for the Goursat function and consequently the tangential thermo-elastic stresses on the boundary of the hole are obtained in quadrature in the presence of a uniform heat stream.

The design centering problem seeks for the optimal values for the system designable parameters that maximize the production yield (probability of satisfying the design specifications by the manufactured systems).A new method for design centering and region approximation for a convex and bounded feasible region is introduced. The method finds iteratively a sequence of increasing-volume ellipsoids enclosing tightly selective sets of feasible points. These ellipsoids are found using semidefinite programming problem and known as Löwner-John ellipsoids. The sequence of Löwner-John ellipsoids is well definedin the method to converge to the minimum volume ellipsoid containing the feasible region. The center of the final ellipsoiddefines a design center for the proposed design problem and the ellipsoid itself is considered as a region approximation for the feasible region. Are-use of system simulations is performedin order to minimize the overall computational effort. Numerical and practical examples are considered to show the effectiveness of the new method.

In this paper, the existence of a unique solution of  Volterra-Hammerstein integral equation of the second kind (V-HIESK)  is proved by using Banach fixed point theorem (BFPT) in the space L2(Ω) X C[O,T]  , where  represents the domain of integration of the variable space and  is the time. Then, different kinds of projection-iteration methods (PIMs) for solving this integral equation in the space  L2(Ω) X C[O,T] are introduced. Finally, we deduced that: this method is quick convergent and the estimating error is better than the approximate error in the method of successive approximation for solving the integral equation numerically.