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Numeric methods for solution of the weakly regular linear and nonlinear evolutionary (Volterra) integral equations of the first kind are proposed. The kernels of such equations have jump discontinuities along the continuous curves (endogenous delays) which start at the origin. In order to linearize these equations the modified Newton–Kantorovich iterative process is employed. Two direct quadrature methods based on the piecewise constant and piecewise linear approximation of the exact solution are proposed for linear solutions. The accuracy of proposed numerical methods is O ( 1 / N ) and O ( 1 / N 2 ) respectively. A certain iterative numerical scheme enjoying the regularization properties is suggested. Furthermore, generalized numerical method for nonlinear equations is adduced. The midpoint quadrature rule in all the cases is employed. In conclusion several numerical examples are applied in order to demonstrate the efficiency of proposed numerical methods. [ABSTRACT FROM AUTHOR]

In this research, an efficient scheme is presented to validate the numerical results and solve the second kind integral equations (IEs). For this reason the homotopy perturbation method (HPM) is illustrated and the stochastic arithmetic is applied to implement the CESTAC 1 1 Controle et Estimation Stochastique des Arrondis de Calculs. method for solving IEs. The accuracy of method is shown by proving a main theorem. Also, the CADNA 2 2 Control of Accuracy and Debugging for Numerical Applications. library is used instead of other usual softwares. Applying the mentioned method, the optimal approximation, iteration, validation of results and any numerical instability can be found whereas the floating-point arithmetic (FPA) has not these properties. Some examples are solved to determine the significance of applying the SA in place of the FPA. [ABSTRACT FROM AUTHOR]