So far, there are no any publications for solving and obtaining a numerical solution of Volterra integro-differential equations in the complex plane by using the finite element method. In this work, we use the linear B-spline finite element method (LBS-FEM) and cubic B-spline finite element method (CBS-FEM) for solving this equation in the complex plane. We also discuss the error and convergence of the method. Furthermore, we give some numerical examples to substantiate efficiency of the proposed method.

In this work, we applied a new method for solving the linear weakly singular mixed Volterra-Fredholm integral equations. We now begin the theoretical study with acquirement of the variational form; in addition, we are using Bernstein spectral Galerkin method to be approximate to my problems. We estimate the error of the method by proved some theorems. Moreover, in the final section, we solved some numerical examples.

Everyone knows about the complicated solution of the nonlinear Fredholm integro-differential equation in general. Hence, often, authors attempt to obtain the approximate solution. In this paper, a numerical method for the solutions of the nonlinear Fredholm integro-differential equation (NFIDE) of the second kind in the complex plane is presented. In fact, by using the properties of Rationalized Haar (RH) wavelet, we try to give the solution of the problem. So far, as we know, no study has yet been attempted for solving the NFIDE in the complex plane. For this purpose, we introduce the continuous integral operator and real valued function. The Banach fixed point theorem guarantees that, under certain assumptions, the integral operator has a unique solution. Furthermore, we give an upper bound for the error analysis. An algorithm is presented to compute and illustrate the solutions for some numerical examples.

Grain growth in Ti6Al4V alloy during fusion welding decreases yield stress and tensile strength. This study examined the effect of mechanical vibration of the work piece during GTAW welding on the mechanical and metallurgical properties of Ti6Al4V. The structures of all welded specimens at different levels of vibration was examined and it was found that, during 330 Hz vibration, grain size in the welded metal zone decreased significantly over that of pieces welded without vibration. GTAW at 330 Hz significantly increased the mechanical properties and produced the highest yield strength, tensile strength and percentage of elon gation. The highest level of hardness in the welded metal zone was achieved under this condition.

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