In this paper, we present an approximate method to solve the solution of the second kind Volterra integral equations. This method is based on a previous scheme, applied by Maleknejad ‎et al., ‎‎[K. Maleknejad ‎and Aghazadeh, Numerical solution of Volterra integral equations of the second kind with convolution kernel by using Taylor-series expansion method, ‎Appl. Math. Comput.‎ (2005)]‎ to gain the approximate solution of the second kind Volterra integral equations with convolution kernel and Maleknejad ‎et al. ‎[K. Maleknejad ‎and‎ T. Damercheli, Improving the accuracy of solutions of the linear second kind volterra integral equations system by using the Taylor expansion method, ‎Indian J. Pure Appl. Math.‎ (2014)] ‎to gain the approximate solutions of systems of second kind Volterra integral equations with the help of Taylor expansion method. The Taylor expansion method transforms the integral equation into a linear ordinary differential equation (ODE) which, in this case, requires specified boundary conditions. Boundary conditions can be determined using the integration technique instead of differentiation technique. This method is more stable than derivative method and can be implemented to obtain an approximate solution of the Volterra integral equation with smooth and weakly singular kernels. An error analysis for the method is provided. A comparison between our obtained results and the previous results is made which shows that the suggested method is accurate enough and more ‎stable.‎ تفاصيل المقالة

In this article, a fast numerical approach is proposed for finding the solution of nonlinear delay differential equations by using hybrid Taylor and Block-pulse Functions (HTBPFs). Firstly, some features of hybrid functions which are a combination of Block-Pulse functions and Taylor polynomials on the interval are introduced [0, 1) . In this spectral approach, the operational matrices of stretch, derivation and coefficient matrices are utilized. Based on these piecewise functions, we transfer delay differential equations (DDEs) into a system of linear or nonlinear algebraic equations. Also, in this numerical approach, it is shown that these operational matrices are sparse which is an effective advantage of the fast implementation of numerical computation. Then, error analysis is done. Finally, three examples are solved to show that the new proposed approach is comparable with other methods of high accuracy and efficiency. تفاصيل المقالة

In this paper, we consider Volterra integral equations of the first kind. Then by extending the modified Block-pulse functions(MBPFs) on the Volterra integral equation of the second kind obtained from Volterra integral equation of the first kind, we obtain the approximate solution. Some theorems are proved to provide an error analysis for proposed method. Numerical examples show that the proposed scheme has a suitable degree of accuracy. تفاصيل المقالة