Solving Second Kind Volterra-Fredholm Integral Equations by Using Triangular Functions (TF) and Dynamical Systems

The method of triangular functions (TF) could be a generalization form of the functions of block-pulse (Bp)‎. ‎The solution of second kind integral equations by using the concept of TF would lead to a nonlinear equations system‎. ‎In this article‎, ‎the obtained nonlinear system has been solved as a dynamical system‎. ‎The solution of the obtained nonlinear system by the dynamical system through the Newton numerical method has got a particular priority‎, ‎in that‎, ‎in this method‎, ‎the number of the unknowns could be more than the number of equations‎. ‎Besides‎, ‎the point of departure of the system could be an infeasible point‎. ‎It has been proved that the obtained dynamical system is stable‎, ‎and the response of this system can be achieved by using of the fourth order Runge-Kutta‎. ‎The results of this method is comparable with the similar numerical methods; in most of the cases‎, ‎the obtained results by the presented method are more efficient than those obtained by other numerical methods‎. ‎The efficiency of the new method will be investigated through examples.

Keywords:

Second kind Fredholm-Volterra integral equations‎ , ‎Nonlinear systems‎ , ‎Dynamical systems‎ , ‎Triangular functions‎ , ‎Block-pulse functions