?triangular functions?

Referring to one of the recent works of the authors, presented in~\cite{differentialbpf}, for numerical solution of linear differential equations, an alternative scheme is proposed in this article to considerably improve the accuracy and efficiency. For this purpose, triangular functions as a set of orthogonal functions are used. By using a special representation of the vector forms of triangular functions and the related operational matrix of integration, solving the differential equation reduces to solve a linear system of algebraic equations. The formulation of the method is quite general, such that any arbitrary linear differential equation may be solved by it. Moreover, the algorithm does not include any integration and, instead, uses just sampling of functions, that results in a lower computational complexity. Also, the formulation of this approach needs no modification when a singularity occurs in the coefficients of differential equation. Some problems are numerically solved by the proposed method to illustrate that it is much more accurate and applicable than the prior method in~\cite{differentialbpf}.

The method of triangular functions (TF) could be a generalization form of the functions of block-pulse (Bp)ý. ýThe solution of second kind integral equations by using the concept of TF would lead to a nonlinear equations systemý. ýIn this articleý, ýthe obtained nonlinear system has been solved as a dynamical systemý. ýThe solution of the obtained nonlinear system by the dynamical system through the Newton numerical method has got a particular priorityý, ýin thatý, ýin this methodý, ýthe number of the unknowns could be more than the number of equationsý. ýBesidesý, ýthe point of departure of the system could be an infeasible pointý. ýIt has been proved that the obtained dynamical system is stableý, ýand the response of this system can be achieved by using of the fourth order Runge-Kuttaý. ýThe results of this method is comparable with the similar numerical methods; in most of the casesý, ýthe obtained results by the presented method are more efficient than those obtained by other numerical methodsý. ýThe efficiency of the new method will be investigated through examples.