triangular functions

Error estimate and rate of convergence are two important topics in the field of numerical analysis. A convenient normed space corresponding to the problem under regard can have better upper bounds. This paper introduces a weighted normed space $L_{p,\omega}$ which from the measure theory point of view, is a special case of $L^{p}$ space. This space is a modification of $L_{p,\alpha}$ space, which is introduced before in \cite{Baghani}. Next, by using $L_{p,\alpha}$-norm, we compute a two-variable upper bound of the triangular function.

In this paper, we propose direct methods to solve linear delay differential equations (DDEs) based on vector forms of Block-Pulse Functions (BPFs) and Triangular Functions (TFs). Operational matrix of integration of BPFs and TFs are applied to transform LDDE to a linear system of algebraic equations. Further, some numerical examples are presented to indicate the reliability and accuracy of these methods. Convergence analysis of the present method has been discussed.

A numerical method for solving linear integral equations of the second kind is formulated. Based on a special representation of vector forms of triangular functions and the related operational matrix of integration, the integral equation reduces to a linear system of algebraic equations. Generation of this system needs no integration, so all calculations can easily be implemented. Numerical results for some examples show that the method has a good accuracy.

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}.

This article proposes a fast and accurate expansion-iterative method for solving second kind linear Volterra integral equations. The method is based on a special representation of vector forms of triangular functions (TFs) and their operational matrix of integration. By using this approach, solving the integral equation reduces to solve a recurrence relation. The approximate solution of integral equation is iteratively produced via the recurrence relation.

This paper introduces a numerical method for solving the vasicek model by using a stochastic operational matrix based on the triangular functions (TFs) in combination with the collocation method. The method is stated by using conversion the vasicek model to a stochastic nonlinear system of $2m+2$ equations and $2m+2$ unknowns. Finally, the error analysis and some numerical examples are provided to demonstrate applicability and accuracy of this method.