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For a long time now, traffic equations have been considered, and different modeling has been done for it. In this article, we work on the macroscopic model, especially the most famous light model. Because these models are among the stiff and shocking problems, theoretical methods do not give good answers to these problems. This paper describes a meshless method to solve the traffic flow equation as a stiff equation. In the proposed method, we also use the exponential time differencing (ETD) method and the exponential time differencing fourth-order Runge-Kutta (ETDRK4). The purpose of this new method is to use methods of the moving least squares (MLS) method and a modified exponential time differencing fourth-order Runge-Kutta scheme. To solve these equations, we use the meshless method MLS to approximate the spatial derivatives and then use method ETDRK4 to obtain approximate solutions. In order to improve the possible instabilities of method ETDRK4, approaches have been stated. The MLS method provided good results for these equations due to its high flexibility and high accuracy and has a moving window and obtains the solution at the shock point without any false oscillations. The technique is described in detail, and a number of computational examples are presented., Author(s): Neda Najafzadeh [1]; Saeid Abbasbandy (corresponding author) [1]; Elyas Shivanian [1] 1. Introduction Car traffic modeling has been around since time immemorial, around the early twentieth century. Many methods [...]

In this paper, the exponential approximation is applied to solve high-order nonlinear differential equations. The main idea of this method is based on the matrix representations of the exponential functions and their derivatives by using collocation points. To indicate the usefulness of this method we employ it for some well-known high-order nonlinear equations like Riccati, Lane-Emden and so on. The numerical approximate solutions are compared with available(existing) exact(analytical) solutions and the comparisons are made with other methods to show the accuracy of the proposed method. For convergence and error analysis of the method, criteria for a number of basis sentences presented. The method has been reviewed by several examples to show its validity and reliability. The reported examples illustrate that the method is appropriately efficient and accurate. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new approach to solve weakly singular fractional delay integrodifferential equations. In the proposed approach, we apply the operational matrices of fractional integration and delay function based on the shifted Chebyshev polynomials to approximate the solution of the considered equation. By approximating the fractional derivative of the unknown function as well as the unknown function in terms of the shifted Chebyshev polynomials and substituting these approximations into the original equation, we obtain a system of nonlinear algebraic equations. We present the convergence analysis of the proposed method. Finally, to show the accuracy and validity of the proposed method, we give some numerical examples. [ABSTRACT FROM AUTHOR]