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In this article, thermal behavior analysis of nonlinear fin problem with power-law heat transfer coefficient is studied to determine temperature distribution. This new supposition for the thermal conductivity, exponential-law temperature dependent, makes it to be nonlinear that is a general case in some sense. It is shown that the governing fin equation, that is, a nonlinear second-order differential equation, is exactly solvable with proper boundary conditions. To this purpose, the order of differential equation is reduced and then is converted into a total differential equation by multiplying a proper integration operant. An exact analytical solution is given to advance physical meaning, and the existence of unique solution for some specific values of the parameters of the model is demonstrated. The results are shown graphically. It is observed that fin efficiency is decreasing with respect to the power-law mode for heat transfer.

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 the present paper, the relatively new method of Radial Basis Function-Generated Finite Difference (RBF-FD) is used to solve a class of Partial Differential Equations (PDEs) with Dirichlet and Robin boundary conditions. For this approximation, Polyharmonic Splines (PHS) are used alongside Polynomials. This combination has many benefits. On the other hand, Polyharmonic Splines have no shape parameter and therefore relieve us of the hassle of calculating the optimal shape parameter. As the first problem, a two-dimensional Poisson equation with the Dirichlet boundary condition is investigated in various domains. Then, an elliptic PDE with the Robin boundary condition is solved by the proposed method. The results of numerical studies indicate the excellent efficiency, accuracy and high speed of the method, while for these studies, very fluctuating and special test functions have been used. [ABSTRACT FROM AUTHOR]

In this paper, the problem of determining heat transfer from convecting-radiating fin of triangular and concave parabolic shapes is investigated.We consider one-dimensional, steady conduction in the fin and neglect radiative exchange between adjacent fins and between the fin and its primary surface. A novel intelligent computational approach is developed for searching the solution. In order to achieve this aim, the governing equation is transformed into an equivalent problem whose boundary conditions are such that they are convenient to apply reformed version of Chebyshev polynomials of the first kind. These Chebyshev polynomials based functions construct approximate series solution with unknown weights. The mathematical formulation of optimization problem consists of an unsupervised error which is minimized by tuning weights via interior point method. The trial approximate solution is validated by imposing tolerance constrained into optimization problem. Additionally, heat transfer rate and the fin efficiency are reported.

This work studies the existence and the uniqueness of the solution to a kind of high-order nonlinear fractional integrodifferential equations involving Rieman-Liouville fractional derivative. The boundary condition is of integral type which entangles ending point of the domain. First, the unique exact solution is extracted in terms of Green's function for the linear fractional differential equation and then Banach contraction mapping theorem is applied to prove the main result in the case of general nonlinear source term. Furthermore, our main result is demonstrated by an illustrative example to show its legitimacy and applicability. [ABSTRACT FROM AUTHOR]