Numerical solution of time-fractional telegraph equation by using a new class of orthogonal polynomials

‎In this article‎, ‎an efficient numerical method based on a new class of orthogonal polynomials‎, ‎namely Chelyshkov polynomials‎, ‎has been presented to approximate solution of time-fractional telegraph (TFT) equations‎. ‎The fractional operational matrix of the Chelyshkov polynomials along with the typical collocation method is used to reduces TFT equations to a system of algebraic equations‎. ‎The error analysis of the proposed collocation method is also investigated‎. ‎A comparison with other published results confirms that the presented Chelyshkov collocation approach is efficient and accurate for solving TFT equations‎. ‎Illustrative examples are included to demonstrate the efficiency of the Chelyshkov method‎.

1D Generalised Burgers-Huxley: Proposed Solutions Revisited and Numerical Solution Using FTCS and NSFD Methods

In this paper, we obtain the numerical solution of a 1-D generalised Burgers-Huxley equation under specified initial and boundary conditions, considered in three different regimes. The methods are Forward Time Central Space (FTCS) and a non-standard finite difference scheme (NSFD). We showed the schemes satisfy the generic requirements of the finite difference method in solving a particular problem. There are two proposed solutions for this problem and we show that one of the proposed solutions contains a minor error. We present results using FTCS, NSFD, and exact solution as well as show how the profiles differ when the two proposed solutions are used. In this problem, the boundary conditions are obtained from the proposed solutions. Error analysis and convergence tests are performed.

Multivalue Collocation Methods for Ordinary and Fractional Differential Equations

The present paper illustrates some classes of multivalue methods for the numerical solution of ordinary and fractional differential equations. In particular, it focuses on two-step and mixed collocation methods, Nordsieck GLM collocation methods for ordinary differential equations, and on two-step spline collocation methods for fractional differential equations. The construction of the methods together with the convergence and stability analysis are reported and some numerical experiments are carried out to show the efficiency of the proposed methods.

Numerical Solution of Three-Dimensional Transient Heat Conduction Equation in Cylindrical Coordinates

Heat equation is a partial differential equation used to describe the temperature distribution in a heat-conducting body. The implementation of a numerical solution method for heat equation can vary with the geometry of the body. In this study, a three-dimensional transient heat conduction equation was solved by approximating second-order spatial derivatives by five-point central differences in cylindrical coordinates. The stability condition of the numerical method was discussed. A MATLAB code was developed to implement the numerical method. An example was provided in order to demonstrate the method. The numerical solution by the method was in a good agreement with the exact solution for the example considered. The accuracy of the five-point central difference method was compared with that of the three-point central difference method in solving the heat equation in cylindrical coordinates. The solutions obtained by the numerical method in cylindrical coordinates were displayed in the Cartesian coordinate system graphically. The method requires relatively very small time steps for a given mesh spacing to avoid computational instability. The result of this study can provide insights to use appropriate coordinates and more accurate computational methods in solving physical problems described by partial differential equations.