Fractional calculus has been used in many fields, such as engineering, population, medicine, fluid mechanics and different fields of chemistry and physics. These fields were found to be best described using fractional differential equations (FDEs) to model their processes and equations. One of the well-known methods for solving fractional differential equations is the Shifted Legendre operational matrix (LOM) method. In this article, I proposed a numerical method based on Shifted Legendre polynomials for solving a class of fractional differential equations. A fractional order operational matrix of Legendre polynomials is also derived where the fractional derivatives are described by the Caputo derivative sense. By using the operational matrix, the initial and boundary equations are transformed into the products of several matrixes and by scattering the coefficients and the products of matrixes. I got a system of linear equations. Results obtained by using the proposed method (LOM) presented here show that the numerical method is very effective and appropriate for solving initial and boundary value problems of fractional ordinary differential equations. Moreover, some numerical examples are provided and the comparison is presented between the obtained results and those analytical results achieved that have proved the method's validity.
In this paper, the classical fourth-order Runge-Kutta method and Heun's method are applied to initial value problems for system of ordinary differential equations in nonlinear cases which we reach it by Painleve analysis, focusing our interest in Ito coupled nonlinear partial differential system. The equations are solved by scheme of one step methods. Numerical results for the velocity in three dimensions are obtained and reported graphically for various temperatures to show interesting aspects of the solution.
In this article, the classical fourth-order Runge Kutta method and Heun's method are applied to initial value problems for system of ordinary differential equations in nonlinear cases, which we reach it by painaleve analysis, focusing our interest on special case which leads to Travelling wave solution of Ito coupled nonlinear partial differential system.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.