The Application And Generalization Of A Differential Quadrature Algorithm For Time-fractional Telegraph Equation

Fractional partial differential equations with good memories and all-hereditary, com-pared to integer-order differential equations, fractional differential equations can better simulate some of the power system and the natural process of physical phenomena, thus in fluid mechanics, mechanics of materials, financial science, engineering and the natural sciences in question applied more widely. Most of the analytical solution of fractional differential equations is to rely on complex transform and special functions obtained, this result is difficult to approximate calculations. So more and more scholars begin to re-search Fractional partial differential equation numerical problems. Its numerical solution are many, but most of them account for a large space, computing capacity, difficult to use in practical problems. Differential quadrature method has the advantage of overcoming such problems, as:the method is conceptually simple, small calculation amount, fast convergence and high precision, is a numerical solution under constant attention. Cur-rently almost no scholars apply differential quadrature method into the fractional order differential equations. In this paper, to take full advantage of the differential quadrature method, and study the using of differential quadrature method in a class of time frac-tional partial differential equations. This paper is divided into four parts.The first part is the introduction,and describes the origin and significance of the study and the main objectives of the study and the research question, and gives the basic knowledge required for the research process, such as:the nature of the definition and the types of fractional derivatives, fractional calculus.The second part is a variety of numerical solution (such as:Gs(p) algorithm, R2algorithm and L algorithm) portfolio, and gives numerical method for solving fractional order partial differential equations; then showed that a combination of various methods is unconditional stability/unconditional convergence, and get the convergence order Gs(p) algorithm is1, R2algorithm2and L algorithm is2-α order; Finally, verify the ac-curacy and convergence of numerical methods and analyze the advantages of differential quadrature method has been well-preserved in the combination.The third part is the promotion of these methods to other equations, this article discusses three time fractional partial differential equations:the telegraph equation, d-iffusion equation, the class of two-dimensional heat conduction equation with variable coefficients.The fourth part is the improvisation of Gs(p) algorithm, by taking p as the special values and discreting on the special points to improve the convergence order to2bands, and improve accuracy.Finally, we find improved research methods are best practices,with the convergence of high-order accuracy. This paper also presents future research directions and issues to be improved.