The Solutions For A Sort Of Fractional Differential Equations

In fractional-order differential equations there be at least one fractional differential operator. Fractional-order calculus has extended to traditional calculus. The former is more suitable to describe the real world, is a strong tool to study the nonlinear problem, and plays an increasingly important role in many areas. More and more fractional-order differential equations have emerged, which have drawn the attentions of many scholar. To solve fractional-order differential equations we sometime can use traditional methods, for example, Laplace transformation. However, it is common knowledge that calculating inverse transformation is difficult. Furthermore, the difficulty is bigger for fractional-order differential equations.This paper is devoted to the fractional damped vibration equation (the traditional first-order derivative at time, instead by using v ( 0 < v <2)fractional derivative). First finite-dimensional systems were discussed, namely, the ordinary differential equations. By Laplace transformation and complicated calculus, the solutions were obtained. The second, infinite-dimensional systems were discussed, namely, the partial differential equations. By the methods of separation of variables and characteristic expansion, the solutions for some related problems were obtained.