In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie＇s modified Riemann–Liouville derivative. The efficiency and power of t...In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie＇s modified Riemann–Liouville derivative. The efficiency and power of this approach are demonstrated by applying it successfully to some important fractional differential equations, namely, the time fractional Burgers, fractional Sawada–Kotera, and fractional coupled mKdV equation. A variety of new exact solutions to these equations under study are constructed.展开更多
In this paper, we apply Malliavin calculus to discuss when the solutions of stochastic differen-tial equations (SDE's) with time-dependent coefficients have smooth density. Under Hormander'scondition,we conclu...In this paper, we apply Malliavin calculus to discuss when the solutions of stochastic differen-tial equations (SDE's) with time-dependent coefficients have smooth density. Under Hormander'scondition,we conclude that the solutions of the SDE's have smooth density. As a consequence,we get the hypoellipticity for inhomogeneous differential operators.展开更多
基金Supported by the National Natural Science Foundation of China under Grant Nos.11101332,11201371,11371293 the Natural Science Foundation of Shaanxi Province under Grant No.2015JM1037
文摘In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie＇s modified Riemann–Liouville derivative. The efficiency and power of this approach are demonstrated by applying it successfully to some important fractional differential equations, namely, the time fractional Burgers, fractional Sawada–Kotera, and fractional coupled mKdV equation. A variety of new exact solutions to these equations under study are constructed.
基金The project supported by National Natural Science Foundation of China Crant 18971061
文摘In this paper, we apply Malliavin calculus to discuss when the solutions of stochastic differen-tial equations (SDE's) with time-dependent coefficients have smooth density. Under Hormander'scondition,we conclude that the solutions of the SDE's have smooth density. As a consequence,we get the hypoellipticity for inhomogeneous differential operators.
