Application of a new method for nonlinear partial di erential equations of fractional order arising in uid mechanics

To illustrate the basic idea of the improved EFM, we take the following nonlinear fractional partial di erential equation in the form: N(u; ux; uy; uxx; uyy; :::;D tu;D x u;D xxu; :::) = ۰; ۰ < ۱: (۱) Using a transformation u(x; y; t) = u( ); = kx+my+ nt 􀀀( + ۱) ; (۲) where k;m and n are constants to be determined later, we can rewrite equation Eq. (۱) in the following nonlinear ODE M(u; ku۰;mu۰; k۲u۰۰;m۲u۰۰; :::; nu۰; :::) = ۰; (۳) where the prime denotes derivative with respect to . If possible, integrate Eq. (۳) term by term one or more times.