Substituting equation (36) into equation (3), we have the first symmetry reduced equation of equation (3) varphi_{xixixieta}+4varphi_{xixi}+4varphi_{xieta}+4varphi _{xi}varphi_{xieta}+2varphi_{xixi} varphi_{eta}+3varphi_{yy}=0.
The mathematical modeling of various physical processes, ranging from physics to biology, where spatial heterogeneity has a primary role, is reduced to nonlinear evolution equations with variable diffusion or dispersion.
With the approximations, the evolution equation reduces to a system of ordinary first order differential equations for axial components of affine molecular deformation tensor.
It is convenient to reduce (1.1) to an evolution equation of the first order in time left { textstylebegin{array}{l} u_{t}=v, v_{t}=-A^{2}u-A^{2}v-bu^-f u)+q(x)dot{W}=-A^{2}u-A^{2}v-bu^-f uuad u_{t}=-A^{2}u-A^{2}v-bu^-f uinOmega, end{array}=-A^{2}u-A^{2}v-bu^-f u
One can then use the Ott/Antonsen (OA) ansatz [29, 30] to reduce the dynamics of this evolution equation to a single complex equation for the evolution of the order parameter (zequivint_{0}^{2pi}p(theta,t e^{itheta},dtheta): frac{dz}{dt}=i eta+kI) (1+z)^{2}/2-i 1-z)^{2}/2-i 1-zga z+H/2-bar {H}z^{2}/2-i 1-z where ω and H are as above and (I=3/2-(z+bar{z})+(z^{2}/2=i{z}^{2})/4).
First, the approach reduces the particle swarm evolution equations and gets an evolution equation without velocity.
If the evolution equation is given by the inviscid Euler equation, the equation is reduced to frac{partial omega}{partial t} = 0 (18).
The proof is based on the semigroup approach (see [4, 12]) that can be used to reduce problem (1.1 - 1.7 1.1 - 1.7stoant initiabstract problem for a finitialder evalueion equation.
You might want to look at it.' " That was the stochastic Loewner evolution equation, now often called Schramm-Loewner evolution.
First, a 2D mass balance field ms x, y) is estimated using the thickness evolution equation.
