User:Lty

=General Nonlinear PDEs= n: independent variable; {xi}={x1,```,xn}

e.g. 1D case with time variable

(1D time dependent problem)

(x,t)=(x1,x2)

x1=x spatial variable

x2=t temporal variable

3D time dependent problems

(x,y,z,t)=(x1,```,x4)

Unknown functions: u1,u2,```

e.g. Navier-Stokes Equation in 3D

ui ({xj}) i=1,2,3; j=1,2,3,4  x4 is time

(x1,x2,x3)=(x,y,z)

(u1,u2,u3) velocity field along (x,y,z) coordinates

1: unknown function u

n: independent variable {xi} i=1,```,n

mth partial derivative:$$a/b$$

{i1,```,im} subset of m indices among n possible indices

i1,```,im=1,```,n