ODE Ordinary Differential Equation - one independent variable
second order - written many ways:
a(x)*ΔU + b(x)*∇U + c(x)*U = f(x)
a(x)*∇2U + b(x)*∇U + c(x)*U = f(x)
a(x)*d2U/dx2 + b(x)*dU/dx + c(x)*U = f(x)
a(x)*U''(x) + b(x)*U'(x) + c(x)*U(x) = f(x)
a(x)*Uxx(x) + b(x)*Ux(x,y) + c(x)*U(x) = f(x)
x is independent variable
the solution is U(x)
U'(x) is derivative of U(x) with respect to x, written Ux, etc.
functions a(x), b(x), c(x) and f(x) must be known
enough initial conditions must be known for a unique solution
Further definition is needed for choice of coordinate system:
Cartesian, Cylindrical, Polar, Toroidal, etc.
PDE Partial Differential Equation - more than one independent variable
a(x,y)*ΔU + b(x,y)*∇U + c(x,y)*ΔU +
d(x,y)*∇U + e(x,y)*∇U + g(x,y)*U = f(x,y) ambiguous
p(x,y)*ΔU + q(x,y)*∇U + r(x,y)*U = f(x,y) typical
a(x,y)*∇2U + b(x,y)*∇2U + c(x,y)*∇2U +
d(x,y)*∇ + e(x,y)*∇U + g(x,y)*U = f(x,y) ambiguous
p(x,y)*∇2U + q(x,y)*∇U + r(x,y)*U = f(x,y) typical
a(x,y)*∂2U/∂x2 + b(x,y)*∂2U/∂x∂y +
c(x,y)*∂2U/∂y2 + d(x,y)*∂U/∂x +
e(x,y)*∂U/∂y + g(x,y)*U = f(x,y)
a(x,y)*Uxx(x,y) + b(x,y)*Uxy(x,y) + c(x,y)*Uyy(x,y) +
d(x,y)*Ux(x,y) + e(x,y)*Uy(x,y) + g(x,y)*U(x,y) = f(x,y)
x and y are independent variables
the solution is U(x,y)
Ux(x,y) is partial derivative of U(x,y) with respect to x,
written Ux, Note: U' is ambiguous because it could be Ux or Uy
Uxy(x,y) is partial derivative of U(x,y) with respect to x and
partial derivative with respect to y,
written Uxy,
functions a(x,y), b(x,y), c(x,y), d(x,y), e(x,y),
f(x,y) and g(x,y) must be known and computable.
Enough boundary condition must be known for a unique solution.
in three dimensions x,y becomes x,y,z
in four dimensions x,y becomes x,y,z,t
in n dimensions the variables are in Rn
and boundary conditions Ω are given.
Further definition is needed for choice of coordinate system:
Cartesian, Cylindrical, Polar, Toroidal, etc.
Methods for computing numerical solutions replace the continuous
variable x with discrete values x1, x2, x3, ... , x_nx
Uniformly spaced values may be used, given xmin, xmax and h,
xmin, xmin+h, xmin+2h, ... , xmax nx=1+(xmax-xmin)/h
Discrete values are also used for y, z, and t
The numerical solution is computed at the discrete values of
the independent variables U(x1), U(x2), U(x3), ... , U(x_nx)
or U(x1,y1), U(x1,y2), U(x2,y2), ... , U(x_nx,y_ny), etc.
The solution is a set of numbers, often written as
U1, U2, ... U_xn or U[1,1], U[1,2], U[2,2], ... U[nx,ny]
A numerical solution may use a uniform grid or a
nonuniform grid.
A numerical solution method may be iterative, computing
a closer approximation at each step.
A numerical solution may set up a system of linear equations
to solve for the solution values.
A numerical solution may use a combination of iterative and
linear equations to solve for the solution values.
Not that a non linear differential equation will create a
non linear system of equations to solve. Very difficult!
All of the above statements apply to a set of methods,
generally called discretization.
FEM Finite Element Method is a set of methods for finding the
numerical solution of a differential equation. Within FEM
there are various sub methods including the Galerkin method.
FEM may use equally spaced or variable spaced values for
the independent variables.
FEM may also use triangles, quads, or other polygons for two
independent variables.
FEM may also use tetrahedrons or other solids for three
independent variables. And four dimensional objects for four
independent variables.
see FEM lecture for the rest of the explanation.
cs455_l32.html Finite Element Method