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Lecture 27a, Differential Equations Definitions


Some simplified definitions and terminology related to differential equations:

Given a function  y = f(x)
In plain text, without mathematical symbols, the derivative of
f(x) with respect to x may be written many ways:

  df(x)/dx = df/dx = f'(x) = f'

  or for y=f(x)  dy/dx = y'(x) = y'

  Plain "d" will be used also for the partial derivative symbol.

The fundamental theorem of calculus is:

   Given f(x) is continuous on the interval a <= x <= b and
   F(x) is the indefinite integral of f(x) then
   integral from x=a to x=b of f(x)dx = F(b) - F(a)

Derivative f(x) = dF(x)/dx = F'(x) = F'

   Note that any letters, upper or lower case, may be used for any
   function or variable.

Second Derivative is simply the derivative of the first derivative

   d^2 f(x)/dx^2 = d (df(x)/dx)/dx = f''(x) = f''

Ordinary Differential Equation (ODE)
     A differential equation with only one independent variable.
     Example:  d^2 f(x)/dx^2 = -f(x)  f(0)=0, f'(0)=1 thus  f(x)=sin(x)

Partial Differential Equation (PDE)
     A differential equation with more than one independent variable.
     Example: dU/dx + dU/dy = f(x,y) given f(x,y)=x+y then U(x,y)=xy

Dimension of a differential equation is the number of independent
     variables. Typically the independent variables are:
     x for one dimension
     x,y for two dimensions
     x,y,z for three dimensions
     x,y,z,t for four dimensions, t usually being time.
     x,y,z,u,v,w,t seven independent variables for fluid problems

With many combinations, we use a shorthand, for example for
a function U(x,y,z):
Uxxx is notation for third derivative with respect to x
Uxy  is notation for partial derivative with respect to x and
     with respect to y
Uyzz is notation for partial derivative with respect to y and
     second partial derivative with respect to z


Order of a differential equation is the highest derivative appearing.
     Example: First order:  dF/dx + dF/dy + F(x,y)^3 + x^4 + y^5 = 0
     Example: Second order: d^2 F/dx^2 = F''(x)     Uxx
     Example: Third order:  d^3 F/dx^3 = F'''(x)    Uxxx
     Example: Fourth order: d^4 F/dx^4 = F''''(x)   Uxxxx
     Example: Fourth order: d^4 F/dx^2dydz = F''''(x,y,z) = Uxxyz

Degree of a differential equation is the highest power of any derivatives.
     Example: First degree(linear):  d^2 F/dx^2 + (dF/dx) = 0
     Example: Second degree(quadratic):  (d^3 F/dx^3) + (dF/dx)^2 = 0
     Example: Third degree(cubic):  (d^2 F/dx^2)^3 + (d^4 F/dx^4) = 0
     Example: Third degree: (d^2 F/dxdy)^3 + d^4 F/dxdydzdt = f(x,y,z,t)
     Example: Fourth degree (dF/dx)*(dF/dy)*(dF/dz)*(dF/dt) = 0

     Note: A "linear" differential equation has highest degree one
           for all orders. No U'*U, U^2, U'*U'' etc.
           The solution methods that use solving a linear system of
           equations only work with linear differential equations.
           Higher degree differential equations require solving
           non-linear systems of equations, see Newtons method.

     Note: The Right Hand Side, RHS, is the part to the right of
           the equal sign. This may be zero or a computable function,
           yet must not contain any derivatives or the solution function.
           All derivatives and the solution must be to left of equal sign.
           Example 7*U'(x,y,z)+x*U''(x,y,z)-z*U(x,y,z) = c(x,y,z)^3+x

Initial value differential equation problems have values given at one end
     of the domain.

Boundary value differential equation problems have values given at all ends
     of the domain. 
     One dimensional has two values.
     Two dimensional has values on an enclosing path. e.g. square, circle
     Three dimensional has values on an enclosing surface. e.g. cube, sphere
     Four dimensional has values on an enclosing volume. e.g. hyper cube

     Dirichlet boundary values are the values at a point on the boundary.
     Neumann boundary values are the derivative at a point on the boundary,
     typically the first derivative in the direction of the outward normal.

Mixed value partial differential equation problems may have some variables
     initial value and some variables boundary value. Often the time
     variable is given only as an initial value.

Types of second order, first degree, partial differential equations
in two variables, very common, thus named:

  Given  A d^2 U/dx^2 + 2B d^2 U/dxdy + C d^2 U/dy^2 + other terms = f(x,y)

Parabolic  when B^2 = A C   e.g. Diffusion equation
                            one unique real characteristic
                            system has one zero eigenvalue, others
                            all positive or all negative

Elliptic   when B^2 < A C   e.g. Laplace's equation
                            two unique complex characteristics
                            system has eigenvalues all positive or all negative

Hyperbolic when B^2 > A C   e.g. Wave equation
                            two unique real characteristics
                            system has no zero eigenvalues and at least one
                            positive and one negative

    Parabolic Diffusion equation: B=0, C=0
                                  k d^2 V/dx^2 - dV/dt = 0

    Elliptic Laplace's equation:  B=0, A>0, C>0
                                  d^2 V/dx^2 + d^2 V/dy^2 = 0

    Hyperbolic Wave equation:     B=0, A>0, C<0
                                  d^2 V/dx^2 - 1/c^2 d^2 V/dt^2 = 0

The above definitions are motivated by the equation of a cone
cut by a plane at various angles, giving the conic sections:
   parabola,  ellipse  and  hyperbola.

The definitions are extended by some authors to first order
equations such as  dV/dt + a dV/dx = 0 to be called a hyperbolic
one-way wave equation.

The three example equations above are called "homogeneous" because
no term has an independent variable.
More general equations have a forcing function that would replace
the "0" with  f(x,y) or f(x,t) and thus have an inhomogeneous equation.

Using web math symbols:

Laplace Equation is  ΔU = 0  or  ∇2U = 0  or

                     ∂2U/∂x2 + ∂2U/∂y2 + ∂2U/∂z2 = 0

Poisson Equation is  ΔU = f  or  ∇2U = f  or

                     ∂2U/∂x2 + ∂2U/∂y2 + ∂2U/∂z2 = f(x,y,z)


Some terminology is used for various methods of numerical solution
of differential equations.

The Finite Difference Method, FDM, replaces the continuous differential
operators with finite difference approximations. The order of
the approximation may be checked by substitution of the Taylor series.
The FDM is explicit if the solution at the next cell can be expressed
entirely in terms of previously computed cells. Our "nderiv".
The FDM is implicit if the solution at the next group of cells must
be represented as a set of simultaneous equations based on a
previous group of cells.

The Finite Element Method, FEM, develops a system of simultaneous
equations for the solution at every cell.
The FEM is explicit if one solution of the simultaneous equations
yields the solution of the differential equation.
The FEM is iterative if each solution of the simultaneous equations
yields the next approximation to the solution of the differential equation.

more definitions on ODE and PDE


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