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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
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)
Example: Third order: d^3 F/dx^3 = F(x)
Degree of a differential equation is the highest power of the highest order.
Example: First degree(linear): d^2 F/dx^2 + (dF/dx)^3 = 0
Example: Second degree(quadratic): (d^2 F/dx^2)^2 (dF/dx)^3 = 0
Initial value differential equation problems have values given at one end
of the domain.
Boundary value differential equation problems have values given at both ends
of the domain.
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 equation in two variables:
Given A d^2 U/dx^2 + 2B d^2 U/dxdy + C d^2 U/dy^2 + other terms = 0
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
and 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.
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.
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.
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