Difference Equation vs Differential Equation
A natural phenomenon may be described mathematically by functions of a number of independent variables and parameters. Especially when they are expressed by a function of spatial position and time it results in equations. The function may change with the change in the independent variables or the parameters. An infinitesimal change happening in the function when one of its variables is changed is called the derivative of that function.
A differential equation is any equation which contains derivatives of a function as well as the function itself. A simple differential equation is that of Newton’s Second Law of Motion. If an object of mass m is moving with acceleration ‘a’ and being acted on with force F then Newton’s Second Law tells us that F=ma. Here again, ‘a’ varies with time, we can rewrite ‘a’ as; a= dv/dt; v is velocity. Velocity is function of space and time, that is v=ds/dt; therefore ‘a’= d2s/dt2.
Keeping these in mind we can rewrite Newton’s second law as a differential equation;
‘F’ as a function of v and t – F(v,t)= mdv/dt, or
‘F’ as a function of s and t – F(s, ds/dt, t)=m d2s/dt2
There are two types of differential equations; ordinary differential equation, abbreviated by ODE or partial differential equation, abbreviated by PDE. Ordinary differential equation will have ordinary derivatives (derivatives of only one variable) in it. Partial differential equation will have differential derivatives (derivatives of more than one variable) in it.
e.g. F= m d2s/dt2 is an ODE, whereas α2 d2u/dx2 = du/dt is a PDE, it has derivatives of t and x.
Difference equation is same as differential equation but we look at it in different context. In differential equations, the independent variable such as time is considered in the context of continuous time system. In discrete time system, we call the function as difference equation.
Difference equation is a function of differences. The differences in the independent variables are three types; sequence of number, discrete dynamical system and iterated function.
In sequence of numbers the change is generated recursively using a rule to relate each number in the sequence to previous numbers in the sequence.
Difference equation in a discrete dynamical system takes some discrete input signal and produce output signal.
Difference equation is an iterated map for iterated function. E.g., y0, f(y0), f(f (y0)), f(f(f(y0))),….is the sequence of an iterated function. The f(y0) is the first iterate of y0. The k-th iterate will be denoted by fk(y0).