Part1: differential and discrete rate of change
Is well known, a function \ (f (x) \) necessary condition is continuously differentiable. For the domain non-close function, apparently no derivative at all. However, recall the definition of derivative
\ [Y '= \ lim _ {\ Delta x \ to0} \ frac {\ Delta} {\ Delta x} = \ frac {\ mathrm {d} and {} \ {d} mathrm x} \]
We have a set of \ (n-\) membered point set \ (\ {(x_1, F (x_1)), (x_2, F (x_2)), \ DOTS, (x_n, F (x_n)) \} \) , and \ (x_1, x_2, \ dots , x_n \) configured to \ (D \) is the difference between the arithmetic sequence, defined
\ [\ Delta y_i = y_i-y_ {i-1}, i = 2,3, \ dots, n \]
Called differential (-difference) , and
This corresponds precisely with the differential. Likewise, the definition of
\[ y'_i=\frac{\Delta y_i}d=\frac{y_i-y_{i-1}}d,i=2,3,\dots,n \]
Called discrete change rate (Rate of Change Discrete) , just echoes derivative Thus, calculus was extended to the set of discrete points. Discrete geometric meaning of the rate of change is the slope of a straight line connecting adjacent points. When the \ ( F \) is a polynomial, is also a linear differential operator, and will reduce the order of the polynomial \ (1 \) .
Accordingly, we can also define high-order and higher-order discrete differential rate of change:
\ [\ Delta ^ k y_i = \ Delta ^ {k-1} y_i- \ Delta ^ {k-1} y_ {i-1}, i = 2,3, \ dots, n, k> 1; \\ and ^ {(k)} _ i = \ frac {\ Delta ^ ky d}, i = 2,3, \ dots, n, k \ ge1. \]
In particular, we define: a set of sequences of \ (0 \) order differential is equal to its own.
Can be found, each seeking a difference, less a sequence of elements, but also corresponds to the "reduced order" meaning the differential of the same. However, due to the different sequence of discrete variables, the difference, the results are different. Therefore, we will defined above difference It referred to the difference defined, and forward difference is:
\ [\ Delta y_i = y_ {i + 1} -y_i, i = 1,2, \ dots, n-1 \]
Can also be defined central difference is:
\[ \Delta y_i=\frac12(y_{i+1}-y_{i-1}),i=2,3,\dots,n-1 \]
Part2: prefix and prefix and the weighted
Recalls the definition of integration:
\[ \int_{\alpha}^{\beta}f(x)\mathrm{d}x=\lim_{\lambda \to 0}\sum_{i=1}^n f(\xi)\Delta x_i \]
Accordingly, the definitions of the variables prefix and (prefix sum) of:
\ [\ Sigma (y_i) = \ sum_ {j = 1} ^ and y_j + C \]
This corresponds to the indefinite integral calculus. Obviously, prefixes, and forward difference is the inverse operation, defined accordingly
\[ \sigma(y)_l^r=\sum_{j=l}^r y_j+C \]
Referred to with the prefix and the right , corresponding to the definite integral. Obviously, \ (Y \) of any one of sub-segments and can be expressed as its prefix and the difference corresponds to any of a definite integral can be expressed as a function of the difference of the original we also can be defined accordingly suffix and (suffix SUM) :
\[ \sigma(y_i)=\sum_{j=i}^n y_j+C \]
Backward difference is the inverse operation.
Part3: Differential Equations
To be completed.