Finite Difference by Polynomials

The other way for approximating a derivative is to represent the function through a polynomial. The coefficients of the polynomial are obtained by the substitution of a dependent variable from a series of equally spaced points of the independent variables. As such, the approximate values of the derivatives are calculated from the polynomial.

Consider a second-order polynomial as

where A, B and C are constants and we assume the origin at x*. Thus, xi=0, xi+j = Дх and xi+2 = 2Дx and so on. The values of function f for corresponding xi are f(xi) =f, f(xi+j) =f+j

and f(xi + 2) =f + 2.

Thus,

and

From Equations 5.42 through 5.44, it follows that and

To compute the first derivative of f, one has the following:

At x{ = 0, we have Hence, Equation 5.45 will be

which is same as the second-order accurate forward difference expression obtained by using Taylor series expansion. Here, one may note that this approximation is classified as second-order accurate for df/dx, since d3f/dx3 vanishes just as in the accuracy analysis of the Taylor series expansion. The second derivative of f may be calculated as

In a similar way, one may write the following:

and is consistent with the first-order finite difference approximation given by Equation 5.38. Considering the spacing of points i, i + 1 and i + 2 as non-identical, a general form of the finite difference approximation of the derivative procedure at x{=0, x{+x = Дх and xi+2 = (1 + а)Дх is as follows.

Here,

and

Consequently,

and

Therefore,

which is a second-order approximation. The second derivative of f is obtained as As a result,

which is a first-order expression. In the same manner, similar relations for backward and central difference approximations may also be obtained.

 
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