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This assignment solution was submitted to Amar Sharma for Finite Element Method course at Aligarh Muslim University. It includes: Hermite, Polynomials, Property, Stations, Kronecker, Delta, Cubic, Interpolation
Typology: Exercises
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Derive the following Hermite polynomials;
As we know the following property of Hermite polynomial for two stations,
For I, p = 1, 2
For k, r = 0, 1, 2, ……… …… ……….., N
Where x is the value of xp at pth^ station and 𝛅mn is the Kronecker delta having the property,
δmn = 0 if m ≠ n
δmn = 1 if m = n
By using above described properties, we can easily determine the desired polynomial, as following;
{ as I ≠ p}
{ as I = p and k = r}
{ as I ≠ p and k ≠ r}
Similarly,
The cubic interpolation model is given below,
……………………………………… (1)
Differentiating it w.r.t x, so that we may easily use the above calculated conditions,
………………………………….………. (2)
Now, by using above calculated conditions, we can easily find the unknown coefficients, as following;
At x = x 1 = 0, equations 1 & 2 becomes respectively,
At x = x 2 = l,
………………………………………….. (3)
Differentiating it w.r.t x, so that we may easily use the above calculated conditions,
…………………………… (6)
Now, by using above calculated conditions, we can easily find the unknown coefficients, as following;
At x = x 1 = 0, equations 5 & 6 becomes respectively,
At x = x 2 = l,
…………………………… (7)
…………….…………(8)
By solving equation 7 & 8 simultaneously, we get
Put these values in equation 5, we get
Rearranging it, we get
Hence proved
