User:Eml5526.s11.team7.jin/HW4

=Problem 4.5: Basis Functions Satisfying Constraint Breaking Solution=

Given
Possible basis functions are given as:

Polynomial: $${{F}_{P}}=\left\{ {{x}^{j}},\text{ }j=0,1,2,\ldots \right\}$$

Cosine: $${{F}_{C}}=\left\{ \cos jx,\text{ }j=0,1,2,\ldots \right\}$$

Sine: $${{F}_{S}}=\left\{ 1,\text{ }\sin jx,\text{ }j=1,2,\ldots \right\}$$

Fourier: $${{F}_{F}}=\left\{ \cos jx,\text{ }j=0,1,2,\ldots ;\text{ }\sin kx,\text{ }k=1,2,\ldots \right\}$$

Exponential: $${{F}_{E}}=\left\{ {{e}^{jx}},\text{ }j=0,1,2,\ldots \right\}$$

For $${{\Gamma }_{g}}=\left\{ \beta \right\}$$, and each of the above families of basis functions, i.e., $${{\Gamma }_{I}}=\left\{ {{b}_{j}}(x),\text{ }j=0,1,\ldots \right\},\text{ }I\in \left\{ P,C,S,F,E \right\}$$

Find
Find corresponding family $${{\bar{F}}_{I}}=\left\{ {{{\bar{b}}}_{j}} \right\}$$ satisfying constraint breaking solution (CBS), i.e., $${{b}_{0}}(\beta )\ne 0$$ and $${{b}_{i}}(\beta )=0,\text{ }i=1,2,\ldots ,n$$. Let $$\Omega =\left] \alpha ,\beta \right[=\left] -2,4 \right[$$.

1) $${{\bar{b}}_{0}}(x)=\text{const}$$, plot $${{b}_{j}}$$ and $${{\bar{b}}_{j}}$$ for $$j=1,2,3$$.

2) Show $$\left\{ {{e}^{jx}},\text{ }j=0,1,2 \right\}$$ linear independent on $$\Omega $$.

Part 1
The corresponding family $${{\bar{F}}_{I}}=\left\{ {{{\bar{b}}}_{j}} \right\}$$ satisfying CBS:

Polynomial:

$${{F}_{P}}=\left\{ {{\left( x-4 \right)}^{j}},\text{ }j=0,1,2,\ldots \right\}$$

Cosine:

$${{F}_{C}}=\left\{ 1,\text{ }\cos \left( j\left( x-4 \right)+\frac{\pi }{2} \right),\text{ }j=1,2,\ldots \right\}$$

Sine:

$${{F}_{S}}=\left\{ 1,\text{ }\sin \left( j\left( x-4 \right) \right),\text{ }j=1,2,\ldots \right\}$$

Fourier:

$${{F}_{F}}=\left\{ 1,\text{ }\cos \left( j\left( x-4 \right)+\frac{\pi }{2} \right),\text{ }j=1,2,\ldots ;\text{ }\sin \left( k\left( x-4 \right) \right),\text{ }k=1,2,\ldots \right\}$$

Exponential:

$${{F}_{E}}=\left\{ 1,\text{ }{{e}^{j\left( x-4 \right)}}-1,\text{ }j=1,2,\ldots \right\}$$

The following figures show the polynomial basis functions (j=1,2,3):





The following figures show the cosine basis functions (j=1,2,3):





The following figures show the sine basis functions (j=1,2,3):





The following figures show the fourier basis functions (j=1,2,3):





The following figures show the exponential basis functions (j=1,2,3):





Part 2
The exponential basis function is $${{b}_{j}}=\left\{ {{e}^{jx}},\text{ }j=0,1,2 \right\}$$,

Thus the Gram matrix is:

$$\Gamma =\left[ \begin{matrix} 6 & {{e}^{4}}-{{e}^{-2}} & \frac{1}{2}\left( {{e}^{8}}-{{e}^{-4}} \right) \\ {{e}^{4}}-{{e}^{-2}} & \frac{1}{2}\left( {{e}^{8}}-{{e}^{-4}} \right) & \frac{1}{3}\left( {{e}^{12}}-{{e}^{-6}} \right) \\ \frac{1}{2}\left( {{e}^{8}}-{{e}^{-4}} \right) & \frac{1}{3}\left( {{e}^{12}}-{{e}^{-6}} \right) & \frac{1}{4}\left( {{e}^{16}}-{{e}^{-8}} \right) \\ \end{matrix} \right]$$

And the determinant of Gram matrix is:

$$\det \left( \Gamma \right)=1.1145\times {{10}^{9}}\ne 0$$

Thus, the exponential basis function is linear independent on $$\Omega $$.

Authors
Jiang Jin