3.2. Summation by parts

The sum of Eqs. (28) or (31), with weights 2χð Þ v;Δ , results in

$$\begin{aligned} &\sum\_{j=n}^{m} 2\chi(\upsilon,\Delta)h\_{j+1}(\mathrm{Dg})\_j + \sum\_{j=n}^{m} 2\chi(\upsilon,\Delta)\mathbf{g}\_{j-1}(\mathrm{Dh})\_j \\ &= \sum\_{j=n}^{m} 2\chi(\upsilon,\Delta)(\mathrm{Dgh})\_j \\ &= \mathbf{g}\_{m+1}h\_{m+1} + \mathbf{g}\_mh\_m - \mathbf{g}\_nh\_n - \mathbf{g}\_{n-1}h\_{n-1} \end{aligned} \tag{33}$$

These derivatives also have the exponential function as one of their eigenvectors, and we can generate expressions for higher derivatives with higher powers of the derivative matrix.

> 1 h2

<sup>2</sup>χð Þ <sup>v</sup>;<sup>Δ</sup> � <sup>h</sup><sup>2</sup> � <sup>h</sup><sup>1</sup>

� <sup>e</sup>�v<sup>Δ</sup> h1 

h1h<sup>2</sup>

<sup>1</sup> � <sup>e</sup>�v<sup>Δ</sup> 2χð Þ v;Δ

¼ � ð Þ <sup>D</sup><sup>h</sup> <sup>j</sup> hj�<sup>1</sup>hjþ<sup>1</sup>

> <sup>e</sup><sup>v</sup><sup>Δ</sup> � <sup>1</sup> 2χð Þ v;Δ

þ

1 h1 þ 1 h2 :

> 1 hN�<sup>1</sup> þ 1 hN

<sup>1</sup> � <sup>e</sup>�v<sup>Δ</sup> h1

ð Þq , including the edge points. For

Matrices Which are Discrete Versions of Linear Operations

http://dx.doi.org/10.5772/intechopen.74356

, (41)

: (42)

(40)

29

3.4. The derivative of the inverse of functions

For central and last points, we find that

3.5. The derivative of the ratio of functions

¼ 1 h2

¼ 1 h2

> D g h

N ¼ 1 hN

D g h 1 ¼ 1 h2

the first point, we have

Δ ¼ 0.

It is possible to give an expression for the derivative of h�<sup>1</sup>

D 1 h 

D 1 h 

N

ð Þ <sup>D</sup><sup>g</sup> <sup>1</sup> <sup>þ</sup> <sup>g</sup><sup>1</sup> <sup>D</sup> <sup>1</sup>

ð Þ <sup>D</sup><sup>g</sup> <sup>1</sup> � <sup>g</sup><sup>1</sup>

ð Þ <sup>D</sup><sup>g</sup> <sup>1</sup> <sup>þ</sup> <sup>g</sup><sup>1</sup> � ð Þ <sup>D</sup><sup>h</sup> <sup>1</sup>

h1h<sup>2</sup>

D g h j

1

<sup>¼</sup> <sup>1</sup> 2χð Þ v;Δ

<sup>¼</sup> <sup>1</sup>

¼ � ð Þ <sup>D</sup><sup>h</sup> <sup>1</sup> h1h<sup>2</sup> þ

> D 1 h

¼ � ð Þ <sup>D</sup><sup>h</sup> <sup>N</sup> hN�<sup>1</sup>hN

product of functions and obtain what the derivative of a ratio of functions is

h 

1 þ g1 h2

h1h<sup>2</sup> þ

ð Þ <sup>D</sup><sup>h</sup> <sup>1</sup> <sup>þ</sup> <sup>g</sup><sup>1</sup> h1

> <sup>¼</sup> ð Þ <sup>D</sup><sup>g</sup> <sup>j</sup> hj�<sup>1</sup>

ð Þ <sup>D</sup><sup>g</sup> <sup>N</sup> � gN�<sup>1</sup>

hN�<sup>1</sup>hN

j

þ

The derivatives for the first and last points coincide with the derivative for central points when

Now, we take advantage of the derivative for the inverse of a function and the derivative of a

<sup>e</sup>�v<sup>Δ</sup> � <sup>1</sup> 2χð Þ v;Δ

1 h1 þ

ð Þ Dh <sup>j</sup> hjþ<sup>1</sup>hj�<sup>1</sup>

ð Þ <sup>D</sup><sup>h</sup> <sup>N</sup> <sup>þ</sup> gN�<sup>1</sup>

hN�<sup>1</sup>

<sup>e</sup><sup>v</sup><sup>Δ</sup> � <sup>1</sup>

<sup>1</sup> � <sup>e</sup>�v<sup>Δ</sup> h2

þ g1 h2 <sup>e</sup>�v<sup>Δ</sup> � <sup>1</sup> 2χð Þ v; Δ

, (44)

<sup>2</sup>χð Þ <sup>v</sup>;<sup>Δ</sup> , (45)

(43)

1 2χð Þ v;Δ

> <sup>1</sup> � <sup>e</sup>�v<sup>Δ</sup> <sup>2</sup>χð Þ <sup>v</sup>;<sup>Δ</sup> ,

> > � gjþ<sup>1</sup>

or

$$\begin{aligned} &\sum\_{j=n}^{m} \mathsf{2}\chi(\boldsymbol{v}, \boldsymbol{\Delta}) \mathsf{g}\_{j+1}(\boldsymbol{\Delta}\boldsymbol{h})\_{j} + \sum\_{j=n}^{m} \mathsf{2}\chi(\boldsymbol{v}, \boldsymbol{\Delta}) \boldsymbol{h}\_{j-1}(\boldsymbol{\Delta}\boldsymbol{g})\_{j} \\ &\boldsymbol{g} = \mathsf{g}\_{m+1}\boldsymbol{h}\_{m+1} + \mathsf{g}\_{m}\boldsymbol{h}\_{m} - \mathsf{g}\_{n}\boldsymbol{h}\_{n} - \mathsf{g}\_{n-1}\boldsymbol{h}\_{n-1}. \end{aligned} \tag{34}$$

This is the discrete version of the integration by parts theorem for continuous variable functions, a very useful result.

## 3.3. Second derivatives

Expressions for higher order derivatives are obtained through the powers of DN. For instance, for the first two points, the second derivative is

$$\left(\mathrm{D}^{2}g\right)\_{1} = \frac{\left(e^{-2v\Lambda} - 1\right)\mathrm{g}\_{1} - e^{-v\Lambda}\mathrm{g}\_{2} + \mathrm{g}\_{3}}{4\chi^{2}(v,\Lambda)} = \frac{(\mathrm{Dg})\_{2} - e^{-v\Lambda}(\mathrm{Dg})\_{1}}{2\chi(v,\Lambda)},\tag{35}$$

$$\left(\mathrm{D}^{2}\mathrm{g}\right)\_{2} = \frac{e^{-v\Delta}\mathrm{g}\_{1} - 2\mathrm{g}\_{2} + \mathrm{g}\_{4}}{4\chi^{2}(v,\Delta)} = \frac{(\mathrm{D}\mathrm{g})\_{3} - (\mathrm{D}\mathrm{g})\_{1}}{2\chi(v,\Delta)},\tag{36}$$

For inner points we get

$$\left(\mathbf{D}^2 \mathbf{g}\right)\_j = \frac{\mathbf{g}\_{j-2} - \mathbf{2}\mathbf{g}\_j + \mathbf{g}\_{j+2}}{4\chi^2(v, \Delta)} = \frac{(\mathbf{D}\mathbf{g})\_{j+1} - (\mathbf{D}\mathbf{g})\_{j-1}}{2\chi(v, \Delta)}, \quad \mathbf{3} \preceq j \le \mathbf{N} - \mathbf{3}, \tag{37}$$

and for the last two points of the mesh, we find

$$(\mathrm{D}^2 g)\_{N-1} = \frac{\mathcal{g}\_{N-3} - 2\mathcal{g}\_{N-1} + e^{v\Lambda}\mathcal{g}\_N}{4\chi^2(v,\Delta)} = \frac{(\mathrm{Dg})\_N - (\mathrm{Dg})\_{N-2}}{2\chi(v,\Delta)},\tag{38}$$

$$\begin{split} \left(\mathbf{D}^2 \mathbf{g}\right)\_N &= \frac{\mathbf{g}\_{N-2} - e^{\nu \Delta} \mathbf{g}\_{N-1} + (e^{2\nu \Delta} - 1)\mathbf{g}\_N}{4\chi^2(\upsilon, \Delta)} \\ &= \frac{e^{\upsilon \Delta} (\mathbf{D} \mathbf{g})\_N - (\mathbf{D} \mathbf{g})\_{N-1}}{\chi\_2(\upsilon, 2\Delta)}. \end{split} \tag{39}$$

These derivatives also have the exponential function as one of their eigenvectors, and we can generate expressions for higher derivatives with higher powers of the derivative matrix.
