3. The matrix D<sup>N</sup> represents a derivation

Let us consider a partition, P Nð Þ≔ qi <sup>N</sup> <sup>1</sup> , qi ∈ ℝ, of N equally spaced points qi of the interval ½ � a; b ∈ ℝ, a < b, with the same separation Δ ¼ ð Þ b � a =ð Þ N � 1 between them.

The rows of the result of the multiplication of the derivative matrix D<sup>N</sup> and a vector g≔ g1; g2;…; gn <sup>T</sup> are

$$(\mathbf{D}\_{N}\mathbf{g})\_{j} = \frac{\mathbf{g}\_{j+1} - \mathbf{g}\_{j-1}}{2\chi(v,\Delta)}, \quad j = 1, 2, \dots, N,\tag{21}$$

∣D2j∣ ¼ 0, ∣D2jþ<sup>1</sup>∣ ¼ 2sinhð Þ vΔ : (26)

Matrices Which are Discrete Versions of Linear Operations

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

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

(27)

27

(29)

(30)

(32)

, (28)

, (31)

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

2 þ v2 <sup>4</sup> <sup>Δ</sup> <sup>þ</sup> <sup>O</sup> <sup>Δ</sup><sup>3</sup> ,

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

2 þ v2 <sup>4</sup> <sup>Δ</sup> <sup>þ</sup> <sup>O</sup> <sup>Δ</sup><sup>3</sup> ,

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

<sup>2</sup> � <sup>v</sup><sup>2</sup> <sup>4</sup> <sup>Δ</sup> <sup>þ</sup> <sup>O</sup> <sup>Δ</sup><sup>3</sup> ,

<sup>2</sup> � <sup>v</sup><sup>2</sup>

<sup>4</sup> <sup>Δ</sup> <sup>þ</sup> <sup>O</sup> <sup>Δ</sup><sup>3</sup> :

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

Hence, only the matrices with an odd dimension have an inverse.

vectors defined on the partition. A set of such expressions is

ð Þ <sup>D</sup>gh <sup>1</sup> <sup>¼</sup> <sup>g</sup>2h<sup>2</sup> � <sup>e</sup>�v<sup>Δ</sup>g1h<sup>1</sup>

2χð Þ v; Δ

<sup>2</sup>χð Þ <sup>v</sup>; <sup>Δ</sup> <sup>þ</sup> <sup>g</sup><sup>1</sup>

¼ h2ð Þ Dg <sup>1</sup> þ g1ð Þ Dh <sup>1</sup> þ g1h<sup>2</sup>

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

<sup>≈</sup> hNð Þ <sup>D</sup><sup>g</sup> <sup>N</sup> <sup>þ</sup> gN�<sup>1</sup>ð Þ <sup>D</sup><sup>h</sup> <sup>N</sup> <sup>þ</sup> gN�<sup>1</sup>hN � <sup>v</sup>

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

<sup>≈</sup> gNð Þ <sup>D</sup><sup>h</sup> <sup>N</sup> <sup>þ</sup> hN�<sup>1</sup>ð Þ <sup>D</sup><sup>g</sup> <sup>N</sup> <sup>þ</sup> gNhN�<sup>1</sup> � <sup>v</sup>

ð Þ <sup>D</sup>gh <sup>j</sup> <sup>¼</sup> gjþ<sup>1</sup> ð Þ <sup>D</sup><sup>h</sup> <sup>j</sup> <sup>þ</sup> hj�<sup>1</sup> ð Þ <sup>D</sup><sup>g</sup> <sup>j</sup>

ð Þ <sup>D</sup>gh <sup>j</sup> <sup>¼</sup> hjþ<sup>1</sup> ð Þ <sup>D</sup><sup>g</sup> <sup>j</sup> <sup>þ</sup> gj�<sup>1</sup> ð Þ <sup>D</sup><sup>h</sup> <sup>j</sup>

<sup>¼</sup> <sup>g</sup>2h<sup>2</sup> � <sup>e</sup>�v<sup>Δ</sup>g1h<sup>2</sup>

ð Þ <sup>D</sup>gh <sup>N</sup> <sup>¼</sup> hNð Þ <sup>D</sup><sup>g</sup> <sup>N</sup> <sup>þ</sup> gN�<sup>1</sup>ð Þ <sup>D</sup><sup>h</sup> <sup>N</sup> <sup>þ</sup>

ð Þ Dgh <sup>1</sup> ¼ g2ð Þ Dh <sup>1</sup> þ h1ð Þ Dg <sup>1</sup> þ g2h<sup>1</sup>

ð Þ Dgh <sup>N</sup> ¼ gNð Þ Dh <sup>N</sup> þ hN�<sup>1</sup>ð Þ Dg <sup>N</sup> þ gNhN�<sup>1</sup>

A second set of equalities is

3.1. The derivative of a product of vectors

Next, we will derive some properties of these finite differences derivatives.

There are two equivalent expressions for the finite differences derivative of a product of

where <sup>g</sup>0≔e�v<sup>Δ</sup>g<sup>1</sup> and gNþ<sup>1</sup>≔e<sup>v</sup><sup>Δ</sup>gN. We recognize these expressions as the second order derivatives of the function g xð Þ at the mesh points, but instead of dividing by twice the separation Δ between the mesh points, there is the function χð Þ v;Δ in the denominator. This function makes it possible that the exponential function be an eigenvector of the matrix DN.

The values <sup>g</sup><sup>0</sup> <sup>¼</sup> <sup>e</sup>�v<sup>Δ</sup>g<sup>1</sup> and gNþ<sup>1</sup> <sup>¼</sup> <sup>e</sup><sup>v</sup><sup>Δ</sup>gN extend the original interval ½ � <sup>a</sup>; <sup>b</sup> to ½ � <sup>a</sup> � <sup>Δ</sup>; <sup>b</sup> <sup>þ</sup> <sup>Δ</sup> so that we have well defined the second order derivatives at all the points of the initial partition, including the edges of the interval. When g xð Þ is the exponential function, we have <sup>g</sup><sup>0</sup> <sup>¼</sup> <sup>e</sup>v xð Þ <sup>1</sup>�<sup>Δ</sup> and gNþ<sup>1</sup> <sup>¼</sup> <sup>e</sup>v xð Þ <sup>N</sup>þ<sup>Δ</sup> , i.e., they are the values of the exponential function evaluated at the points of the extension.

Thus, we define finite differences derivatives for any function g xð Þ defined on the partition as

$$(\mathrm{Dg})\_1 = \frac{\mathcal{g}\_2 - e^{-v\Lambda}\mathcal{g}\_1}{2\chi(v,\Lambda)},\tag{22}$$

$$(\mathrm{Dg})\_j = \frac{\mathcal{g}\_{j+1} - \mathcal{g}\_{j-1}}{2\chi(v, \Delta)} \, \, \, \, \tag{23}$$

$$(\mathrm{Dg})\_N = \frac{e^{\upsilon \Delta} g\_N - g\_{N-1}}{2\chi(\upsilon, \Delta)},\tag{24}$$

to be used on the first, central, and last points of the partition.

The determinant of the derivative matrix is not always zero, and in fact, it is [see Eqs. (4) and (9)]

$$|\overline{\mathbf{D}}\_N| = 2\sinh(v\Delta)\, F\_N(\mathbf{0}).\tag{25}$$

But, since F2jþ<sup>1</sup> ¼ 1, and F2<sup>j</sup> ¼ 0, then

Matrices Which are Discrete Versions of Linear Operations http://dx.doi.org/10.5772/intechopen.74356 27

$$|\overline{\mathbf{D}}\_{2j}| = 0, \quad |\overline{\mathbf{D}}\_{2j+1}| = 2\sinh(\upsilon\Delta). \tag{26}$$

Hence, only the matrices with an odd dimension have an inverse.

Next, we will derive some properties of these finite differences derivatives.

## 3.1. The derivative of a product of vectors

The vector that we will be interested on is the one which is the exponential function (19) with

The rows of the result of the multiplication of the derivative matrix D<sup>N</sup> and a vector

where <sup>g</sup>0≔e�v<sup>Δ</sup>g<sup>1</sup> and gNþ<sup>1</sup>≔e<sup>v</sup><sup>Δ</sup>gN. We recognize these expressions as the second order derivatives of the function g xð Þ at the mesh points, but instead of dividing by twice the separation Δ between the mesh points, there is the function χð Þ v;Δ in the denominator. This function makes

The values <sup>g</sup><sup>0</sup> <sup>¼</sup> <sup>e</sup>�v<sup>Δ</sup>g<sup>1</sup> and gNþ<sup>1</sup> <sup>¼</sup> <sup>e</sup><sup>v</sup><sup>Δ</sup>gN extend the original interval ½ � <sup>a</sup>; <sup>b</sup> to ½ � <sup>a</sup> � <sup>Δ</sup>; <sup>b</sup> <sup>þ</sup> <sup>Δ</sup> so that we have well defined the second order derivatives at all the points of the initial partition, including the edges of the interval. When g xð Þ is the exponential function, we have <sup>g</sup><sup>0</sup> <sup>¼</sup> <sup>e</sup>v xð Þ <sup>1</sup>�<sup>Δ</sup> and gNþ<sup>1</sup> <sup>¼</sup> <sup>e</sup>v xð Þ <sup>N</sup>þ<sup>Δ</sup> , i.e., they are the values of the exponential function evaluated at the points

Thus, we define finite differences derivatives for any function g xð Þ defined on the partition as

ð Þ <sup>D</sup><sup>g</sup> <sup>1</sup> <sup>¼</sup> <sup>g</sup><sup>2</sup> � <sup>e</sup>�v<sup>Δ</sup>g<sup>1</sup>

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

ð Þ <sup>D</sup><sup>g</sup> <sup>N</sup> <sup>¼</sup> <sup>e</sup><sup>v</sup><sup>Δ</sup>gN � gN�<sup>1</sup>

The determinant of the derivative matrix is not always zero, and in fact, it is [see Eqs. (4) and (9)]

to be used on the first, central, and last points of the partition.

But, since F2jþ<sup>1</sup> ¼ 1, and F2<sup>j</sup> ¼ 0, then

<sup>1</sup> , qi ∈ ℝ, of N equally spaced points qi of the interval

<sup>2</sup>χð Þ <sup>v</sup>;<sup>Δ</sup> , j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>,…, N, (21)

<sup>2</sup>χð Þ <sup>v</sup>;<sup>Δ</sup> , (22)

<sup>2</sup>χð Þ <sup>v</sup>; <sup>Δ</sup> , (23)

<sup>2</sup>χð Þ <sup>v</sup>;<sup>Δ</sup> , (24)

∣DN∣ ¼ 2sinhð Þ vΔ FNð Þ0 : (25)

<sup>N</sup>

½ � a; b ∈ ℝ, a < b, with the same separation Δ ¼ ð Þ b � a =ð Þ N � 1 between them.

ð Þ <sup>D</sup>N<sup>g</sup> <sup>j</sup> <sup>¼</sup> gjþ<sup>1</sup> � gj�<sup>1</sup>

it possible that the exponential function be an eigenvector of the matrix DN.

eigenvalue v.

26 Matrix Theory-Applications and Theorems

g≔ g1; g2;…; gn <sup>T</sup>

of the extension.

3. The matrix D<sup>N</sup> represents a derivation

Let us consider a partition, P Nð Þ≔ qi

are

There are two equivalent expressions for the finite differences derivative of a product of vectors defined on the partition. A set of such expressions is

$$\begin{aligned} (\mathbf{D}gh)\_1 &= \frac{g\_2 h\_2 - e^{-v\Delta} g\_1 h\_1}{2\chi(v, \Delta)} \\ &= \frac{g\_2 h\_2 - e^{-v\Delta} g\_1 h\_2}{2\chi(v, \Delta)} + g\_1 \frac{e^{-v\Delta} h\_2 - h\_2 + h\_2 - e^{-v\Delta} h\_1}{2\chi(v, \Delta)} \\ &= h\_2 (\mathbf{Dg})\_1 + g\_1 (\mathbf{Dh})\_1 + g\_1 h\_2 \frac{e^{-v\Delta} - 1}{2\chi(v, \Delta)} \\ &= h\_2 (\mathbf{Dg})\_1 + g\_1 (\mathbf{Dh})\_1 + g\_1 h\_2 \left[ -\frac{v}{2} + \frac{v^2}{4} \Delta + O(\Delta^3) \right], \end{aligned} \tag{27}$$
 
$$(\mathbf{Dg}h)\_j = h\_{j+1} (\mathbf{Dg})\_j + g\_{j-1} (\mathbf{Dh})\_{j}. \tag{28}$$

$$\begin{split} (\mathbf{D}\mathbf{g}h)\_N &= h\_N (\mathbf{D}\mathbf{g})\_N + \mathbf{g}\_{N-1} (\mathbf{D}h)\_N + \frac{1 - e^{\eta \Delta}}{2\chi(v, \Delta)} \mathbf{g}\_{N-1} h\_N \\ &\approx h\_N (\mathbf{D}\mathbf{g})\_N + \mathbf{g}\_{N-1} (\mathbf{D}h)\_N + \mathbf{g}\_{N-1} h\_N \left[ -\frac{v}{2} - \frac{v^2}{4} \Delta + \mathcal{O}(\Delta^3) \right]. \end{split} \tag{29}$$

A second set of equalities is

$$\begin{split} \mathbf{(Dgh)}\_{1} &= \mathbf{g}\_{2} (\mathbf{Dh})\_{1} + h\_{1} (\mathbf{Dg})\_{1} + \mathbf{g}\_{2} h\_{1} \frac{e^{-\upsilon \Delta} - 1}{2\chi (\upsilon, \Delta)} \\\\ &= \mathbf{g}\_{2} (\mathbf{Dh})\_{1} + h\_{1} (\mathbf{Dg})\_{1} + \mathbf{g}\_{2} h\_{1} \left[ -\frac{\upsilon}{2} + \frac{\upsilon^{2}}{4} \Delta + O(\Delta^{3}) \right], \end{split} \tag{30}$$

$$(\mathrm{Dg}h)\_{\circ} = \mathrm{g}\_{\circ+1}(\mathrm{Dh})\_{\circ} + h\_{\circ-1}(\mathrm{Dg})\_{\circ} \tag{31}$$

$$\begin{split} \mathbf{(Dgh)}\_{N} &= \mathbf{g}\_{N} (\mathbf{Dh})\_{N} + h\_{\mathrm{N}-1} (\mathbf{Dg})\_{N} + \mathbf{g}\_{N} h\_{\mathrm{N}-1} \frac{1 - e^{\eta \Delta}}{2 \chi (v, \Delta)} \\ &\approx \mathbf{g}\_{N} (\mathbf{Dh})\_{N} + h\_{\mathrm{N}-1} (\mathbf{Dg})\_{N} + \mathbf{g}\_{N} h\_{\mathrm{N}-1} \left[ -\frac{v}{2} - \frac{v^{2}}{4} \Delta + O \left( \Delta^{3} \right) \right], \end{split} \tag{32}$$
