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# 数学代写|有限元方法代写finite differences method代考|Integration by parts formulae

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## 数学代写|有限元代写Finite Element Method代考|Integration by parts formulae

Let $p, q, u, v$, and $w$ be sufficiently differentiable functions of the coordinate $x$. Then the following integration by parts formulae hold:
$$\int_a^b w \frac{d}{d x}\left(p \frac{d u}{d x}\right) d x=-\int_a^b p \frac{d w}{d x} \frac{d u}{d x} d x-w(a)\left(p \frac{d u}{d x}\right){x=a}+w(b)\left(p \frac{d u}{d x}\right){x=b}$$

\begin{aligned} \int_a^b v \frac{d^2}{d x^2}\left(q \frac{d^2 w}{d x^2}\right) d x= & \int_a^b q \frac{d^2 v}{d x^2} \frac{d^2 w}{d x^2} d x \ & -v(a)\left[\frac{d}{d x}\left(q \frac{d^2 w}{d x^2}\right)\right]{x=a}+v(b)\left[\frac{d}{d x}\left(q \frac{d^2 w}{d x^2}\right)\right]{x=b} \ & +\left(\frac{d v}{d x}\right){x=a}\left(q \frac{d^2 w}{d x^2}\right){x=a}-\left(\frac{d v}{d x}\right){x=b}\left(q \frac{d^2 w}{d x^2}\right){x=b} \end{aligned}
These relations can easily be established.
To establish the relation in Eq. (2.2.26), we begin with the identity
$$\frac{d}{d x}\left(w \cdot p \frac{d u}{d x}\right)=\frac{d w}{d x} p \frac{d u}{d x}+w \frac{d}{d x}\left(p \frac{d u}{d x}\right)$$
Therefore, we have
\begin{aligned} \int_a^b w \frac{d}{d x}\left(p \frac{d u}{d x}\right) d x & =\int_a^b \frac{d}{d x}\left(w \cdot p \frac{d u}{d x}\right) d x-\int_a^b p \frac{d w}{d x} \frac{d u}{d x} d x \ & =-w(a)\left(p \frac{d u}{d x}\right){x=a}+w(b)\left(p \frac{d u}{d x}\right){x=b}-\int_a^b p \frac{d w}{d x} \frac{d u}{d x} d x \end{aligned}

## 数学代写|有限元代写Finite Element Method代考|Definition of a matrix

Consider the system of linear algebraic equations
\begin{aligned} & b_1=a_{11} x_1+a_{12} x_2+a_{13} x_3 \ & b_2=a_{21} x_1+a_{22} x_2+a_{23} x_3 \ & b_3=a_{31} x_1+a_{32} x_2+a_{33} x_3 \end{aligned}
We see that there are nine coefficients $a_{i j}, i, j=1,2,3$ relating the three coefficients $\left(b_1, b_2, b_3\right)$ to $\left(x_1, x_2, x_3\right)$. The form of these linear equations suggests writing down the coefficients $a_{i j}$ (jth components in the ith equation) in the rectangular array
$$\mathbf{A}=\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{array}\right]$$
This rectangular array $\mathbf{A}$ of numbers $a_{i j}$ is called a matrix, and the quantities $a_{i j}$ are called the elements of matrix $\mathbf{A}$.

If a matrix has $m$ rows and $n$ columns, we will say that is $m$ by $n(m \times n)$, the number of rows always being listed first. The element in the $i$ th row and $j$ th column of a matrix $\mathbf{A}$ is generally denoted by $a_{i j}$, and we will sometimes designate a matrix by $\mathbf{A}=[A]=\left[a_{i j}\right]$. A square matrix is one that has the same number of rows as columns. An $n \times n$ matrix is said to be of order $n$. The elements of a square matrix for which the row number and the column number are the same (that is, $a_{i i}$ for any fixed $i$ ) are called diagonal elements. A square matrix is said to be a diagonal matrix if all of the off-diagonal elements are zero. An identity matrix or its unit matrix, denoted by $\mathbf{I}=[I]$, is a diagonal matrix whose elements are all 1’s. Examples of diagonal and identity matrices are:
$$\left[\begin{array}{rrrr} 6 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 3 & 0 \ 0 & 0 & 0 & -2 \end{array}\right], \quad \mathbf{I}=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{array}\right]$$
If the matrix has only one row or one column, we will normally use only a single subscript to designate its elements. For example,
$$\mathbf{X}=\left{\begin{array}{l} x_1 \ x_2 \ x_3 \end{array}\right}, \quad \mathbf{Y}=\left{\begin{array}{lll} y_1 & y_2 & y_3 \end{array}\right}$$
denote a column matrix and a row matrix, respectively. Row and column matrices can be used to denote the components of a vector.

## 数学代写|有限元代写Finite Element Method代考|Integration by parts formulae

$$\int_a^b w \frac{d}{d x}\left(p \frac{d u}{d x}\right) d x=-\int_a^b p \frac{d w}{d x} \frac{d u}{d x} d x-w(a)\left(p \frac{d u}{d x}\right){x=a}+w(b)\left(p \frac{d u}{d x}\right){x=b}$$

\begin{aligned} \int_a^b v \frac{d^2}{d x^2}\left(q \frac{d^2 w}{d x^2}\right) d x= & \int_a^b q \frac{d^2 v}{d x^2} \frac{d^2 w}{d x^2} d x \ & -v(a)\left[\frac{d}{d x}\left(q \frac{d^2 w}{d x^2}\right)\right]{x=a}+v(b)\left[\frac{d}{d x}\left(q \frac{d^2 w}{d x^2}\right)\right]{x=b} \ & +\left(\frac{d v}{d x}\right){x=a}\left(q \frac{d^2 w}{d x^2}\right){x=a}-\left(\frac{d v}{d x}\right){x=b}\left(q \frac{d^2 w}{d x^2}\right){x=b} \end{aligned}

$$\frac{d}{d x}\left(w \cdot p \frac{d u}{d x}\right)=\frac{d w}{d x} p \frac{d u}{d x}+w \frac{d}{d x}\left(p \frac{d u}{d x}\right)$$

\begin{aligned} \int_a^b w \frac{d}{d x}\left(p \frac{d u}{d x}\right) d x & =\int_a^b \frac{d}{d x}\left(w \cdot p \frac{d u}{d x}\right) d x-\int_a^b p \frac{d w}{d x} \frac{d u}{d x} d x \ & =-w(a)\left(p \frac{d u}{d x}\right){x=a}+w(b)\left(p \frac{d u}{d x}\right){x=b}-\int_a^b p \frac{d w}{d x} \frac{d u}{d x} d x \end{aligned}

## 数学代写|有限元代写Finite Element Method代考|Definition of a matrix

\begin{aligned} & b_1=a_{11} x_1+a_{12} x_2+a_{13} x_3 \ & b_2=a_{21} x_1+a_{22} x_2+a_{23} x_3 \ & b_3=a_{31} x_1+a_{32} x_2+a_{33} x_3 \end{aligned}

$$\mathbf{A}=\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{array}\right]$$

$$\left[\begin{array}{rrrr} 6 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 3 & 0 \ 0 & 0 & 0 & -2 \end{array}\right], \quad \mathbf{I}=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{array}\right]$$

$$\mathbf{X}=\left{\begin{array}{l} x_1 \ x_2 \ x_3 \end{array}\right}, \quad \mathbf{Y}=\left{\begin{array}{lll} y_1 & y_2 & y_3 \end{array}\right}$$

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