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## 数学代写|数值方法作业代写numerical methods代考|SAIRA and Shuhuang Xiang

Boundary element method and finite element method are intensively eminent numerical approaches to evaluate partial differential equations (PDEs), which appear in variety of disciplines from engineering to astronomy and quantum mechanics [1-5]. Although these methods lead PDEs to Fredholm integral equations or Voltera integral equations, but these kind of integral equations posses integrals of oscillatory, Cauchy-singular, logarithmic singular, weak singular kernel functions. However, these classical methods are failed to approximate the integrals constitute kernel functions of highly oscillation and logarithmic singularity.
This paper aims at approximation of the integral
$$I^\alpha[f]=f_{-1}^1 \frac{f(x) \log (x-\alpha) e^{i k x}}{x-t} d x,$$
where $t \in(-1,1), k \gg 1, \alpha \in[-1,1], f(x)$ is relatively smooth function. For integral (1) the developed strategy for logarithmic singularity $\log (x-\alpha)$ is valid for $\alpha \in[-1,1]$. In particular, the highly oscillatory integral, $\int_{-1}^1 f(x) e^{i k x} d x$ has been computed by many methods such as asymptotic expansion, Filon method, Levin collocation method and numerical steepest descent method [6-10]. For instant, Dominguez et al. [11] for function $f(x)$ with integrable singularities have proposed an error bound, calculated in Sobolev spaces $H^m$, for composite Filon-Clenshaw-Curtis quadrature. Error bound depends on the derivative of $f(x)$ and length of the interval $M$, for some $C_1(f)$ defined as $E_N \leq C_1(f)\left(\frac{1+|\log (k)|}{k^{1+\beta}}\right)^r(\log M)^{1+\beta-r}\left(\frac{1}{M}\right)^{N+1-r}$ for $\beta \in(-1,0), r \in[0,1+\beta]$

On the other hand, one methodology for numerical evaluation of integral $f_{-1}^1 \frac{f(x) e^{i k x}}{x-t} d x$ is replacing $f(x)$ by different kind of polynomials [12,13]. Another technique is based on analytic continuation of the integral if the integrand $f(x)$ is analytic in the complex region [14]. As far as for $k=0$ solution methods and properties of the solution for relative non-homogenous integrals have been discussed by using Brestain polynomials and Chebyshev polynoimals of all four kinds in [3,15].

For integral $\int_{-1}^1 f(x) \log \left((x-\alpha)^2\right) e^{i k x} d x$ Clenshaw-Curtise rule is applied for numerical calculation. Wherein the convergence rate is independent of $k$ but depends on the number of nodes of quadrature rule and function $f(x)$ [16]. Furthermore, Piessense and Branders [17] established the Clenshaw-Curtis quadrature rule, relies on the recurrence relation for $\int_{-1}^1 f(x) e^{i k x}(x+1)^a \log (x+1) d x$. They replaced the nonoscillatory and nonsingular part of the integrand by Chebyshev series. Chen [18] presented the numerical approximation of the integral $I[f]=f_{-1}^1 \frac{f(x)^{k i x x}}{(x+1)^\mu(x-1)^\beta \prod_{m=1}^n\left(x-\tau_m \gamma^{m m}\right.} d x$, with $\alpha, \beta<1, a<\gamma_m<b$ and $\gamma_m \leq 1$. For analytic function $f(x)$ the integral was rewritten in the form of sum of line integrals, wherein the integrands do not oscillate and decay exponentially.

## 数学代写|数值方法作业代写numerical methods代考|Numerical Methods

In the computation of integral $I^\alpha[f]$, the Clenshaw-Curtis quadrature approach is extensively adopted. The scheme is postulated on interpolating the function $f(x)$ at Clenshaw-Curtis points set $X_{N+1}=\left{x_j=\cos \frac{j \pi}{N}\right}_{j=0}^N$. Writing the interpolation polynomial as basis of Chebyshev series
$$f(x) \approx P_N(x)=\sum_{n=0}^N{ }^n a_n T_n(x),$$
where $T_n(x)$ is the Chebyshev polynomial of first kind of degree $N$ and double prime denotes a sum whose first and last terms are halved, the coefficients
$$a_n=\frac{2}{N} \sum_{j=0}^N f\left(x_j\right) T_n\left(x_j\right)$$ can be computed efficiently by FFT in $O(N \log N)$ operations $[8,9]$. This paper appertains to Clenshaw-Curtis quadrature, which depends on Hermite interpolating polynomial that allow us to get higher order accuracy
$$\tilde{P}\left(x_j\right)=f\left(x_j\right), j=0, \cdots N ; \quad \widetilde{P}(t)=f(t) .$$
For any fixed $t$, we can elect felicitous $N$ such that $t \notin\left{x_j\right}_{j=0}^N$ and rewrite Hermite interpolating polynomial of degree $N+1$ in terms of Chebyshev series
$$\tilde{P}{N+1}(x)=\sum{n=0}^{N+1} c_n T_n(x)$$
$c_n$ can be calculated in $O(N)$ operations once if $a_n$ are known [13,21]. Finally Clenshaw-Curtis quadrature for integral $I^\alpha[f]$ is defined as
\begin{aligned} I_{N+1}^\alpha[f] &=\sum_{n=0}^{N+1} c_n f_{-1}^1 \frac{T_n(x) \log (x-\alpha) e^{i k x}}{x-t} d x \ &=\sum_{n=0}^{N+1} c_n D_n^\alpha(k, t) \end{aligned}
where
$$D_n^\alpha(k, t)=f_{-1}^1 \frac{T_n(x) \log (x-\alpha) e^{i k x}}{x-t} d x$$
more specifically $D_n^a(k, t)$ are called the modified moments. Efficiency of the Clenshaw-Curtis quadrature depends on the fast computation of the moments. In ensuing sub-section, we deduce the recurrence relation for $D_n^a(k, t)$.

# 数值方法代考

## 数学代写|数值方法作业代写numerical methods代考|SAIRA and Shuhuang Xiang

$$I^\alpha[f]=f_{-1}^1 \frac{f(x) \log (x-\alpha) e^{i k x}}{x-t} d x,$$

## 数学代写|数值方法作业代写numerical methods代考|Numerical Methods

$$f(x) \approx P_N(x)=\sum_{n=0}^N{ }^n a_n T_n(x),$$

$$a_n=\frac{2}{N} \sum_{j=0}^N f\left(x_j\right) T_n\left(x_j\right)$$

$$\tilde{P}\left(x_j\right)=f\left(x_j\right), j=0, \cdots N ; \quad \widetilde{P}(t)=f(t) .$$

$$\tilde{P} N+1(x)=\sum n=0^{N+1} c_n T_n(x)$$
$c_n$ 可以计算在 $O(N)$ 如果操作一次 $a_n$ 已知 $[13,21]$ 。最后 Clenshaw-Curtis 求积积分 $I^\alpha[f]$ 定义为
$$I_{N+1}^\alpha[f]=\sum_{n=0}^{N+1} c_n f_{-1}^1 \frac{T_n(x) \log (x-\alpha) e^{i k x}}{x-t} d x \quad=\sum_{n=0}^{N+1} c_n D_n^\alpha(k, t)$$

$$D_n^\alpha(k, t)=f_{-1}^1 \frac{T_n(x) \log (x-\alpha) e^{i k x}}{x-t} d x$$

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