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# 数学代写|数值分析代写Numerical analysis代考|MATH2200 Handling Discontinuities in Differential Equations

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## 数学代写数值分析代写Numerical analysis代考|Handling Discontinuities in Differential Equations

So far, the methods for solving initial value problems
$$y^{\prime}(x)=f(x, y(x)), \quad y\left(x_0\right)=y_0,$$
assumed that the right hand side $f$ of the differential equation and, therefore, its solution $y(x)$ is sufficiently often differentiable. However, there are many applied problems where $f$ and $y$ are discontinuous. These discontinuities are often due to the underlying tasks, e.g., in friction and contact problems of mechanics, in process engineering if one admits different charges of input. But they may also be caused by unavoidable simplifications of a model, e.g. if one represents discrete tabular data found in experiments by an interpolating function of low order of differentiability.
Here, we consider initial value problems of the form
(7.2.18.1) \begin{aligned} y^{\prime}(x) &=f_n(x, y(x)), \quad x_{n-1}<x<x_n, \quad n=1,2, \ldots, \ y\left(x_0\right) &=y_0 . \end{aligned}
We assume that the initial values $y_n^{+}=\lim {x \rightarrow x_n^{+}} y(x)$ at the points of discontinuity, the “switching points” $x_n$, are determined by known transition functions $\phi_n(x, y)$ as follows: $$y_n^{+}=\phi_n\left(x_n, y_n^{-}\right)$$ where $y_n^{-}=\lim {x \rightarrow x_n^{-}} y(x)$.

## 数学代写|数值分析代写Numerical analysis代考|Sensitivity Analysis of Initial-Value Problems

Initial value problems often depend on real parameters $p=\left(p_1, \ldots, p_{n_p}\right)$,
\begin{aligned} y^{\prime}(x ; p) &=f(x, y(x ; p), p), \quad x_0<x<x_1, \ y\left(x_0 ; p\right) &=y_0(p) \end{aligned}
Possibly, even small changes of these parameters may produce large changes in the solution $y(x ; p)$. Thus, it is important to carefully investigate parameter dependence. This may add valuable insight into the process described by the differential equation. Mainly, such an investigation requires computing, along with a solution trajectory $y(x ; p)$, also its sensitivity (condition numbers) given by the matrix
$$\left.\frac{\partial y(x ; p)}{\partial p}\right|_p$$
If the parameters are genuine model parameters, then this information allows to judge the quality of the model. Sensitivities play also a dominant role in the solution of optimal control problems, such as parametrization of control functions or parameter identification [Heim and von Stryk (1996), Engl et al. (1999)]. There are several numerical methods to compute these sensitivities:

• approximation by difference quotients,
• solution of the sensitivity equations,
With the first method, the sensitivities are approximated by
$$\left.\frac{\partial y(x ; p)}{\partial p_i}\right|_p \approx \frac{y\left(x ; p+\Delta p_i e_i\right)-y(x ; p)}{\Delta p_i}, \quad i=1, \ldots, n_p$$

## 数学代写数值分析代写Numerical analysis代考|Handling Discontinuities in Differential Equations

$$y^{\prime}(x)=f(x, y(x)), \quad y\left(x_0\right)=y_0,$$

(7.2.18.1)的初始直问题 $y^{\prime}(x)=f_n(x, y(x)), \quad x_{n-1}<x<x_n, \quad n=1,2, \ldots, y\left(x_0\right) \quad=y_0$.

$$y_n^{+}=\phi_n\left(x_n, y_n^{-}\right)$$

## 数学代写|数值分析代写Numerical analysis代考|Sensitivity Analysis of InitialValue Problems

$$y^{\prime}(x ; p)=f(x, y(x ; p), p), \quad x_0<x<x_1, y\left(x_0 ; p\right) \quad=y_0(p)$$

$$\left.\frac{\partial y(x ; p)}{\partial p}\right|_p$$

• 差商逼近，
• 灵敏度方程的解,
• 伴随方程的解。
使用第一种方法，灵敏度近似为
$$\left.\frac{\partial y(x ; p)}{\partial p_i}\right|_p \approx \frac{y\left(x ; p+\Delta p_i e_i\right)-y(x ; p)}{\Delta p_i}, \quad i=1, \ldots, n_p$$

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