MY-ASSIGNMENTEXPERT™可以为您提供sydney MATH3078 Partial Differential Equations偏微分方程的代写代考和辅导服务！
这是悉尼大学 偏微分方程的代写成功案例。
MATH3078课程简介
The aim of this unit is to introduce some fundamental concepts of the theory of partial differential equations (PDEs) arising in Physics, Chemistry, Biology and Mathematical Finance. The focus is mainly on linear equations but some important examples of nonlinear equations and related phenomena re introduced as well. After an introductory lecture, we proceed with first-order PDEs and the method of characteristics. Here, we also nonlinear transport equations and shock waves are discussed. Then the theory of the elliptic equations is presented with an emphasis on eigenvalue problems and their application to solve parabolic and hyperbolic initial boundary-value problems. The Maximum principle and Harnack’s inequality will be discussed and the theory of Green’s functions.
Prerequisites
At the completion of this unit, you should be able to:
- LO1. demonstrating the ability to recognize different types of partial differential equations: “linear” or “nonlinear”, “order of the given equation”, “homogeneous” or “inhomogeneous”, and if it concerns 2nd-order equations, whether they are of “elliptic”, “parabolic”, or “hyperbolic” type
- LO2. demonstrating the conceptional understanding of how to apply different methods for solving different types of partial differential equations. Those methods include the use of classical ODE-concepts to solve PDEs
- LO3. understanding the definitions, main theorem, and corollaries of Green’s functions and Poisson kernel
- LO4. be fluent with “change of variable” into polar, cylindrical and spherically coordinates and to be able to compute partial derivatives in these coordinates
- LO5. develop an appreciation and strong working knowledge of the theory and application of elementary partial differential equations
- LO6. be fluent in using generalized Fourier transforms to solve parabolic and hyperbolic initial boundary value problems where the spatial variable might be of more than one variable
MATH3078 Partial Differential Equations HELP（EXAM HELP， ONLINE TUTOR）
For each pair of matrices $A$ and $B$, compute both $A B$ and $B A$.
a) $A=\left[\begin{array}{ll}1 & 2 \ 2 & 4\end{array}\right], B=\left[\begin{array}{cc}2 & -1 \ -1 & \frac{1}{2}\end{array}\right]$.
b) $A=\left[\begin{array}{ccc}1 & 3 & -1 \ 2 & -1 & 0\end{array}\right], B=\left[\begin{array}{cc}1 & 2 \ 2 & 3 \ 6 & 11\end{array}\right]$.
Find the inverse of the matrix $A$ made from the first 4 rows of Pascal’s triangle. After that, guess without computing the inverse of the matrix $B$ made from the first 5 rows of Pascal’s triangle and verify that you have guessed right. $A=\left[\begin{array}{llll}1 & 0 & 0 & 0 \ 1 & 1 & 0 & 0 \ 1 & 2 & 1 & 0 \ 1 & 3 & 3 & 1\end{array}\right]$,
$$
B=\left[\begin{array}{lllll}
1 & 0 & 0 & 0 & 0 \
1 & 1 & 0 & 0 & 0 \
1 & 2 & 1 & 0 & 0 \
1 & 3 & 3 & 1 & 0 \
1 & 4 & 6 & 4 & 1
\end{array}\right]
$$
a) Assume $A^3=A \cdot A \cdot A$ is the identity. Can you find a formula for $A^{-1}$.
b) Find a transformation in the plane which has the property that $A^7=1$. Find $A^{-1}$.
a) Assume $A$ is small enough so that $B=1+A+A^2+A^3+\ldots$. converges. Verify that $B$ is the inverse of $1-A$.
b) Let $A$ be the $3 \times 3$ matrix which has $1 / 10$ in every entry. Use Mathematica to compute $1+A+\ldots+A^{10}$ and compare it with the inverse of $1-A$.
Optional: lets experiment a bit! Make matrix plots of the inverses of the following $1000 \times 1000$ matrices
a) $A_{n m}=\cos (n m+3)$
b) $A_{n m}=\cot (2.1+n+m)$
c) $A_{n m}=1.4+n+m$
d) $A_{n m}=1.1-n^2-m^2$.
Here is how to get a random $1000 \times 1000$ matrix in Mathematica:
MatrixPlot $[$ Inverse $[$ Table $[$ Random []$,{\mathrm{n}, 1000},{\mathrm{m}, 1000}$
MY-ASSIGNMENTEXPERT™可以为您提供SYDNEY MATH3078 PARTIAL DIFFERENTIAL EQUATIONS偏微分方程的代写代考和辅导服务！
