Problem 1.

Consider the solution of the linear advection equation
$$\frac{\partial \rho}{\partial t}+a \frac{\partial \rho}{\partial x}=0 .$$
For $a=2, x \in[0,5], t \in[0,5]$, plot contours and three-dimensional surfaces of $\rho(x, t)$ for the following initial conditions:
(a) $\rho(x, 0)=\sin (\pi x)$
(b) $\rho(x, 0)=H(x)-H(x-1)$, where $H(x)$ is the Heaviside ${ }^4$ unit step function.

Problem 2.

Solve the Volterra equation
$$a+\int_0^t e^{b s} u(s) d s=a e^{b t}$$
Hint: Differentiate.
Find any and all eigenvalues $\lambda$ and associated eigenfunctions $y$ which satisfy
$$y(x)=\lambda \int_0^1 \frac{x}{s} y(s) d s .$$
Find a numerical approximation to the first six eigenvalues and eigenfunctions of
$$y(x)=\lambda \int_0^1 \cos (10 x s) y(s) d s .$$
Use sufficient resolution to resolve the eigenvalues to three digits of accuracy. Plot on a single graph the first six eigenfunctions.

## MATH0011 Mathematical Methods 2

Year:
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Normal student group $(s)$ :
Value:
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Lecturer:
2023-2024
MATH0011
$4(\mathrm{UG})$
UG: Year 1 Mathematics degrees
15 credits $(=7.5$ ECTS credits $)$
2
The final weighted mark for the module is given by: $20 \%$ programming component. $5 \%$ assessed homework (vector calculus). $75 \%$ unseen exam, including one question on the programming part. In order to pass the module you must have at least $40 \%$ in both the examination and the final weighted mark.
MATH0010
Prof $\mathrm{H}$ Wilson and Prof R Halburd
Course Description and Objectives
This module consists of two parts: first an introduction to programming for the mathematical sciences (4 weeks) and then an introduction to multivariable calculus (5 weeks).
Programming is becoming an increasingly important tool for mathematicians both in industry and in research. In view of this, some elements of basic scientific computation should be part of any modern undergraduate curriculum in Mathematics. Multivariable calculus on the other hand is of fundamental importance in a variety of fields of pure and applied mathematics such as electromagnetism, fluid mechanics, differential geometry and integration theory.
The aim of the first part is to introduce the students to the ideas of computer programming and its uses in scientific computing for science and its applications. The programming language Python will be considered in the course, but the underlying principles are general. Students should learn how to write accurate programs for the computational solution of mathematical problems.
The aim of the second part is to introduce the students to the ideas of the calculus of several variables, and to develop their understanding of functions of several variables, their derivatives and integrals.
Recommended Texts
The Python Tutorial https://docs.python. org/3/tutorial/ takes you through the functionality of Python with lots of examples. The best way to learn Python is to practise: Project Euler https://projecteuler.net/ is an online collection of over 600 mathematical programming problems that make excellent practice. Finally, Langtangen’s book A Primer on Scientific Programming with Python https://hplgit.github.io/primer.html/doc/pub/half/book.pdf gives an introduction to using Python with a focus on using Python for mathematical problems; however, please note that many of the problems featured in this book are outside the scope of this course.
For the vector calculus part, there are many excellent texts available. In particular, Calculus (Gilbert Strang), of which chapters 13-15 are relevant to this course, is free online at https://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/ You may also like the Schaum’s Outline Series Advanced Calculus; Vector Calculus (Paul Matthews) or Advanced Engineering Mathematics (Erwin Kreyszig).

Fourier analysis代写

## 离散数学代写

Partial Differential Equations代写可以参考一份偏微分方程midterm答案解析