recentl I have read Geroid De Barra theory of measure and integration. It was a very nice experience because the book is well written but with interesting exercises well explained and solution. Is there a simlar books who covers my phisical mathematical course like geroid de barra's book?
topicsI will approach are wave equations, caucjy global problems, d'alamber formula ,duhamel formula ,Cauchy-Dirichlet e Cauchy-Neumann, Fourier series, distributions, lapce and poisson equation green functions , heat equation
these are some problem that the orofessor gave last year $$ \begin{cases}\frac{\partial^2 u}{\partial t^2}=4 \frac{\partial^2 u}{\partial x^2} & \text { in } \mathbb{R}_{+} \times \mathbb{R}_{+} \\ u(x, 0)=x^{2 n}, \quad \frac{\partial u}{\partial t}(x, 0)=\sin x & \text { for} x \geq 0 \\ u(0, t)=0 \quad \text { for } t \geq 0\end{cases} $$ discuss the regularity of solution $n \in \mathbb{N}$.
Exercise 1. find with fourier series the solution to following CauchyDirichlet omogeneneus problem $$ \left\{\begin{array}{lll} \frac{\partial^2 u}{\partial t^2}=4 \frac{\partial^2 u}{\partial x^2} & \text { in } \quad(0, \pi) \times \mathbb{R}_{+} \\ u(x, 0)=\sin x \cos x, & \frac{\partial u}{\partial t}(x, 0)=\sin x \cos x \cos (2 x) & \text { for } \quad x \in[0, \pi] \\ u(0, t)=u(\pi, t)=0 & \text { for } \quad t \geq 0 \end{array}\right. $$ Is the solution periodic respect to time?
Exercise 1. find the solution to the following Poisson problem in circular crown $$ \begin{aligned} & \text { circolare } \Omega:=\left\{x \in \mathbb{R}^2: 1<|x|<2\right\}, \\ & \qquad \begin{cases}\Delta u+|x|^4=0 & \text { in } \Omega \\ u(x)=0 & \text { for }|x|=1, \\ u(x)=x_1^2-x_2^2 & \text { for }|x|=2 .\end{cases} \end{aligned} $$
Exercise 5. find with Fourier method the formal solution of the following Laplace roblem in the disk $B_R(0) \subset \mathbb{R}^2$, $$ \begin{cases}\Delta u=0 & \text { in } B_R(0) \\ u(x)=1 & \text { for }|x|=R \text { with } x_2>0, \\ u(x)=0 & \text { for }|x|=R \text { with } x_2 \leq 0,\end{cases} $$
Exercise 2. find with the Fourier method the solution of the following CauchyNeumann problem for diffusion equation, $$ \begin{cases}\frac{\partial u}{\partial t}=3 \frac{\partial^2 u}{\partial x^2}+t-\cos (\pi x) & \text { in }(0,1) \times \mathbb{R}_{+}, \\ u(x, 0)=\cos (2 \pi x) & \text { for } x \in[0,1], \\ \frac{\partial u}{\partial x}(0, t)=\frac{\partial u}{\partial x}(1, t)=0 & \text { for } t \geq 0 .\end{cases} $$ how does this quntit change $M(t)=\int_0^1 \mathrm{~d} x u(x, t) ?$
Esercizio 5. find the solution of the following diffusion euqation $\Omega_R=B_R(0)=\left\{x \in \mathbb{R}^3:|x|<R\right\},{ }^2$ $$ \begin{cases}\frac{\partial u}{\partial t}=\Delta u & \text { in } \Omega_R \times \mathbb{R}_{+}, \\ u(x, 0)=|x|^{-1} \sin (\pi|x| / R) \cos (\pi|x| / R) & \text { for } x \in \bar{\Omega}_R . \\ u(x, t)=0 & \text { for }(x, t) \in \partial \Omega_R \times[0,+\infty) .\end{cases} $$
thanks to everyone