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6.E: Parabolic Equations (Exercises)
https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Miersemann)/6%3A_Parabolic_Equations/6.E%3A_Parabolic_Equations_(Exercises)
Solve the initial-boundary value problem (rotationally symmetric solution in a ball): find $c(r,t)$ on $(0,R)\times(0,\infty)$ of \begin{eqnarray} \label{equr} \frac{\partial c}{\partial t}&=&\fra...Solve the initial-boundary value problem (rotationally symmetric solution in a ball): find $c(r,t)$ on $(0,R)\times(0,\infty)$ of $\begin{eqnarray} \label{equr} \frac{\partial c}{\partial t}&=&\frac{1}{r^2}\frac{\partial}{\partial r}\left(Dr^2\frac{\partial c}{\partial r}\right)-kc\\ \label{initial} c(r,0)&=&h(r),\ 0<r<R,\\ \label{bc1r} c(R,t)&=&c_0\qquad \mbox{(boundary condition)},\\ \label{bc2r} \sup_{0<r<R,\;0<t<T}|c(r,t)|&<&\infty \qquad \mbox{(boundary condition)}, \end{eqnarray}$ where…
3.4: Systems of Second Order
https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Miersemann)/3%3A_Classification/3.4.0%3A_Systems_of_Second_Order
We assume $A^{kl}=A^{lk}$ , which is no restriction of generality provided $u\in C^2$ is satisfied. As in the previous sections, the classification follows from the question whether or not we can cal...We assume $A^{kl}=A^{lk}$ , which is no restriction of generality provided $u\in C^2$ is satisfied. As in the previous sections, the classification follows from the question whether or not we can calculate formally the solution from the differential equations, if sufficiently many data are given on an initial manifold. If there is a solution $\chi$ with $\nabla\chi\not=0$ , then it is possible that second derivatives are not continuous in a neighborhood of $\mathcal{S}$ .
4: Hyperbolic Equations
https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Miersemann)/4%3A_Hyperbolic_Equations
Here we consider hyperbolic equations of second order, mainly wave equations.
3.2: Quasilinear Equations of Second Order
https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Miersemann)/3%3A_Classification/3.2.0%3A_Quasilinear_Equations_of_Second_Order
Here we consider the equation \sum_{i,j=1}^na^{ij}(x,u,\nabla u)u_{x_ix_j}+b(x,u,\nabla u)=0 in a domain $\Omega\subset\mathbb{R}$ , where $u:\ \Omega\mapsto\mathbb{R}^1$ . We assume that \(a^{ij}=a...Here we consider the equation \sum_{i,j=1}^na^{ij}(x,u,\nabla u)u_{x_ix_j}+b(x,u,\nabla u)=0 in a domain $\Omega\subset\mathbb{R}$ , where $u:\ \Omega\mapsto\mathbb{R}^1$ . We assume that $a^{ij}=a^{ji}$ . As in the previous section we can derive the characteristic equation \sum_{i,j=1}^na^{ij}(x,u,\nabla u)\chi_{x_i}\chi_{x_j}=0. In contrast to linear equations, solutions of the characteristic equation depend on the solution considered. Erich Miersemann (Universität Leipzig)
6.3: Maximum Principle
https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Miersemann)/6%3A_Parabolic_Equations/6.3%3A_Maximum_Principle
The initial value problem $u_t-\triangle u=0$ in $D_T$ , $u(x,0)=f(x)$ , $x\in\mathbb{R}^n$ , has a unique solution in the class defined by $u\in C(\overline{D_T})$ , $u_t$ , $u_{x_ix_k}$ exi...The initial value problem $u_t-\triangle u=0$ in $D_T$ , $u(x,0)=f(x)$ , $x\in\mathbb{R}^n$ , has a unique solution in the class defined by $u\in C(\overline{D_T})$ , $u_t$ , $u_{x_ix_k}$ exist and are continuous in $D_T$ and $|u(x,t)|\le Me^{a|x|^2}$ . where $a^{ij}\in C(D_T)$ are real, $a^{ij}=a^{ji}$ , and the matrix $(a^{ij})$ is non-negative, that is,
7: Elliptic Equations of Second Order
https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Miersemann)/7%3A_Elliptic_Equations_of_Second_Order
Here we consider linear elliptic equations of second order, mainly the Laplace equation Solutions of the Laplace equation are called potential functions or harmonic functions. The general elliptic equ...Here we consider linear elliptic equations of second order, mainly the Laplace equation Solutions of the Laplace equation are called potential functions or harmonic functions. The general elliptic equation for a scalar function $u(x)$ , $x\in\Omega\subset\mathbb{R}^n$ , is where the matrix $A=(a^{ij})$ is real, symmetric and positive definite. If $A$ is a constant matrix, then a transform to principal axis and stretching of axis leads to
Front Matter
https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Miersemann)/3%3A_Classification/00%3A_Front_Matter
3.1: Linear Equations of Second Order
https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Miersemann)/3%3A_Classification/3.1.0%3A_Linear_Equations_of_Second_Order
The condition ( $\ref{nonchar}$ ) is satisfied for each $\chi$ with $\nabla\chi\not=0$ if the quadratic matrix $(a^{ij}(x))$ is positive or negative definite for each $x\in\Omega$ , which is equiva...The condition ( $\ref{nonchar}$ ) is satisfied for each $\chi$ with $\nabla\chi\not=0$ if the quadratic matrix $(a^{ij}(x))$ is positive or negative definite for each $x\in\Omega$ , which is equivalent to the property that all eigenvalues are different from zero and have the same sign. Consider the case that the (real) coefficients $a^{ij}$ in equation ( $\ref{linsecond}$ ) are {\it constant}. We recall that the matrix $A=(a^{ij})$ is symmetric, that is, $A^T=A$ .
1.1: Examples
https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Miersemann)/1%3A_Introduction/1.1%3A_Examples
is independent of the curve which connects the points $P_0$ with $P_1$ in a simply connected domain $\Omega\subset\mathbb{R}^2$ is that the partial differential equation (condition of integrabil...is independent of the curve which connects the points $P_0$ with $P_1$ in a simply connected domain $\Omega\subset\mathbb{R}^2$ is that the partial differential equation (condition of integrability) It is known from the theory of functions of one complex variable that the real part $u$ and the imaginary part $v$ of a differentiable function $f(z)$ are solutions of the Laplace equation
2.1: Linear Equations
https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Miersemann)/2%3A_Equations_of_First_Order/2.1%3A_Linear_Equations
From the theory of ordinary differential equations it follows (Theorem of Picard-Lindelöf) that there is a unique solution in a neighbourhood of $t=0$ provided the functions $a_1,\ a_2$ are in \(C...From the theory of ordinary differential equations it follows (Theorem of Picard-Lindelöf) that there is a unique solution in a neighbourhood of $t=0$ provided the functions $a_1,\ a_2$ are in $C^1$ . Then an integral is $y/x$ , $x\not=0$ , and for a given $C^1$ -function the function $u=H(x/y)$ is a solution of the differential equation.
7.2.1: Conclusions from the Representation Formula
https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Miersemann)/7%3A_Elliptic_Equations_of_Second_Order/7.2.1%3A_Conclusions_from_the_Representation_Formula
\int_{\partial B_\rho(x)}\frac{1}{r^{n-2}}\frac{\partial u}{\partial n_y}\ dS_y&=&\frac{1}{\rho^{n-2}}\int_{\partial B_\rho(x)}\frac{\partial u}{\partial n_y}\ dS_y\\ We recall that a domain \(\Omega\...\int_{\partial B_\rho(x)}\frac{1}{r^{n-2}}\frac{\partial u}{\partial n_y}\ dS_y&=&\frac{1}{\rho^{n-2}}\int_{\partial B_\rho(x)}\frac{\partial u}{\partial n_y}\ dS_y\\ We recall that a domain $\Omega\in\mathbb{R}^n$ is called connected if $\Omega$ is not the union of two nonempty open subsets $\Omega_1$ , $\Omega_2$ such that $\Omega_1\cap\Omega_2=\emptyset$ . Assume $\Omega$ is connected and bounded, and $u\in C^2(\Omega)\cap C(\overline{\Omega})$ is harmonic in $\Omega$ .

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