Mathematics

Questions tagged [linear-pde]

This tag is for questions relating to linear partial differential equations, in which the dependent variable (and its derivatives) appear in terms with degree at most one

Definition: A partial differential equation is said to be linear if the dependent variable and its partial derivatives occur only in the first degree and are not multiplied.

  • A first-order PDE for an unknown function $~u(x,y)~$ is said to be linear if it can be expressed in the form $$a(x,y)~\frac{\partial}{\partial x}u(x,y)+b(x,y)~\frac{\partial}{\partial y}u(x,y)+c(x,y)~u(x,y)=d(x,y)$$
  • The general linear second-order PDE in two independent variables has the form $$Au_{xx}+2Bu_{xy}+Cu_{yy}+\cdots\text{(lower order terms)}=0$$where the coefficients $~A,~ B, ~C,\cdots~$ may depend upon $~x~$ and $~y~$.
  • Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables.

Examples: Common examples of linear PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Klein–Gordon equation, and Poisson's equation.

Reference:

https://en.wikipedia.org/wiki/Partial_differential_equation

568 questions
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Need good reference or a proof on regularity of solution to Neumann problem

Let $\Omega\subset \mathbb{R}^d $ be a bounded open subset ( $d\in \mathbb{N}$ ) and denote $\partial\Omega$ its boundary which we assume to be Lipschitz. The classical inhomogeneous Neumann problem for Laplace operator associate to data…
Guy Fsone
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Boundary value problem for Laplace's equation / probability a BMP leaves cone via the ice cream rather than the biscuit

Suppose I start a Brownian motion $X_t$ at position $(0,0,\varepsilon)$ in $\mathbb R^3$ . Suppose also I have a cone with its apex at the origin, given by $U=\{(x,y,z):\sqrt{x^2+y^2}< z\tan\alpha\}$ (recall $\alpha$ is the angle of incidence). Let…
Lavender
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9
votes
3 answers

Can every closed differential form be expressed via constant coefficients?

Let $M$ be a smooth $n$ dimensional manifold, and let $1 \le k < n$ . Let $\omega \in \Omega^k(M)$ be a closed $k$ -form on $M$ . Let $p \in M$ . Do there exist coordinates around $p$ , such that $\omega=a_{i_1i_2\dots i_k}dx^{i_1} \wedge dx^{i_2}…
Asaf Shachar
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How to solve a system of PDE $u_t+u_x=v, v_t+v_x=-u$

Solve the following initial value problem: $$u_t+u_x=v, \\v_t+v_x=-u, \\u(0,x)=u_0(x), \\ v(0,x)=v_0(x).$$ I did not learn any method to solve a system of PDE so I guess there is a "trick". So far we've learned how to solve PDE's of the form:…
catch22
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Why can we pass limit under integral sign in proof of solving Poisson's equation? (Evans PDE)

On page 23 of Lawrence Evans' Partial Differential Equations text (2nd edition) he claims that $$\frac{ f( x + he_i - y) - f( x-y)}{h} \to \frac{ \partial f}{ \partial x_i} ( x-y)$$ uniformly on $\mathbb{R}^n$ as $h \to 0$ . So $$\frac{ \partial u}{…
kam
  • 312
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1 answer

Are distributional (local) solutions to the heat equation smooth?

I have thought about this apparently simple problem. Question: Let $\Omega\subset \mathbb{R}_{x,y}^2$ be an open subset. Let's suppose we have $u\in\mathscr{D}'(\Omega)$ that satisfies $$ (\partial_y-\partial_x^2)u=0.$$ Is it true that $u\in…
Lorenzo Pompili
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7
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A possible error in Villani's monograph “Hypocoercivity”

I think there is an (possible) error in Villani's monograph titled " Hypocoercivity ". To be specific, in page 62 (the first snapshot), he defined a new inner product $((\cdot,\cdot))$ as in (8.1). Then in the Theorem 40 below (page 64, with the…
Fei Cao
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2 answers

Is it possible to solve a hyperbolic moving boundary problem?

J. L. Davies says in his book, " The basic principle in PDEs is that boundary value problems are associated with elliptic equations while initial value problems, mixed problems, and problems with radiation effects at boundaries are associated with…
Nanashi No Gombe
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6
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Bessel satisfies linear pde but can't find any references for it

The form $$\Phi_s(p)= \int_0^\infty e^{-px} e^{-s/x} \, dx = 2\sqrt{\frac sp} K_1(2\sqrt{sp})$$ is a standard representation for the $K_\nu(\cdot)$ Bessel function ( $\nu=1$ ). It appears in analytic number theory in some very deep, beautiful, and…
J. Zimmerman
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1 answer

warping functions obey diff eq. does that imply $g_t$ obeys same diff eq?

Consider $(M,g_{t})$ equipped with a $1$ -parameter family of warped metrics for real parameter $t>0$ $$g_{t} = \frac{1}{\phi_t(u)^{2}}\ du^{2} + \phi_t(u)\ dv^{2}$$ and suppose that the warping function obeys the linear equation $$ t…
J. Zimmerman
  • 1,250
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1 answer

Equivalence of two harmonic problems on different domains

I want to solve \begin{cases} \Delta u = 0,&\text{ in }\mathbb{R}^3\setminus B_1(0) \\ u=0,&\text{ as }\Vert x\Vert\rightarrow +\infty \\ u=1,&\text{ on }\partial B_1(0). \end{cases} I know that the solution to this problem is $$ \bar{u}(x) = \Vert…
Dadeslam
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How to transform a PDE into canonical form

Question: $$4U_{xx} +12U_{xy} +9U_{yy}=0$$ I would like to transform this pde into canonical form. I know that the pde is a parabolic type but I am unsure how to proceed with rewriting it without cross-derivatives.
ViB
  • 145
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Can one show that the solution to this elliptic PDE is a decreasing function?

Assume $\boldsymbol{g}:\mathbb{R}_{+}^{n}\to [0,1]^{n}$ and $\boldsymbol{h}:\mathbb{R}_{+}^{n}\to \mathbb{R}^{n}$ continuous functions, and that if I know $\boldsymbol{g}$ to be decreasing* then I can deduce $\boldsymbol{h}$ strictly increasing. Now…
Ali
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6
votes
1 answer

Two PDE for one unknown?

Let $x \in (0,L)$ , $t \in (0,T)$ , and let $f_1 = f_1(x,t) \in \mathbb{R}$ , $f_2 = f_2(x,t) \in \mathbb{R}$ , $u^0 = u^0(x) \in \mathbb{R}$ and $g= g(t) \in \mathbb{R}$ be continuous functions. My question is: Can we find a function $u = u(x,t)…
user344045
  • 479
6
votes
1 answer

Method of characteristics for a first-order linear PDE: $D_t u + xD_x u = t^3$

In short terms, I must solve a certain first-order PDE. I applied the method of caracteristics to find the integral curves and found an answer that indeed satisfies the equation. However, Wolfram Mathematica gives a different solution. Comments on…
Danilo Gregorin Afonso
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