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In finite element method for second order elliptic problem with neumann boundary value, is the solution weakly satisfies the boundary conditions?

You would need extra assumptions on the domain and the boundary data $g$. In particular if $g$ is the trace of a function that does not belong to $H^2$, then $u$ will not be regular enough to be a ...
Ellya's user avatar
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Solution for the Second order ODE with Dirichlet boundary condition

This might be useful. For a general inhomogenous 2nd ODE like $$y''(t)-(\alpha+\beta)y'(t)+\alpha\beta y(t)=r(t),$$ of which the characteristic or eigen equation can be written as $(\mathfrak{D}-\...
MathArt's user avatar
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Solution of the parabolic PDE using Green's function

The Green's function G(x⃗,t;ξ⃗,θ)G(x,t;ξ ​,θ) for the parabolic PDE is defined by the equation: Δξ⃗G(x⃗,t;ξ⃗,θ)−a∂G∂θ=δ(x⃗−ξ⃗)δ(t−θ),Δξ ​​G(x,t;ξ​,θ)−a∂θ∂G​=δ(x−ξ​)δ(t−θ), where GG satisfies the ...
Groovy's user avatar
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Troubles with solving a Laplace equation

Notice that $$ C_1e^{2n\pi}+C_2e^{-2n\pi}=0 \implies C_2=-C_1e^{4n\pi}, \tag{1} $$ hence $$ Y=C_1e^{2ny}-C_1e^{4n\pi-2ny}=C_1e^{2n\pi}\left(e^{2n(y-\pi)}-e^{-2n(y-\pi)}\right) =C\sinh(2n(y-\pi)). \tag{...
Gonçalo's user avatar
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If $f(x) = Ae^{x} + Be^{-x}$ and $f(1) = 0$, then $f(x) = C\sinh(x - 1)$

If $f(1) = 0$, one gets that \begin{align*} f(1) = Ae^{1} + Be^{-1} = 0 \Longleftrightarrow Ae^{2} + B = 0 \Longleftrightarrow B = - Ae^{2} \end{align*} Consequently, one has that \begin{align*} f(x) &...
Átila Correia's user avatar
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Regularity of weak solution of elliptic equation with nonlinear Neumann boundary

Finally got the answer in the book of Grisvard Elliptic Problems in Nonsmooth Domains. First we find a solution $u_1$ solving $\Delta u_1=-u_1$ on $\Omega$ and $\partial u_1/\partial n = -g(u)$ by ...
mnmn1993's user avatar
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Confusion about one initial/boundary value problem for heat equation

So, solving this question via conformal invariance turned out to be quite doable. Identifying $\mathbb{R}^2$ with $\mathbb{C}$ via $z = x + i y$ and letting $\xi = f(z) = ( e^{-i \pi/2} ( z - (1+i) ) )...
tsnao's user avatar
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Use method of characteristics to solve $u_x(x,y)+u_y(x,y)=(u(x,y))^2$, $u(x,0)=x$

Using characteristic lines we have $$ \dfrac{dx}{1}=\dfrac{dy}{1}=\dfrac{du}{u^2} $$ Using the equation for $dx$ and $dy$, after integration we find that $$ x=y+C $$ Then, using the equation for $du$ ...
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Separation of variables method with $u(0,x)=5e^{x^2}-e^{-10x^2};x>0$

This is a linear first-order PDE, so we can solve it using characteristic lines $$ \dfrac{dt}{x}=\dfrac{dx}{1} \qquad \implies \qquad t=\dfrac12x^2+C. $$ Thus, the general solution is $$ u(t,x)=f\big(...
W2S's user avatar
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Compatibility of Initial/Boundary Conditions in a Convection-Diffusion Problem?

You are correct that this will have an effect. The effect of this will be that $u(x,t)$ will not be continuous exactly at the point $(0,0)$. However, many PDEs are well-defined even for initial/...
whpowell96's user avatar
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2 votes

Minimizing the functional $J[y]=\int_0^1 (\frac{1}{2}y'^2+yy'+y'+y)dx$ with undetermined boundary values

OP's variational problem is ill-posed since the boundary values are not specified/undetermined/arbitrary/free. E.g. if we choose a constant function $x\mapsto y(x)=c$, then OP's functional takes the ...
Qmechanic's user avatar
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