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Questions tagged [partial-differential-equations]

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Is the viscosity solution of Hamilton-Jacobi equation of practical use in optimal control?

My understanding is, given an optimal control problem, one can show that the optimal cost satisfies a Hamilton-Jacobi PDE and use dynamic programming to figure out the optimal control. However, sometimes this PDE has no strong solution, and the deep…
Isley
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Question regarding evaluating 1D heat transfer PDE

My question regards cooling of an object using 1D-heat transfer with fixed surface temperatures. First I need to find the solution to this PDE: Based on the conditions, I worked out that the temperature of the interface of heat transfer is Ts and…
collproj
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Finite difference discretization of the Cauchy-Riemann PDEs

I made a forward fd-discretization of the Cauchy-Riemann PDEs but I am struggling to implement this in python. I have a quadratic mesh with heighτ = $2*\pi$. The dirichlet boundary conditions are at $u(x,0) = f(x) = \cos(x)$ and $v(x,0) = g(x) =…
Rico227
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Internal Tent Temperature

Here is the problem I am working on: "Consider a perfectly sealed polygonal tent with the sun directly overhead. The solar irradiance of a surface 90° to the sun’s rays is 1,000 W/m2. However, the solar irradiance on sides of the tent is 65° from…
Tramory
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Why stopping at the first derivative in conservation equations?

Oftentimes we, as engineers, have to write energy/mass/momentum balances to derive the governing equations for a quantity $\phi(x,t)$ of a physical system. One of many derivations of such balances involves writing a net sum of fluxes $\Phi(x,t)$…
TheVal
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Pre requisites for a course in vibration for a Mechanical Engineering Bachelors Degree

Can anyone please tell me what parts of differential equations, linear and partial I need to revise properly before I take a course in vibrations , in Dynamics, our course in vibrations mainly deals with Free , Forced and Torsional Vibrations , so…
Sergeant Afanasiev
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Solution to wave equation in a stretched string with one end fixed and the other subject to periodic input

I have a uniform string of length $L$, linear mass density $\mu$, subject to tension $T$. It satisfies the wave equation: $$\frac{\partial^2 y}{\partial t^2}=c^2\frac{\partial^2 y}{\partial x^2}$$ where $y(x,t)$ is its displacement and…
JP19774028
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Are equilibrium values of a differential equation uniquely dependent on its constants?

I have a dynamic model which undergoes two distinct stages. The system starts out with certain initial conditions and once a specific point is reached in stage 1, these ending conditions are used as the initial conditions for stage 2, and the…
alcopo63q
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Stability analysis of pressure relief valve using jacobian matrix

I'm modeling a pressure relief valve in order to find the limits of stability using the Jacobian matrix at an equilibrium point. The equations are the folowing: $$ \ddot{y} = \frac{1}{m}(A\cdot p\space - k\cdot y \space -d_a \cdot \dot{y})…
Tarti
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Interpretation of dx/dt and u in CFD

I am currently studying CFD and I have a question about how I should interpret the term u and dx/dt. I would like to find that out in the textbook, but I could not. When I solve 1D Euler Equation with Mass conservation , then I get the result . In…
Sang MuMu
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In FEM, what is the difference between a single element with a quadratic shape function and two elements with linear shape functions?

Using Finite Element Analysis to obtain a Weak form of a PDE, what is the difference between the two cases: A single element with a quadratic shape function Two elements with linear shape functions. Thank you for any insights you can…
Eggart
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Optimal control of the gradient type PDE

I recently encountered the following optimal control problem. The purpose of the system is to find the parameter $x$ at which the maximum or minimum of the function $f$ a is reached. $x$ is unknown to us in advance. We have gradient type…
dtn
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Obtaining equation from real life

As I was studying about differential equation,then I got a question in my mind that we are taking out general solutions for a differential equation. Like for example - Q) $\frac{dy}{dx} = \left[\frac{1}{(1+x^3)}\right] – \left[\frac{3x^2}{1 +…
user87284
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