Questions tagged [partial-differential-equations]
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Partial derivative of change in state with respect to multiplicative faults
I am reading the book "Model-Based Fault Diagnosis Techniques" by Steven X. Ding. Here, they describe a system with the equation $$ ẋ = (A + ΔA_F)x +(B + ΔB_F)u+E_f f$$ Then, $$ ΔA_F = ...
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Understanding Momentum conservation at a Control Volume
I am taking a course in Fluid Dynamics, and we discussed the conservation of momentum in class:
$$\dot{M}_{out,x} = \rho u (dydz) u|_{x+dx} + \rho v (dxdz) u|_{x+dx} + \rho v (dxdy) u|_{x+dx}$$
I ...
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Is a vibrating Euler beam supposed to behave in this way?
I am a student of mathematics and for my modelling seminar, I need to dip a little bit into engineering in order to model the movements of a suspension bridge under influence of wind. This will be ...
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Simulation of the direct piezoelectric effect using FEM
I have to simulate the direct piezoelectric effect, i.e. the generation of electrical charge under an applied mechanical strain using FEM with FEniCS. My geometry is a freestanding quadratic prism (a ...
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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 ...
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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 ...
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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.
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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 ...
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Pre requisites for a course in vibration for a Mechanical Engineering Bachelors Degree [closed]
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 ...
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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}$$
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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 ...
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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 $...
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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 ...
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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{...
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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, ...
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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, ...
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