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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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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, $$ ΔA_F = A_i θ{_A}{_i} $$ is given Then, they proceed with finding the…
Dipsy2000
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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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The Heat Generation and heat flux both in General Heat Conduction Equation

How would I incorporate both heat generated within and element and heat flux from external to the element in GHCE. For example, Let there be a thick wall of thickness 'L', with internal heat generation of Q_g. There is also heat flux flowing from…
HungryChicken412
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Utilizing Engineering Stack Exchange: Laplace Transform case study

I'm a new student to engineering. I've been investigating different means for learning material and investigating the discipline. I'm already familiar with using different professional organizations for reading new articles and referencing past…
Mitchell
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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 modeled as a one-dimensional vibrating beam and so I…
Eriol
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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…
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…
ayr
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