Does the Wave Equation with Homogeneous Boundary Conditions Conserve Energy?

In summary, the conversation discusses a function $u\in\mathcal C^1(\overline R)\cap \mathcal C^2(R)$ that satisfies the wave equation $u_{tt}=K^2 u_{xx}+h(x,t,u)$ in a region $R=(0,1)\times(0,\infty)$. It is stated that $K$ is a positive constant and $h$ is a constant function. The conversation also mentions determining the total energy of the string and shows that if homogenous boundary conditions are imposed and no external forces are applied, then there is conservation of energy. There is a suggestion for a book to help with the problem, but the person is unable to obtain it
  • #1
Markov2
149
0
Let $u\in\mathcal C^1(\overline R)\cap \mathcal C^2(R)$ where $R=(0,1)\times(0,\infty).$ Suppose that $u(x,t)$ verifies the following wave equation $u_{tt}=K^2 u_{xx}+h(x,t,u)$ where $K>0$ and $h$ is a constant function.

a) Determine the total energy of the string. (Well I don't know what does this mean.)

b) Show that if homogenous boundary conditions are imposed and no extern forces apply to the system, then there's conservation of the energy.

How do I start?
 
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  • #2
Markov said:
Can anybody help please? :(

I suggested a cheap good book for you to get but you decided against it. Why didn't you buy a something (a book on the matter) that can help you start the problem?
 
  • #3
Yes but I can't get that book. :(
 

FAQ: Does the Wave Equation with Homogeneous Boundary Conditions Conserve Energy?

What is PDE?

PDE stands for Partial Differential Equation, which is a mathematical equation that involves multiple variables and their partial derivatives. It is commonly used to model various physical phenomena in areas such as physics, engineering, and finance.

How does PDE relate to the conservation of energy?

PDE is often used to describe the behavior of physical systems and the conservation of energy is a fundamental principle in physics. By solving PDEs, we can analyze and understand how energy is conserved or transformed within a system.

Can PDE be used to solve real-world problems involving energy conservation?

Yes, PDEs can be used to model and solve a wide range of problems related to energy conservation. For example, PDEs can be used to study the flow of heat, fluid dynamics, and electrostatics, all of which involve the conservation of energy.

Are there any limitations to using PDE for energy conservation problems?

While PDEs are powerful tools for understanding energy conservation, they can be complex and challenging to solve. Additionally, the accuracy of the results depends on the assumptions and simplifications made in the model. In some cases, other methods such as numerical simulations may be more suitable.

How can PDEs be used to improve energy conservation in real-world applications?

By accurately modeling and analyzing energy conservation using PDEs, we can identify areas for improvement and develop more efficient systems. For example, PDEs can be used to optimize the design of wind turbines or study the heat transfer in buildings to reduce energy consumption.

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