Question

Find a solution to the heat transfer type equation student submitted image, transcription available below constudent submitted image, transcription available belowque sea cero-->(i.e T(x=pi,-pi,t)=0) en x=pi y x=-pi y que sea igual astudent submitted image, transcription available belowen t=0 -->(i.e T(x,t=0)=sen(x)[1+cos(x)])

Answer 1

To satisfy the given conditions, the solution to the heat transfer equation is
`\(T(x, t) = \sin(x)[1 + \cos(x)] \cdot e^(-t)\)`.

Step-by-step explanation:

The heat transfer equation is a partial differential equation (PDE) that describes the distribution of heat in a given region over time. The solution provided,
`\(T(x, t) = \sin(x)[1 + \cos(x)] \cdot e^(-t)\)`, meets the specified boundary and initial conditions.

Firstly, to ensure
`\(T(x=\pi, t) = 0\)` and
`\(T(x=-\pi, t) = 0\)`, the sine term in the solution ensures that these conditions are satisfied at
`\(x=\pi\)` and
`\(x=-\pi\)`, respectively.

Secondly, for the initial condition
`\(T(x, t=0) = \sin(x)[1 + \cos(x)]\)`, the given solution incorporates this initial temperature distribution at time t=0.

The exponential term
`\(e^(-t)\)` in the solution accounts for the decay of temperature over time, ensuring that the temperature distribution approaches zero as time progresses. This is a common feature in heat transfer problems where the system tends to reach thermal equilibrium.

In summary, the provided solution
`\(T(x, t) = \sin(x)[1 + \cos(x)] \cdot e^(-t)\)` satisfies the specified boundary conditions at
`\(x=\pi\)` and
`\(x=-\pi\)`, as well as the initial condition at
`\(t=0\)`, making it a suitable solution to the given heat transfer equation with the provided constraints.

Complete Question: What is the solution to the heat transfer equation given the conditions
`\(T(x=\pi, t) = 0\), \(T(x=-\pi, t) = 0\)`, and
`\(T(x, t=0) = \sin(x)[1 + \cos(x)]\)`?