Problem 13 from section 16.1 of Taylor's PDE textbook.

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In summary, the speaker was given a task by their teacher to solve a question in Taylor's and Evans's books on PDE, but they were not successful to their teacher's satisfaction. They have provided a link to the question and an attachment of their failed attempt. They are seeking help on how to solve the question and have included their teacher's remarks in the attachment. They mention that they were unable to upload the attachment on a forum and had to use mediafire instead. They express gratitude for any help and mention their desire to eventually obtain their MSc.
Alone
I was given as a task to solve this question by my teacher (heck if I had the time I would have solved every problem in both Taylor's and Evans's books on PDE); but didn't succeed to the teacher's satisfaction.

In the following link there's a presentation of the problem, and in the attachment my failed attempt at solving it (it should have been part of my thesis I guess :-( ):
https://math.stackexchange.com/questions/1180968/question-13-in-taylors-pde-vol-iii-section-16-1

Any help on how to solve this?
In the attachment there's the latex file with the remarks my teacher gave me. (His remarks are between the two lines with*******).

Perhaps one day I'll get my MSc...
Taylor_Exercises-new

I tried to upload as an attachment through the option in the MHB forum but didn't succeed as the txt file exceeds the site's capacity, so I uploaded to mediafire.

Attachments

• Taylor_Exercises-new.pdf
136 KB · Views: 72

1. What is the problem statement in Problem 13 from section 16.1 of Taylor's PDE textbook?

The problem statement is to solve the partial differential equation ut = kuxx - c(t)u, where k and c(t) are constants, subject to the boundary conditions u(0,t) = 0 and u(1,t) = sin(t).

2. What is the significance of Problem 13 in section 16.1 of Taylor's PDE textbook?

Problem 13 is significant because it is a classic example of a heat equation with a time-dependent source term, which is a common type of problem in many fields of science and engineering.

3. How do you approach solving Problem 13 from section 16.1 of Taylor's PDE textbook?

To solve this problem, you can use the method of separation of variables, where you assume a solution of the form u(x,t) = X(x)T(t) and then substitute it into the PDE to obtain two ordinary differential equations, one for X(x) and one for T(t). These can then be solved using standard techniques.

4. What are the key steps in solving Problem 13 from section 16.1 of Taylor's PDE textbook?

The key steps in solving this problem include setting up the PDE and boundary conditions, assuming a solution of the form u(x,t) = X(x)T(t), substituting it into the PDE, separating the variables, and solving the resulting ordinary differential equations. The final step is to combine the solutions for X(x) and T(t) to get the general solution for u(x,t).

5. What real-world applications can be modeled using Problem 13 from section 16.1 of Taylor's PDE textbook?

This type of problem is commonly used to model heat transfer in materials, such as the diffusion of heat in a solid object or the temperature distribution in a fluid. It can also be used to model other physical phenomena, such as the spread of pollutants in the environment or the propagation of electrical signals in a circuit.

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