A rectangular plate {0 x a, 0 y b} initially has a hot

Chapter 12, Problem 12

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QUESTION:

A rectangular plate {0 x a, 0 y b} initially has a hot spot at its center so that its initial temperature distribution is u(x, y,0)= M(x a 2, y b 2). Its edges are maintained at zero temperature. Let k be the diffusion constant. Find the temperature at any later time in the form of a series.

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QUESTION:

A rectangular plate {0 x a, 0 y b} initially has a hot spot at its center so that its initial temperature distribution is u(x, y,0)= M(x a 2, y b 2). Its edges are maintained at zero temperature. Let k be the diffusion constant. Find the temperature at any later time in the form of a series.

ANSWER:

Step 1 of 4

Series can be expressed as

                                     

It has to satisfy the initial condition

                                     

Multiply equation by ,

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