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Heat Problem on the Circle

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So I am able to get to u(x,t) = sum_{n=-infty}^{infty} Cexp(iBx - ktB^2) + Dexp(-iBx - ktB^2)

because the general series is sum_{n=-infty}^{infty} X(x)T(t)

You have 1 coefficient in your general series, yet I have 2. How do I get it into 1?

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Solution Summary

A Heat Problem on the Circle is investigated.

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Partial Differential Equations : Heat Equations

1) Let A(x,y) be the area of a rectangle not degenerated of dimensions x and y, in a way that the rectangle is inside a circle of a radius of 10. Determine the domain and the range of this function.

2) The wave equation (c^2 ∂^2 u / ∂ x^2 = ∂^2 u / ∂ t^2) and the heat equation (c ∂^2 u / ∂ x^2 = ∂ u / ∂ t) are two of the most important equations of physics (c is a constant). They are called partial differential equations. Show the following:

a) u = cos x cos ct and u = e^x cosh ct satisfies the wave equation.

b) u = e^-ct sin x and u = t^-1/2 e^[(-x^2)/(4ct)] satisfies the heat equation.

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