Question

Drive heat equation in polar form. b. Also verify part (a) with counter example.​

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

For straightforwardness, how about we guess that your disk is the unit disk.

The heat condition is uₓ=kδu. Consistent state implies that the temperature u doesn't change; subsequently, uₓ=0 and you are left with Laplace's situation:

Δu=0 subject to u(1,θ)=f(θ).

The solution may then be composed:

                   u(r,θ)=a0 / 2 + (n=1 to ∞) ∑ r^n (aₙ cos(nθ) + bₙ sin(nθ) ),

where aₙ and bₙ are the Fourier coefficients of f, as advocated underneath.

On the other hand, the aₙs and bₙs might be supplanted with their essential portrayals, the request for summation and incorporation might be flipped, and (after a ton of disentanglement) this transforms into a fundamental portrayal of the Poisson indispensable equation:

        u(r,θ) = 1 / 2π (-π to π) ∫ f(ϕ) (R²−r²) / ( R²+r²−2 R r cos(θ−ϕ) ) dϕ