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Question: mathematical physics pls use steps and explain...

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Mathematical Physics
Pls use steps and explain
1. Diffusion with insulated boundary conditions Consider the following eigenvalue problem dr2 (a) Find the nontrivial solutions of (1). Consider λ > 0,A-0 and λ--s < (b) Show that the nontrivial solutions of (1) are orthogonal on 0<<L with the inner product defined as (f(x), g(x)) = Next consider f(z)g(z)dz み! ot (c) Find the temperature u(r, t) that satisfies (2)-(4) above using separation of variables. Your results from (a) and (b) above should be useful. (d) Find the steady state temperat ure distribution, (), by finding limu, t) (e) Solve for the steady state solution us() directly from (2)-(4) and show that it equivalent to your answer in (d) above. First solve (2)-(3) for us() Then, by integrating (2) from -0 to z-L, show that u(z,t)dz = constant and evaluate the constant using the initial condition (4)
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