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Calculus - Geometry

user_striner Created by 6 mon ago

Cards in this deck(16)

Assume a product form of solution. T(x,t)=Φ(x)G(t)

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Substitute this solution into the PDE. dG/dt(Φ)=(DG)d²Φ/dx²

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Get variables on the same side. dG/dt(1/DG)=(1/Φ)d²Φ/dx²

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Set equal to -λ

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Create 2 ODEs. dG/dt=-λDG, and d²Φ/dx²= -λΦ

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Solve first derivative using separation of variables. G=Ce^(-λDt)

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Use corresponding T(x,0) initial condition

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Set up an SLEVP eigenvalue problem with σ(x)=p(x)=1 and q(x)=0. Simplify to use later with an initial condition

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Solve r²+λ to r=sqrt(λ)i to get Φ in the form Acos(sqrt(λ)x)+Bsin(sqrt(λ)x)

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Use initial condition where x=0 to find if A or B are equal to 0

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Solve for λn if required

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After finding ΦG apply principe of superposition so Σ(ΦG)

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Use initial condition setting t=0, e^(-λDt)=1 when t=0

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Multiply by sin(nπx/L) or cos(nπx/L) on both sides depending on which is better by the principal of orthogonality

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Use final initial condition to solve for A or B. Plug in all knows into the final formula and make sure to include a Σ in your final answer.

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Φ and G cannot equal 0 at any time, λ>0, A or B can equal 0, but not both in the same equation

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