Consider the diffusion equation ut = boundary conditions kuza for k > 0 on (-1,1] with...

 

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Consider the diffusion equation ut = boundary conditions kuza for k > 0 on (-1,1] with Robin Uz(-1, t) – au(-1,t) = 0 = uz(1, t) + au(1, t). (a) If a > 0 show that the energy E(t) = L, [u(x, t)]² dx is decreasing. (b) If a is much smaller than 0, show by example that energy may increase or decrease. Hint: Consider solutions of the form h(t) cosh(bx). %3D Consider the diffusion equation ut = boundary conditions kuza for k > 0 on (-1,1] with Robin Uz(-1, t) – au(-1,t) = 0 = uz(1, t) + au(1, t). (a) If a > 0 show that the energy E(t) = L, [u(x, t)]² dx is decreasing. (b) If a is much smaller than 0, show by example that energy may increase or decrease. Hint: Consider solutions of the form h(t) cosh(bx). %3D