Question

The lossy heat equation.

Show that the general solution to vₜ =γvₓₓ −αv,α,γ>0 is v=e⁻ᵃᵗ u(t,x) where u(t,x) is a solution of uₜ=γuₓₓ

Answer 1

To demonstrate that v=e⁻ᵃᵗu(t,x) satisfies the lossy heat equation vₜ =γvₓₓ −αv, we must show that e⁻ᵃᵗu(t,x) reduces the original equation to the standard heat equation uₜ=γuₓₓ when substituted.

Step-by-step explanation:

The question concerns the solution to the lossy heat equation vₜ = γvₓₓ −αv, where α and γ are positive constants. To show that v=e⁻ᵃᵗ u(t,x) is a solution to this equation, we can substitute v into the lossy heat equation and see that the dependency on u(t,x) transforms the equation into a form of the standard heat equation uₜ=γuₓₓ. Thus, the function u(t,x) must be a solution to the heat equation without loss, while the exponential term e⁻ᵃᵗ accounts for the loss of heat over time due to the term −αv in the original equation.