Lista de Exercicios Resolucao da Equacao de Difusao do Calor Boyce-Diprima

Lista 13 Equação de difusão do calor 2 extraídos do livro de Boyce e DiPrima In each of Problems 1 through 8 find the steadystate solution of the heat conduction equation α2 uxx ut that satisfies the given set of boundary conditions 1 u0t 10 u50t 40 2 u0t 30 u40t 20 3 ux0t 0 uLt 0 4 ux0t 0 uLt T 5 u0t 0 uxLt 0 6 u0t T uxLt 0 7 ux0t u0t 0 uLt T 8 u0t T uxLt uLt 0 9 Let an aluminum rod of length 20 cm be initially at the uniform temperature of 25C Suppose that at time t 0 the end x 0 is cooled to 0C while the end x 20 is heated to 60C and both are thereafter maintained at those temperatures 20 Consider the problem α2 uxx ut 0 x Lt 0 u0t 0 uxLt γ uLt 0 t 0 ux0 f x 0 x L a Let uxt Xx Tt and show that X λ X 0 X0 0 XL γ XL 0 48 and T λ α2 T 0 where λ is the separation constant b Assume that λ is real and show that problem 48 has no nontrivial solutions if λ 0 c If λ 0 let λ μ2 with μ 0 Show that problem 48 has nontrivial solutions only if μ is a solution of the equation μ cosμ L γ sinμ L 0 49 d Rewrite equation 49 as tanμ L μγ Then by drawing the graphs of y tanμ L and y μγ for μ 0 on the same set of axes show that equation 49 is satisfied by infinitely many positive values of μ denote these by μ1 μ2 μn ordered in increasing size e Determine the set of fundamental solutions unxt corresponding to the values μn found in part d 15 Consider a uniform bar of length L having an initial temperature distribution given by fx 0 x L Assume that the temperature at the end x 0 is held at 0C while the end x L is insulated so that no heat passes through it a Show that the fundamental solutions of the partial differential equation and boundary conditions are unxt e 2n12 π2 α2 t 4L2 sin2 n 1 π x 2L n 123 b Find a formal series expansion for the temperature uxt uxt Σn1 cn unxt that also satisfies the initial condition ux0 fx Hint Even though the fundamental solutions involve only the odd sines it is still possible to represent f by a Fourier series involving only these functions See Problem 39 of Section 104 An External Heat Source Consider the heat conduction problem in a bar that is in thermal contact with an external heat source or sink Then the modified heat conduction equation is ut α2 uxx sx 50 where the term sx describes the effect of the external agency sx is positive for a source and negative for a sink Suppose that the boundary conditions are u0t T1 uLt T2 51 and the initial condition is ux0 fx 52 Problems 21 through 23 deal with this kind of problem 21 Write uxt vx wxt where v and w are the steadystate and transient parts of the solution respectively State the boundary value problems that vx and wxt respectively satisfy Observe that the problem for w is the fundamental heat conduction problem discussed in Section 105 with a modified initial temperature distribution