Lista de Exercícios Resolvendo a Equação de Difusão do Calor - Boyce DiPrima

Lista 12 Equação de difusão do calor extraídos do livro de Boyce e DiPrima Problems In each of Problems 1 through 6 determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations If so find the equations 1 2 3 4 5 6 7 8 9 10 11 12 xx u 0 0 0 px rxuo 0 Find the solution of the heat conduction problem 100xx u 0 x 1 t 0 u0 t 0 u1 t 0 ux 0 sin2πx sin5πx 0 x 1 Find the solution of the heat conduction problem u xx 4 u 0 x 2 t 0 u0 t 0 u2 t 0 t 0 ux 0 2 sinπx 2 sinπx 4 sin2πx 0 x 2 Consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0 C for all t 0 In each of Problems 9 through 12 find an expression for the temperature ux t if the initial temperature distribution in the rod is the given function Suppose that α2 1 9 ux0 50 0 x 40 10 ux 0 x 0x 20 40 x 20 x 40 11 ux 0 0 0 x 10 50 10 x 30 0 30 x 40 12 ux 0 x 0 x 40 24 In solving differential equations the computations can almost always be simplified by the use of dimensionless variables a Show that if the dimensionless variable ξ x L is introduced the heat conduction equation becomes b Since L2 α2 has the units of time it is convenient to use this quantity to define a dimensionless time variable τ α2 L2 t Then show that the heat conduction equation reduces to 25 Consider the equation au x x bu cu 0 where a b and c are constants a Let ux t e wt wx t where b is constant and find the corresponding partial differential equation for w b If b 0 show that it can be chosen so that the partial differential equation found in part a has no term in w Thus by a change of dependent variable it is possible to reduce equation 25 to the heat conduction equation 26 The heat conduction equation in two space dimensions is c2 u x x u y y u t Assuming that ux y t Xx Yy Tt find ordinary differential equations that are satisfied by Xx Yy and Tt 27 The heat conduction equation in two space dimensions may be expressed in terms of polar coordinates as α2 urt u r u u t Assuming that ur θ t Rr Θθ Tt find ordinary differential equations that are satisfied by Rr Θθ and Tt