Lista de Exercícios - Equação de Laplace - Problemas Resolvidos Boyce e DiPrima

Lista 15 Equação de Laplace extraídos do livro de Boyce e DiPrima Problems 1 a Find the solution ux y of Laplaces equation in the rectangle 0 x a 0 y b that satisfies the boundary conditions u0 y 0 ua y 0 0 y b ux 0 0 ux b gx 0 x a b Find the solution if gx x 0 x a2 a x a2 x a G c For a 3 and b 1 plot u versus x for several values of y and also plot u versus y for several values of x Use enough terms in the Fourier series to accurately approximate the nonhomogeneous boundary condition G d Plot u versus both x and y in three dimensions Also draw a contour plot showing several level curves of ux y in the xyplane 2 Find the solution ux y of Laplaces equation in the rectangle 0 x a 0 y b that satisfies the boundary conditions u0 y 0 ua y 0 0 y b ux 0 hx ux b 0 0 x a N c Let a 5 Find the smallest value of y₀ for which ux y 01 for all y y₀ 9 Show that equation 24 has periodic solutions only if λ is real Hint Let λ μ² where μ ν iσ with ν and σ real 10 Consider the problem of finding a solution ux y of Laplaces equation in the rectangle 0 x a 0 y b that satisfies the boundary conditions uₓ0 y 0 uₓa y fy 0 y b uᵧx 0 0 uᵧx b 0 0 x a This is an example of a Neumann problem a Show that Laplaces equation and the homogeneous boundary conditions determine the fundamental set of solutions u₀x y c₀ uₙx y cₙ coshnπxb cosnπyb n 1 2 3 b By superposing the fundamental solutions of part a formally determine a function u satisfying the nonhomogeneous boundary condition uₓa y fy Note that when uₓa y is calculated the constant term in ux y is eliminated and there is no condition from which to determine c₀ Furthermore it must be possible to express f by means of a Fourier cosine series of period 2b which does not have a constant term This means that ₀ᵇ fy dy 0 is a necessary condition for the given problem to be solvable Finally note that c₀ remains arbitrary and hence the solution is determined only up to this additive constant This is a property of all Neumann problems 11 Find a solution ur θ of Laplaces equation inside the circle r a that satisfies the boundary condition on the circle uᵣa θ gθ 0 θ 2π Note that this is a Neumann problem and that its solution is determined only up to an arbitrary additive constant State a necessary condition on gθ for this problem to be solvable by the method of separation of variables see Problem 10 12 a Find the solution ux y of Laplaces equation in the rectangle 0 x a 0 y b that satisfies the boundary conditions u0 y 0 ua y 0 0 y b uᵧx 0 0 uᵧx b gx 0 x a Note that this is neither a Dirichlet nor a Neumann problem but a mixed problem in which u is prescribed on part of the boundary and its normal derivative on the rest b Find the solution if gx x 0 x a2 a x a2 x a G c Let a 3 and b 1 By drawing suitable plots compare this solution with the solution of Problem 1 13 a Find the solution ux y of Laplaces equation in the rectangle 0 x a 0 y b that satisfies the boundary conditions u0 y 0 ua y fy 0 y b uₓx 0 0 uᵧx b 0 0 x a Hint Eventually it will be necessary to expand fy in a series that makes use of the functions sinπ y2b sin3π y2b sin5π y2b see Problem 39 of Section 104 b Find the solution if fy y2b y G c Let a 3 and b 2 plot several different views of the solution 14 a Find the solution ux y of Laplaces equation in the rectangle 0 x a 0 y b that satisfies the boundary conditions uₓ0 y 0 uₓa y 0 0 y b uₓx 0 0 uᵧx b gx 0 x a b Find the solution if gx 1 x²x a² G c Let a 3 and b 2 plot several different views of the solution 15 Show that Laplaces equation in polar coordinates is uᵣᵣ 1r uᵣ 1r² uᵩϴ 0 Hint Use x r cos θ and y r sin θ and the chain rule 16 Show that Laplaces equation in cylindrical coordinates is uᵣᵣ 1r uᵣ 1r² uᵩϴϴ uzz 0 Hint Use x r cosθ y r sinθ z z and the chain rule 17 Show that Laplaces equation in spherical coordinates is uᵨᵨ 2ρ uᵨ 1ρ² sin² ϕ uᵩϴϴ cot ϕρ² uᵩ 1ρ² uϴϴ 0 Hint Use x ρ sin ϕ cos θ y ρ sin ϕ sin θ z ρ cos θ and the chain rule 18 a Laplaces equation in cylindrical coordinates was found in Problem 15 Show that axially symmetric solutions ie solutions that do not depend on θ satisfy uᵣᵣ 1r uᵣ uzz 0 b Assuming that ur z Rr Zz show that R and Z satisfy the equations r R R λ²r R 0 Z λ²Z 0 Note The equation for R is Bessels equation of order zero with independent variable λr 19 Flow in an Aquifer Consider the flow of water in a porous medium such as sand in an aquifer The flow is driven by the hydraulic head a measure of the potential energy of the water above the aquifer Let R 0 x a 0 z b be a vertical section of an aquifer In a uniform homogeneous medium the hydraulic head ux z satisfies Laplaces equation uₓₓ uzz 0 in R 39 If water cannot flow through the sides and bottom of R then the boundary conditions there are uₓ0 z 0 uₓa z 0 0 z b 40 uzx 0 0 0 x a 41 Finally suppose that the boundary condition at z b is ux b b αx 0 x a 42 where α is the slope of the water table a Solve the given boundary value problem for ux z G b Let a 1000 b 500 and α 01 Draw a contour plot of the solution in R that is plot some level curves of ux z G c Water flows along paths in R that are orthogonal to the level curves of ux z in the direction of decreasing u Plot some of the flow paths