Lista de Exercicios Resolucao Equacao de Laplace 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 m0 y 0 0 y b mx 0 0 ux b gx 0 x a b Find the solution if ux x 0x a2 ax a2 x a 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 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 m0 y 0 ua y 0 0 y b ux 0 hx ux b 0 0 x a 3 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 ux 0 hx ux b 0 0 x a Hint Consider the possibility of adding the solutions of two problems one with homogeneous boundary conditions except for ua y fy and the other with homogeneous boundary conditions except for ux 0 hx b Find the solution if hx 12xa and fy 1 yb c Let a 2 and b 2 Plot the solution in several ways u versus x for a uniform sample of y values u versus y for a uniform sample of x values u versus both x and y and a contour plot 4 Show how to find the solution ux y of Laplaces equation in the rectangle 0 x a 0 y b that satisfies the boundary conditions u0 y ky ua y fy 0 y b ux 0 hx ux b gx 0 x a Hint See Problem 3 5 Find the solution ur θ of Laplaces equation uᵣᵣ 1r uᵣ 1r² uθθ 0 r a 0 θ 2π outside the circle r a that satisfies the boundary condition ua θ fθ 0 θ 2π on the circle Assume that ur θ is singlevalued and bounded for r a 6 a Find the solution ur θ of Laplaces equation in the semicircular region r a 0 θ π that satisfies the boundary conditions ur 0 0 ur π 0 0 r a ua θ fθ 0 θ π Assume that u is singlevalued and bounded in the given region b Find the solution if fθ 6θ r θ c Let a 2 and plot the solution in several ways u versus r u versus θ u versus both r and θ and a contour plot 7 Find the solution ur θ of Laplaces equation in the circular sector 0 r a 0 θ α that satisfies the boundary conditions ur 0 0 ur α 0 0 r a ua θ fθ 0 θ α Assume that u is singlevalued and bounded in the sector and that 0 α 2π 8 a Find the solution ux y of Laplaces equation in the semiinfinite strip 0 x a y 0 that satisfies the boundary conditions u0 y 0 ua y 0 y 0 ux 0 fx 0 x a and the additional condition that uxx y 0 as y b Find the solution if fx xa x 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 ί with ν and ί 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 u0 y 0 ua a ƒy 0 y b ux 0 0 ux 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 uₙx y cₙ cosh nπxb cos nπyb n 1 2 3 b By superposing the fundamental solutions of part a formally determine a function u satisfying the nonhomogeneous boundary condition ua y fy Note that when ua 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 ƒ 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 φ of Laplaces equation inside the circle r a that satisfies the boundary condition on the circle ua θ 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 ux 0 0 ux 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 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 ux 0 0 ux b 0 0 x a Hint Eventually it will be necessary to expand fy in a series that makes use of the functions sin nγ2b sin³ nγ2b sin⁵nγ2b see Problems 39 of Section 104 b Find the solution if fy y2b y 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 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 RrZz show that R and Z satisfy the equations r²R rR λ²r²R 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 r a 0 z b be a vertical section edisciplinasuspbr