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Pu can check your solution by substituting back r into the PDE 2. Explain why the problem has no solution. has an in...
a) Find the solution to the following interior Dirichlet problem with radius R=1 1 PDE Urr + Up t 0 0 <r <1 wee p2 r BC u (1,0) = 10 + 3 sin(0) 10 cos(20) 0 <0 < 27 b) Consider the above problem on the unit square (x,y) domain PDE Urr + Uyy = 0 0<x<1 0<y <1 Transform the solution u(r, 0) from "a)" to the solution u(x, y) for "b)" Use the solution u(x,y) to calculate...
Please show all work and provide and an original solution. We can apply the Method of Separation of Variables to obtain a representation for the solution u u(, t) for the following partial differential equation (PDE) on a bounded domain with homogeneous boundary conditions. The PDE model is given by: u(r, 0) 0, (2,0) = 4. u(0,t)0, t 0 t 0 (a) (20 points) Assume that the solution to this PDE model has the form u(x,t) -X (r) T(t). State...
Using Laplace Equation PDE 42.(a) Solve for u(r, e): That is, the region is an annulus betweenr 1 andr 2. HINT: First draw a picture of it, to get a look at the problem. Now, you should be able to readily get Then, see that you have 27-periodicity, so K n (n-1, 2, ) and D-0, so u (r, θ) A' + B' In r + an infinite series with r's and θ's in it. But look at your picture:...
QUESTION 2 Consmder the problem ди 2k, 0<r< 1, t>O оt and the boundary conditions u(0,t)= 1, u (1,t) = 3, t > 0 (a) Find the equiltbrium solutiou ug (r) (b) Find the solution u (z.t) of the PDE and the boundary condition which also satisfies the mitial condition (,0)-1+++sin (3wx), 0<o< 1 [25]
Can anyone help with this question please? Consider the problem-Δu = 0 in the annulus 2- E R R where 0<F< R with Dirichlet boundary condition if l u(z) = uテ xuR if |x| = R where ur, uRE R Use the general solution u(A log( problem B with A, B R to solve the Consider the problem-Δu = 0 in the annulus 2- E R R where 0
2. Consider the utility maximization problem with n goods (a finite) (a) If the utility function u(c) is strictly concave, increasing, C1, and as- suming interiority of the optimal solution, what is the problem the consumer is solving? What are the FOCs for this problem using an "unconstrained" ap- proach (i.e., variable substitution in "primal" problem)? (b) Do optimal solutions for all goods satisfy "MRS" "price ratio" condition (i.e., MRSy(c) for all (V) i j)? If so, explain why. If...
Determine the solution in the expansion fan. In this problem, we consider the flow of water through a porous medium. Specifically, u, is the water content at depth z and timet measured as a ratio of water-filled pore space to total pore-space. (Thus, 0 u 1.) We assume that u satisfies the conservation law and initial condition (12), + ut = 0 t>0 x<0 a(x, 0) = 0 u(r,0)-0 > In this problem, we consider the flow of water through...
Please prove this solution and explain why y2 can be taken as (x^2)(y1) Problem 2. Find the general solution of the equation Note that one of two linearly independent solutions is yi(r) -e. Solution. Using Abel's formula, we get the following relations for the Wronskian dW pi dW 2r1 On the other hand, Comparing these two expression for W(x), we can take y2 :- r2yı. Correspondingly, the general solution is Problem 2. Find the general solution of the equation Note...
Problem 2. A string of a guitar is fixed at the two ends, x = 0 and r = a. The string is set in motion with initial position f(x) = (h/a)., 0 <r <a, where h > 0, and then it is released with no initial velocity. The displacement u of the string is described by the PDE au 1 au ar2 2 212 0<x<a, t> 0. (i) State the boundary value initial value problem that u satisfies. (ii)...
PDE questions. Please show all steps in detail. 2. Consider the initial-boundary value problem 0